AI Headshot Generator

AI Headshot Generator — hands-on reviews, top picks, pricing, pros and cons and a practical how-to guide on Aizhi.

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  Automatic summarization

  Automatic summarization is the process of shortening a set of data computationally, to create a subset (a summary) that represents the most important or relevant information within the original content. Artificial intelligence (AI) algorithms are commonly developed and employed to achieve this, specialized for different types of data. Text summarization is usually implemented by natural language processing methods, designed to locate the most informative sentences in a given document. On the other hand, visual content can be summarized using computer vision algorithms. Image summarization is the subject of ongoing research; existing approaches typically attempt to display the most representative images from a given image collection, or generate a video that only includes the most important content from the entire collection. Video summarization algorithms identify and extract from the original video content the most important frames (key-frames), and/or the most important video segments (key-shots), normally in a temporally ordered fashion. Video summaries simply retain a carefully selected subset of the original video frames and, therefore, are not identical to the output of video synopsis algorithms, where new video frames are being synthesized based on the original video content. == Commercial products == In 2022 Google Docs released an automatic summarization feature. == Approaches == There are two general approaches to automatic summarization: extraction and abstraction. === Extraction-based summarization === Here, content is extracted from the original data, but the extracted content is not modified in any way. Examples of extracted content include key-phrases that can be used to "tag" or index a text document, or key sentences (including headings) that collectively comprise an abstract, and representative images or video segments, as stated above. For text, extraction is analogous to the process of skimming, where the summary (if available), headings and subheadings, figures, the first and last paragraphs of a section, and optionally the first and last sentences in a paragraph are read before one chooses to read the entire document in detail. Other examples of extraction that include key sequences of text in terms of clinical relevance (including patient/problem, intervention, and outcome). === Abstractive-based summarization === Abstractive summarization methods generate new text that did not exist in the original text. This has been applied mainly for text. Abstractive methods build an internal semantic representation of the original content (often called a language model), and then use this representation to create a summary that is closer to what a human might express. Abstraction may transform the extracted content by paraphrasing sections of the source document, to condense a text more strongly than extraction. Such transformation, however, is computationally much more challenging than extraction, involving both natural language processing and often a deep understanding of the domain of the original text in cases where the original document relates to a special field of knowledge. "Paraphrasing" is even more difficult to apply to images and videos, which is why most summarization systems are extractive. === Aided summarization === Approaches aimed at higher summarization quality rely on combined software and human effort. In Machine Aided Human Summarization, extractive techniques highlight candidate passages for inclusion (to which the human adds or removes text). In Human Aided Machine Summarization, a human post-processes software output, in the same way that one edits the output of automatic translation by Google Translate. == Applications and systems for summarization == There are broadly two types of extractive summarization tasks depending on what the summarization program focuses on. The first is generic summarization, which focuses on obtaining a generic summary or abstract of the collection (whether documents, or sets of images, or videos, news stories etc.). The second is query relevant summarization, sometimes called query-based summarization, which summarizes objects specific to a query. Summarization systems are able to create both query relevant text summaries and generic machine-generated summaries depending on what the user needs. An example of a summarization problem is document summarization, which attempts to automatically produce an abstract from a given document. Sometimes one might be interested in generating a summary from a single source document, while others can use multiple source documents (for example, a cluster of articles on the same topic). This problem is called multi-document summarization. A related application is summarizing news articles. Imagine a system, which automatically pulls together news articles on a given topic (from the web), and concisely represents the latest news as a summary. Image collection summarization is another application example of automatic summarization. It consists in selecting a representative set of images from a larger set of images. A summary in this context is useful to show the most representative images of results in an image collection exploration system. Video summarization is a related domain, where the system automatically creates a trailer of a long video. This also has applications in consumer or personal videos, where one might want to skip the boring or repetitive actions. Similarly, in surveillance videos, one would want to extract important and suspicious activity, while ignoring all the boring and redundant frames captured. At a very high level, summarization algorithms try to find subsets of objects (like set of sentences, or a set of images), which cover information of the entire set. This is also called the core-set. These algorithms model notions like diversity, coverage, information and representativeness of the summary. Query based summarization techniques, additionally model for relevance of the summary with the query. Some techniques and algorithms which naturally model summarization problems are TextRank and PageRank, Submodular set function, Determinantal point process, maximal marginal relevance (MMR) etc. === Keyphrase extraction === The task is the following. You are given a piece of text, such as a journal article, and you must produce a list of keywords or key[phrase]s that capture the primary topics discussed in the text. In the case of research articles, many authors provide manually assigned keywords, but most text lacks pre-existing keyphrases. For example, news articles rarely have keyphrases attached, but it would be useful to be able to automatically do so for a number of applications discussed below. Consider the example text from a news article: "The Army Corps of Engineers, rushing to meet President Bush's promise to protect New Orleans by the start of the 2006 hurricane season, installed defective flood-control pumps last year despite warnings from its own expert that the equipment would fail during a storm, according to documents obtained by The Associated Press". A keyphrase extractor might select "Army Corps of Engineers", "President Bush", "New Orleans", and "defective flood-control pumps" as keyphrases. These are pulled directly from the text. In contrast, an abstractive keyphrase system would somehow internalize the content and generate keyphrases that do not appear in the text, but more closely resemble what a human might produce, such as "political negligence" or "inadequate protection from floods". Abstraction requires a deep understanding of the text, which makes it difficult for a computer system. Keyphrases have many applications. They can enable document browsing by providing a short summary, improve information retrieval (if documents have keyphrases assigned, a user could search by keyphrase to produce more reliable hits than a full-text search), and be employed in generating index entries for a large text corpus. Depending on the different literature and the definition of key terms, words or phrases, keyword extraction is a highly related theme. ==== Supervised learning approaches ==== Beginning with the work of Turney, many researchers have approached keyphrase extraction as a supervised machine learning problem. Given a document, we construct an example for each unigram, bigram, and trigram found in the text (though other text units are also possible, as discussed below). We then compute various features describing each example (e.g., does the phrase begin with an upper-case letter?). We assume there are known keyphrases available for a set of training documents. Using the known keyphrases, we can assign positive or negative labels to the examples. Then we learn a classifier that can discriminate between positive and negative examples as a function of the features. Some classifiers make a binary classification for a test example, while others assign a probability of being a keyphrase. For ins

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  Deep tomographic reconstruction

  Deep Tomographic Reconstruction is a set of methods for using deep learning methods to perform tomographic reconstruction of medical and industrial images. It uses artificial intelligence and machine learning, especially deep artificial neural networks or deep learning, to overcome challenges such as measurement noise, data sparsity, image artifacts, and computational inefficiency. This approach has been applied across various imaging modalities, including CT, MRI, PET, SPECT, ultrasound, and optical imaging == Historical background == Traditional tomographic reconstruction relies on analytic methods such as filtered back-projection, or iterative methods which incrementally compute inverse transformations from measurement data (e.g., Radon or Fourier transform data). However, these approaches are not sufficient for certain imaging techniques such as low-dose CT and fast MRI, or scenarios involving metal artifacts and patient motion. == Use in imaging modalities == === Computed tomography (CT) === In CT, deep learning models can be particularly effective in reducing radiation exposure while maintaining image quality. Deep neural networks can also be able to reconstruct images of fair quality from sparsely sampled data without sacrificing diagnostic performance. Deep learning-based generative AI models can reduce CT metal artifacts. === Magnetic resonance imaging (MRI) === In magnetic resonance imaging (MRI), deep learning can lead to reduced MRI motion artifacts, and increased acquisition speed, referred to as fast MRI. Despite suffering from disadvantages such as lower signal-to-noise ratio (SNR), deep learning can enhance image quality in low field MRI, making these systems clinically viable. === Positron emission tomography (PET) and single-photon emission CT (SPECT) === For PET imaging, deep learning models can provide substantial improvements in low-dose imaging and motion artifact correction. Also, deep learning can help SPECT for generation of attenuation background. A notable technique for PET denoising involves integrating MR data through multimodal networks, which use anatomical information from MRI to enhance PET image quality. === Ultrasound imaging === Deep learning can enhance ultrasound imaging by reducing speckle noise and motion blur. For ultrasound beamforming, deep neural networks can allow superior image quality with limited data at high speed. === Optical imaging and microscopy === Diffuse optical tomography, optical coherence tomography and microscopy can be improved by deep neural networks beyond traditional methods. Furthermore, deep learning can also enhance Photoacoustic imaging (see Deep learning in photoacoustic imaging), addressing challenges like high noise, low contrast, and limited resolution. Deep learning has also been applied to label-free live-cell imaging, where convolutional neural networks predict fluorescence labels from transmitted light images, a technique known as in silico labeling. This method can enable high-throughput, non-invasive cell analysis and phenotyping without the need for traditional fluorescent dyes.

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  Human-in-the-loop

  Human-in-the-loop (HITL) is used in multiple contexts. It can be defined as a model requiring human interaction. HITL is associated with modeling and simulation (M&S) in the live, virtual, and constructive taxonomy. HITL, along with the related human-on-the-loop, are also used in relation to lethal autonomous weapons. Further, HITL is used in the context of machine learning.It is also used in conversational AI to manage complex interactions that require human empathy. == Machine learning == In machine learning, HITL is used in the sense of humans aiding the computer in making the correct decisions in building a model. HITL improves machine learning over random sampling by selecting the most critical data needed to refine the model. == Simulation == In simulation, HITL models may conform to human factors requirements as in the case of a mockup. In this type of simulation, a human is always part of the simulation and consequently influences the outcome in such a way that is difficult if not impossible to reproduce exactly. HITL also readily allows for the identification of problems and requirements that may not be easily identified by other means of simulation. HITL is often referred to as an interactive simulation, which is a special kind of physical simulation in which physical simulations include human operators, such as in a flight or a driving simulator. === Benefits === Human-in-the-loop allows the user to change the outcome of an event or process. The immersion effectively contributes to a positive transfer of acquired skills into the real world. This can be demonstrated by trainees utilizing flight simulators in preparation to become pilots. HITL also allows for the acquisition of knowledge regarding how a new process may affect a particular event. Utilizing HITL allows participants to interact with realistic models and attempt to perform as they would in an actual scenario. HITL simulations bring to the surface issues that would not otherwise be apparent until after a new process has been deployed. A real-world example of HITL simulation as an evaluation tool is its usage by the Federal Aviation Administration (FAA) to allow air traffic controllers to test new automation procedures by directing the activities of simulated air traffic while monitoring the effect of the newly implemented procedures. As with most processes, there is always the possibility of human error, which can only be reproduced using HITL simulation. Although much can be done to automate systems, humans typically still need to take the information provided by a system to determine the next course of action based on their judgment and experience. Intelligent systems can only go so far in certain circumstances to automate a process; only humans in the simulation can accurately judge the final design. Tabletop simulation may be useful in the very early stages of project development for the purpose of collecting data to set broad parameters, but the important decisions require human-in-the-loop simulation. HITL reflects scenarios where human input remains essential despite advances in automation. === Within the virtual simulation taxonomy === Virtual simulations inject HITL in a central role by exercising motor control skills (e.g. flying an airplane), decision making skills (e.g. committing fire control resources to action), or communication skills (e.g. as members of a C4I team). === Examples === Flight simulators Driving simulators Marine simulators Video games Supply chain management simulators Digital puppetry === Misconceptions === Although human-in-the-loop simulation can include a computer simulation in the form of a synthetic environment, computer simulation is not necessarily a form of human-in-the-loop simulation, and is often considered as human-out-of-the loop simulation. In this particular case, a computer model’s behavior is modified according to a set of initial parameters. The results of the model differ from the results stemming from a true human-in-the-loop simulation because the results can easily be replicated time and time again, by simply providing identical parameters. == Weapons == === Taxonomy === Three classifications of the degree of human control of autonomous weapon systems were laid out by Bonnie Docherty in a 2012 Human Rights Watch report. human-in-the-loop: a human must instigate the action of the weapon (in other words not fully autonomous) human-on-the-loop: a human may abort an action human-out-of-the-loop: no human action is involved === Positive human action === In discussions of autonomous weapons and nuclear command and control, the phrase positive human action has been used alongside "human-in-the-loop" to emphasize that a human operator must affirmatively authorize the use of force. Descriptions of the United States Navy's Aegis Combat System have used the phrase in characterizing a requirement for affirmative human action to initiate live firing. A survey of autonomous weapons systems described the Aegis "Auto SM" mode as one in which "the system fully develops the engagement process however engagement requires positive human action". The phrase entered United States federal law in the National Defense Authorization Act for Fiscal Year 2025, which stipulates that artificial intelligence systems not compromise "the principle of requiring positive human actions in execution of decisions by the President with respect to the employment of nuclear weapons".

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  Computer audition

  Computer audition (CA) or machine listening is the general field of study of algorithms and systems for audio interpretation by machines. Since the notion of what it means for a machine to "hear" is very broad and somewhat vague, computer audition attempts to bring together several disciplines that originally dealt with specific problems or had a concrete application in mind. The engineer Paris Smaragdis, interviewed in Technology Review, talks about these systems — "software that uses sound to locate people moving through rooms, monitor machinery for impending breakdowns, or activate traffic cameras to record accidents." Inspired by models of human audition, CA deals with questions of representation, transduction, grouping, use of musical knowledge and general sound semantics for the purpose of performing intelligent operations on audio and music signals by the computer. Technically this requires a combination of methods from the fields of signal processing, auditory modelling, music perception and cognition, pattern recognition, and machine learning, as well as more traditional methods of artificial intelligence for musical knowledge representation. == Applications == Like computer vision versus image processing, computer audition versus audio engineering deals with understanding of audio rather than processing. It also differs from problems of speech understanding by machine since it deals with general audio signals, such as natural sounds and musical recordings. Applications of computer audition are widely varying, and include search for sounds, genre recognition, acoustic monitoring, music transcription, score following, audio texture, music improvisation, emotion in audio and so on. == Related disciplines == Computer Audition overlaps with the following disciplines: Music information retrieval: methods for search and analysis of similarity between music signals. Auditory scene analysis: understanding and description of audio sources and events. Computational musicology and mathematical music theory: use of algorithms that employ musical knowledge for analysis of music data. Computer music: use of computers in creative musical applications. Machine musicianship: audition driven interactive music systems. == Areas of study == Since audio signals are interpreted by the human ear–brain system, that complex perceptual mechanism should be simulated somehow in software for "machine listening". In other words, to perform on par with humans, the computer should hear and understand audio content much as humans do. Analyzing audio accurately involves several fields: electrical engineering (spectrum analysis, filtering, and audio transforms); artificial intelligence (machine learning and sound classification); psychoacoustics (sound perception); cognitive sciences (neuroscience and artificial intelligence); acoustics (physics of sound production); and music (harmony, rhythm, and timbre). Furthermore, audio transformations such as pitch shifting, time stretching, and sound object filtering, should be perceptually and musically meaningful. For best results, these transformations require perceptual understanding of spectral models, high-level feature extraction, and sound analysis/synthesis. Finally, structuring and coding the content of an audio file (sound and metadata) could benefit from efficient compression schemes, which discard inaudible information in the sound. Computational models of music and sound perception and cognition can lead to a more meaningful representation, a more intuitive digital manipulation and generation of sound and music in musical human-machine interfaces. The study of CA could be roughly divided into the following sub-problems: Representation: signal and symbolic. This aspect deals with time-frequency representations, both in terms of notes and spectral models, including pattern playback and audio texture. Feature extraction: sound descriptors, segmentation, onset, pitch and envelope detection, chroma, and auditory representations. Musical knowledge structures: analysis of tonality, rhythm, and harmonies. Sound similarity: methods for comparison between sounds, sound identification, novelty detection, segmentation, and clustering. Sequence modeling: matching and alignment between signals and note sequences. Source separation: methods of grouping of simultaneous sounds, such as multiple pitch detection and time-frequency clustering methods. Auditory cognition: modeling of emotions, anticipation and familiarity, auditory surprise, and analysis of musical structure. Multi-modal analysis: finding correspondences between textual, visual, and audio signals. === Representation issues === Computer audition deals with audio signals that can be represented in a variety of fashions, from direct encoding of digital audio in two or more channels to symbolically represented synthesis instructions. Audio signals are usually represented in terms of analogue or digital recordings. Digital recordings are samples of acoustic waveform or parameters of audio compression algorithms. One of the unique properties of musical signals is that they often combine different types of representations, such as graphical scores and sequences of performance actions that are encoded as MIDI files. Since audio signals usually comprise multiple sound sources, then unlike speech signals that can be efficiently described in terms of specific models (such as source-filter model), it is hard to devise a parametric representation for general audio. Parametric audio representations usually use filter banks or sinusoidal models to capture multiple sound parameters, sometimes increasing the representation size in order to capture internal structure in the signal. Additional types of data that are relevant for computer audition are textual descriptions of audio contents, such as annotations, reviews, and visual information in the case of audio-visual recordings. === Features === Description of contents of general audio signals usually requires extraction of features that capture specific aspects of the audio signal. Generally speaking, one could divide the features into signal or mathematical descriptors such as energy, description of spectral shape etc., statistical characterization such as change or novelty detection, special representations that are better adapted to the nature of musical signals or the auditory system, such as logarithmic growth of sensitivity (bandwidth) in frequency or octave invariance (chroma). Since parametric models in audio usually require very many parameters, the features are used to summarize properties of multiple parameters in a more compact or salient representation. === Musical knowledge === Finding specific musical structures is possible by using musical knowledge as well as supervised and unsupervised machine learning methods. Examples of this include detection of tonality according to distribution of frequencies that correspond to patterns of occurrence of notes in musical scales, distribution of note onset times for detection of beat structure, distribution of energies in different frequencies to detect musical chords and so on. === Sound similarity and sequence modeling === Comparison of sounds can be done by comparison of features with or without reference to time. In some cases an overall similarity can be assessed by close values of features between two sounds. In other cases when temporal structure is important, methods of dynamic time warping need to be applied to "correct" for different temporal scales of acoustic events. Finding repetitions and similar sub-sequences of sonic events is important for tasks such as texture synthesis and machine improvisation. === Source separation === Since one of the basic characteristics of general audio is that it comprises multiple simultaneously sounding sources, such as multiple musical instruments, people talking, machine noises or animal vocalization, the ability to identify and separate individual sources is very desirable. Unfortunately, there are no methods that can solve this problem in a robust fashion. Existing methods of source separation rely sometimes on correlation between different audio channels in multi-channel recordings. The ability to separate sources from stereo signals requires different techniques than those usually applied in communications where multiple sensors are available. Other source separation methods rely on training or clustering of features in mono recording, such as tracking harmonically related partials for multiple pitch detection. Some methods, before explicit recognition, rely on revealing structures in data without knowing the structures (like recognizing objects in abstract pictures without attributing them meaningful labels) by finding the least complex data representations, for instance describing audio scenes as generated by a few tone patterns and their trajectories (polyphonic voices) and acoustical contours drawn by a tone (c

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  Cache language model

  A cache language model is a type of statistical language model. These occur in the natural language processing subfield of computer science and assign probabilities to given sequences of words by means of a probability distribution. Statistical language models are key components of speech recognition systems and of many machine translation systems: they tell such systems which possible output word sequences are probable and which are improbable. The particular characteristic of a cache language model is that it contains a cache component and assigns relatively high probabilities to words or word sequences that occur elsewhere in a given text. The primary, but by no means sole, use of cache language models is in speech recognition systems. To understand why it is a good idea for a statistical language model to contain a cache component one might consider someone who is dictating a letter about elephants to a speech recognition system. Standard (non-cache) N-gram language models will assign a very low probability to the word "elephant" because it is a very rare word in English. If the speech recognition system does not contain a cache component, the person dictating the letter may be annoyed: each time the word "elephant" is spoken another sequence of words with a higher probability according to the N-gram language model may be recognized (e.g., "tell a plan"). These erroneous sequences will have to be deleted manually and replaced in the text by "elephant" each time "elephant" is spoken. If the system has a cache language model, "elephant" will still probably be misrecognized the first time it is spoken and will have to be entered into the text manually; however, from this point on the system is aware that "elephant" is likely to occur again – the estimated probability of occurrence of "elephant" has been increased, making it more likely that if it is spoken it will be recognized correctly. Once "elephant" has occurred several times, the system is likely to recognize it correctly every time it is spoken until the letter has been completely dictated. This increase in the probability assigned to the occurrence of "elephant" is an example of a consequence of machine learning and more specifically of pattern recognition. There exist variants of the cache language model in which not only single words but also multi-word sequences that have occurred previously are assigned higher probabilities (e.g., if "San Francisco" occurred near the beginning of the text subsequent instances of it would be assigned a higher probability). The cache language model was first proposed in a paper published in 1990, after which the IBM speech-recognition group experimented with the concept. The group found that implementation of a form of cache language model yielded a 24% drop in word-error rates once the first few hundred words of a document had been dictated. A detailed survey of language modeling techniques concluded that the cache language model was one of the few new language modeling techniques that yielded improvements over the standard N-gram approach: "Our caching results show that caching is by far the most useful technique for perplexity reduction at small and medium training data sizes". The development of the cache language model has generated considerable interest among those concerned with computational linguistics in general and statistical natural language processing in particular: recently, there has been interest in applying the cache language model in the field of statistical machine translation. The success of the cache language model in improving word prediction rests on the human tendency to use words in a "bursty" fashion: when one is discussing a certain topic in a certain context, the frequency with which one uses certain words will be quite different from their frequencies when one is discussing other topics in other contexts. The traditional N-gram language models, which rely entirely on information from a very small number (four, three, or two) of words preceding the word to which a probability is to be assigned, do not adequately model this "burstiness". Recently, the cache language model concept – originally conceived for the N-gram statistical language model paradigm – has been adapted for use in the neural paradigm. For instance, recent work on continuous cache language models in the recurrent neural network (RNN) setting has applied the cache concept to much larger contexts than before, yielding significant reductions in perplexity. Another recent line of research involves incorporating a cache component in a feed-forward neural language model (FN-LM) to achieve rapid domain adaptation.

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  Artificial brain

  An artificial brain (or artificial mind) is software and hardware with cognitive abilities similar to those of the animal or human brain. Research investigating "artificial brains" and brain emulation plays three important roles in science: An ongoing attempt by neuroscientists to understand how the human brain works, known as cognitive neuroscience. A thought experiment in the philosophy of artificial intelligence, demonstrating that it is possible, at least in theory, to create a machine that has all the capabilities of a human being. A long-term project to create machines exhibiting behavior comparable to those of animals with complex central nervous system such as mammals and most particularly humans. The ultimate goal of creating a machine exhibiting human-like behavior or intelligence is sometimes called strong AI. An example of the first objective is the project reported by Aston University in Birmingham, England where researchers are using biological cells to create "neurospheres" (small clusters of neurons) in order to develop new treatments for diseases including Alzheimer's, motor neurone and Parkinson's disease. The second objective is a reply to arguments such as John Searle's Chinese room argument, Hubert Dreyfus's critique of AI or Roger Penrose's argument in The Emperor's New Mind. These critics argued that there are aspects of human consciousness or expertise that can not be simulated by machines. One reply to their arguments is that the biological processes inside the brain can be simulated to any degree of accuracy. This reply was made as early as 1950, by Alan Turing in his classic paper "Computing Machinery and Intelligence". The third objective is generally called artificial general intelligence by researchers. However, Ray Kurzweil prefers the term "strong AI". In his book The Singularity is Near, he focuses on whole brain emulation using conventional computing machines as an approach to implementing artificial brains, and claims (on grounds of computer power continuing an exponential growth trend) that this could be done by 2025. Henry Markram, director of the Blue Brain project (which is attempting brain emulation), made a similar claim (2020) at the Oxford TED conference in 2009. == Approaches to brain simulation == W. Ross Ashby's pioneering work in cybernetics provided an early mathematical framework for understanding adaptive brain-like systems. In his 1952 book Design for a Brain, Ashby proposed that the brain could be modeled as an ultrastable system that maintains equilibrium through continuous adaptation to environmental perturbations. His approach used differential equations and state-space models to describe how neural systems could exhibit purposeful behavior through feedback mechanisms. Ashby's homeostat, a physical machine built in 1948, demonstrated these principles through an electromechanical device with four interconnected units that automatically adjusted their parameters to maintain stability when disturbed. The homeostat represented one of the first attempts to build an artificial system exhibiting brain-like adaptive behavior, influencing subsequent work in adaptive systems, neural networks, and artificial intelligence. Although direct human brain emulation using artificial neural networks on a high-performance computing engine is a commonly discussed approach, there are other approaches. An alternative artificial brain implementation could be based on Holographic Neural Technology (HNeT) non linear phase coherence/decoherence principles. The analogy has been made to quantum processes through the core synaptic algorithm which has strong similarities to the quantum mechanical wave equation. EvBrain is a form of evolutionary software that can evolve "brainlike" neural networks, such as the network immediately behind the retina. In November 2008, IBM received a US$4.9 million grant from the Pentagon for research into creating intelligent computers. The Blue Brain project is being conducted with the assistance of IBM in Lausanne. The project is based on the premise that it is possible to artificially link the neurons "in the computer" by placing thirty million synapses in their proper three-dimensional position. Some proponents of strong AI speculated in 2009 that computers in connection with Blue Brain and Soul Catcher may exceed human intellectual capacity by around 2015, and that it is likely that we will be able to download the human brain at some time around 2050. While Blue Brain is able to represent complex neural connections on the large scale, the project does not achieve the link between brain activity and behaviors executed by the brain. In 2012, project Spaun (Semantic Pointer Architecture Unified Network) attempted to model multiple parts of the human brain through large-scale representations of neural connections that generate complex behaviors in addition to mapping. Spaun's design recreates elements of human brain anatomy. The model, consisting of approximately 2.5 million neurons, includes features of the visual and motor cortices, GABAergic and dopaminergic connections, the ventral tegmental area (VTA), substantia nigra, and others. The design allows for several functions in response to eight tasks, using visual inputs of typed or handwritten characters and outputs carried out by a mechanical arm. Spaun's functions include copying a drawing, recognizing images, and counting. There are good reasons to believe that, regardless of implementation strategy, the predictions of realising artificial brains in the near future are optimistic. In particular brains (including the human brain) and cognition are not currently well understood, and the scale of computation required is unknown. Another near term limitation is that all current approaches for brain simulation require orders of magnitude larger power consumption compared with a human brain. The human brain consumes about 20 W of power, whereas current supercomputers may use as much as 1 MW—i.e., an order of 100,000 more. == Artificial brain thought experiment == Some critics of brain simulation believe that it is simpler to create general intelligent action directly without imitating nature. Some commentators have used the analogy that early attempts to construct flying machines modeled them after birds, but that modern aircraft do not look like birds.

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  EfficientNet

  EfficientNet is a family of convolutional neural networks (CNNs) for computer vision published by researchers at Google AI in 2019. Its key innovation is compound scaling, which uniformly scales all dimensions of depth, width, and resolution using a single parameter. EfficientNet models have been adopted in various computer vision tasks, including image classification, object detection, and segmentation. == Compound scaling == EfficientNet introduces compound scaling, which, instead of scaling one dimension of the network at a time, such as depth (number of layers), width (number of channels), or resolution (input image size), uses a compound coefficient ϕ {\displaystyle \phi } to scale all three dimensions simultaneously. Specifically, given a baseline network, the depth, width, and resolution are scaled according to the following equations: depth multiplier: d = α ϕ width multiplier: w = β ϕ resolution multiplier: r = γ ϕ {\displaystyle {\begin{aligned}{\text{depth multiplier: }}d&=\alpha ^{\phi }\\{\text{width multiplier: }}w&=\beta ^{\phi }\\{\text{resolution multiplier: }}r&=\gamma ^{\phi }\end{aligned}}} subject to α ⋅ β 2 ⋅ γ 2 ≈ 2 {\displaystyle \alpha \cdot \beta ^{2}\cdot \gamma ^{2}\approx 2} and α ≥ 1 , β ≥ 1 , γ ≥ 1 {\displaystyle \alpha \geq 1,\beta \geq 1,\gamma \geq 1} . The α ⋅ β 2 ⋅ γ 2 ≈ 2 {\displaystyle \alpha \cdot \beta ^{2}\cdot \gamma ^{2}\approx 2} condition is such that increasing ϕ {\displaystyle \phi } by a factor of ϕ 0 {\displaystyle \phi _{0}} would increase the total FLOPs of running the network on an image approximately 2 ϕ 0 {\displaystyle 2^{\phi _{0}}} times. The hyperparameters α {\displaystyle \alpha } , β {\displaystyle \beta } , and γ {\displaystyle \gamma } are determined by a small grid search. The original paper suggested 1.2, 1.1, and 1.15, respectively. Architecturally, they optimized the choice of modules by neural architecture search (NAS), and found that the inverted bottleneck convolution (which they called MBConv) used in MobileNet worked well. The EfficientNet family is a stack of MBConv layers, with shapes determined by the compound scaling. The original publication consisted of 8 models, from EfficientNet-B0 to EfficientNet-B7, with increasing model size and accuracy. EfficientNet-B0 is the baseline network, and subsequent models are obtained by scaling the baseline network by increasing ϕ {\displaystyle \phi } . == Variants == EfficientNet has been adapted for fast inference on edge TPUs and centralized TPU or GPU clusters by NAS. EfficientNet V2 was published in June 2021. The architecture was improved by further NAS search with more types of convolutional layers. It also introduced a training method, which progressively increases image size during training, and uses regularization techniques like dropout, RandAugment, and Mixup. The authors claim this approach mitigates accuracy drops often associated with progressive resizing.

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  Kernel embedding of distributions

  In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space Ω {\displaystyle \Omega } on which a sensible kernel function (measuring similarity between elements of Ω {\displaystyle \Omega } ) may be defined. For example, various kernels have been proposed for learning from data which are: vectors in R d {\displaystyle \mathbb {R} ^{d}} , discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song, Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in. The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback–Leibler divergence. However, to estimate these quantities, one must first either perform density estimation, or employ sophisticated space-partitioning/bias-correction strategies which are typically infeasible for high-dimensional data. Commonly, methods for modeling complex distributions rely on parametric assumptions that may be unfounded or computationally challenging (e.g. Gaussian mixture models), while nonparametric methods like kernel density estimation (Note: the smoothing kernels in this context have a different interpretation than the kernels discussed here) or characteristic function representation (via the Fourier transform of the distribution) break down in high-dimensional settings. Methods based on the kernel embedding of distributions sidestep these problems and also possess the following advantages: Data may be modeled without restrictive assumptions about the form of the distributions and relationships between variables Intermediate density estimation is not needed Practitioners may specify the properties of a distribution most relevant for their problem (incorporating prior knowledge via choice of the kernel) If a characteristic kernel is used, then the embedding can uniquely preserve all information about a distribution, while thanks to the kernel trick, computations on the potentially infinite-dimensional RKHS can be implemented in practice as simple Gram matrix operations Dimensionality-independent rates of convergence for the empirical kernel mean (estimated using samples from the distribution) to the kernel embedding of the true underlying distribution can be proven. Learning algorithms based on this framework exhibit good generalization ability and finite sample convergence, while often being simpler and more effective than information theoretic methods Thus, learning via the kernel embedding of distributions offers a principled drop-in replacement for information theoretic approaches and is a framework which not only subsumes many popular methods in machine learning and statistics as special cases, but also can lead to entirely new learning algorithms. == Definitions == Let X {\displaystyle X} denote a random variable with domain Ω {\displaystyle \Omega } and distribution P {\displaystyle P} . Given a symmetric, positive-definite kernel k : Ω × Ω → R {\displaystyle k:\Omega \times \Omega \rightarrow \mathbb {R} } the Moore–Aronszajn theorem asserts the existence of a unique RKHS H {\displaystyle {\mathcal {H}}} on Ω {\displaystyle \Omega } (a Hilbert space of functions f : Ω → R {\displaystyle f:\Omega \to \mathbb {R} } equipped with an inner product ⟨ ⋅ , ⋅ ⟩ H {\displaystyle \langle \cdot ,\cdot \rangle _{\mathcal {H}}} and a norm ‖ ⋅ ‖ H {\displaystyle \|\cdot \|_{\mathcal {H}}} ) for which k {\displaystyle k} is a reproducing kernel, i.e., in which the element k ( x , ⋅ ) {\displaystyle k(x,\cdot )} satisfies the reproducing property ⟨ f , k ( x , ⋅ ) ⟩ H = f ( x ) ∀ f ∈ H , ∀ x ∈ Ω . {\displaystyle \langle f,k(x,\cdot )\rangle _{\mathcal {H}}=f(x)\qquad \forall f\in {\mathcal {H}},\quad \forall x\in \Omega .} One may alternatively consider x ↦ k ( x , ⋅ ) {\displaystyle x\mapsto k(x,\cdot )} as an implicit feature mapping φ : Ω → H {\displaystyle \varphi :\Omega \rightarrow {\mathcal {H}}} (which is therefore also called the feature space), so that k ( x , x ′ ) = ⟨ φ ( x ) , φ ( x ′ ) ⟩ H {\displaystyle k(x,x')=\langle \varphi (x),\varphi (x')\rangle _{\mathcal {H}}} can be viewed as a measure of similarity between points x , x ′ ∈ Ω . {\displaystyle x,x'\in \Omega .} While the similarity measure is linear in the feature space, it may be highly nonlinear in the original space depending on the choice of kernel. === Kernel embedding === The kernel embedding of the distribution P {\displaystyle P} in H {\displaystyle {\mathcal {H}}} (also called the kernel mean or mean map) is given by: μ X := E [ k ( X , ⋅ ) ] = E [ φ ( X ) ] = ∫ Ω φ ( x ) d P ( x ) {\displaystyle \mu _{X}:=\mathbb {E} [k(X,\cdot )]=\mathbb {E} [\varphi (X)]=\int _{\Omega }\varphi (x)\ \mathrm {d} P(x)} If P {\displaystyle P} allows a square integrable density p {\displaystyle p} , then μ X = E k p {\displaystyle \mu _{X}={\mathcal {E}}_{k}p} , where E k {\displaystyle {\mathcal {E}}_{k}} is the Hilbert–Schmidt integral operator. A kernel is characteristic if the mean embedding μ : { family of distributions over Ω } → H {\displaystyle \mu :\{{\text{family of distributions over }}\Omega \}\to {\mathcal {H}}} is injective. Each distribution can thus be uniquely represented in the RKHS and all statistical features of distributions are preserved by the kernel embedding if a characteristic kernel is used. === Empirical kernel embedding === Given n {\displaystyle n} training examples { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} drawn independently and identically distributed (i.i.d.) from P , {\displaystyle P,} the kernel embedding of P {\displaystyle P} can be empirically estimated as μ ^ X = 1 n ∑ i = 1 n φ ( x i ) {\displaystyle {\widehat {\mu }}_{X}={\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})} === Joint distribution embedding === If Y {\displaystyle Y} denotes another random variable (for simplicity, assume the co-domain of Y {\displaystyle Y} is also Ω {\displaystyle \Omega } with the same kernel k {\displaystyle k} which satisfies ⟨ φ ( x ) ⊗ φ ( y ) , φ ( x ′ ) ⊗ φ ( y ′ ) ⟩ = k ( x , x ′ ) k ( y , y ′ ) {\displaystyle \langle \varphi (x)\otimes \varphi (y),\varphi (x')\otimes \varphi (y')\rangle =k(x,x')k(y,y')} ), then the joint distribution P ( x , y ) ) {\displaystyle P(x,y))} can be mapped into a tensor product feature space H ⊗ H {\displaystyle {\mathcal {H}}\otimes {\mathcal {H}}} via C X Y = E [ φ ( X ) ⊗ φ ( Y ) ] = ∫ Ω × Ω φ ( x ) ⊗ φ ( y ) d P ( x , y ) {\displaystyle {\mathcal {C}}_{XY}=\mathbb {E} [\varphi (X)\otimes \varphi (Y)]=\int _{\Omega \times \Omega }\varphi (x)\otimes \varphi (y)\ \mathrm {d} P(x,y)} By the equivalence between a tensor and a linear map, this joint embedding may be interpreted as an uncentered cross-covariance operator C X Y : H → H {\displaystyle {\mathcal {C}}_{XY}:{\mathcal {H}}\to {\mathcal {H}}} from which the cross-covariance of functions f , g ∈ H {\displaystyle f,g\in {\mathcal {H}}} can be computed as Cov ⁡ ( f ( X ) , g ( Y ) ) := E [ f ( X ) g ( Y ) ] − E [ f ( X ) ] E [ g ( Y ) ] = ⟨ f , C X Y g ⟩ H = ⟨ f ⊗ g , C X Y ⟩ H ⊗ H {\displaystyle \operatorname {Cov} (f(X),g(Y)):=\mathbb {E} [f(X)g(Y)]-\mathbb {E} [f(X)]\mathbb {E} [g(Y)]=\langle f,{\mathcal {C}}_{XY}g\rangle _{\mathcal {H}}=\langle f\otimes g,{\mathcal {C}}_{XY}\rangle _{{\mathcal {H}}\otimes {\mathcal {H}}}} Given n {\displaystyle n} pairs of training examples { ( x 1 , y 1 ) , … , ( x n , y n ) } {\displaystyle \{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}} drawn i.i.d. from P {\displaystyle P} , we can also empirically estimate the joint distribution kernel embedding via C ^ X Y = 1 n ∑ i = 1 n φ ( x i ) ⊗ φ ( y i ) {\displaystyle {\widehat {\mathcal {C}}}_{XY}={\frac {1}{n}}\sum _{i=1}^{n}\varphi (x_{i})\otimes \varphi (y_{i})} === Conditional distribution embedding === Given a conditional distribution P ( y ∣ x ) , {\displaystyle P(y\mid x),} one can define the corresponding RKHS embedding as μ Y ∣ x = E [ φ ( Y ) ∣ X ] = ∫ Ω φ ( y ) d P ( y ∣ x ) {\displaystyle \mu _{Y\mid x}=\mathbb {E} [\varphi (Y)\mid X]=\int _{\Omega

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  WIPO GREEN

  WIPO GREEN is a World Intellectual Property Organization program established in 2013 that supports global efforts to address climate change and food security through sharing of sustainable technology innovations. == WIPO GREEN database == The WIPO GREEN database is the foundation of the platform. The database is a free, solutions-oriented, global innovation catalog that connects needs for solving environmental or climate change problems with sustainable solutions from prototypes to marketable products available for sale, license, collaborations, knowledge transfer, joint ventures, or collaborations. Green technology innovators can promote their products, businesses, organizations, and governments looking for green technologies can explain their needs and seek collaboration with providers. As of July 2022, WIPO GREEN has over 120,000 technologies, needs and experts, more than 2000 users in 110 countries, and has recorded over 1000 connections made between technology providers and seekers. The database utilizes AI-assisted auto-matching, user uploads tracing and alerts, full-text search for solutions based on long need descriptions, and the Patent2Solution search function for finding commercial applications of a patent, which are some of the unique features of the database. Free registration is required for detailed record view and uploading. All technologies uploaded to the WIPO GREEN database remain the property of the rights holder. It is up to the rights holder and the collaborating parties to structure agreements in the manner they feel is most appropriate and effective. WIPO GREEN does not require that technologies or innovations uploaded to the database be patented or in the process of being patented. Therefore, technology providers can upload their technology while related patent applications are pending. Technology providers are encouraged to upload technology solutions on the WIPO GREEN database and connect with other users to explore partnerships, technology transfers, including funding and licensing opportunities. == Acceleration projects == Acceleration projects work with WIPO GREEN partners and local organizations to explore local challenges and green opportunities for particular environmental needs. These projects are organized annually in different countries or regions around and connect providers and seekers of green technologies. For example, the Latin America Acceleration Project explores innovative new technologies in the region and facilitates green technology exchange between providers and seekers in green opportunities in intensified crop rotation, soil re-carbonization, and forest management in Argentina; zero-till or conservation agriculture in Brazil; and wine production in Chile. In October 2021, a project in Indonesia on palm oil mill effluent (POME), a by-product of palm oil production that emits greenhouse gases and reportedly harms flora and fauna in local rivers, identified viable green solutions to turn the high organic content of POME wastewater into biogas and other environmentally friendly uses. Former projects took place in Cambodia, Indonesia, and the Philippines around wastewater treatment, agriculture, and water technologies. == The Green Technology Book == In November 2022 at UNFCCC COP27, WIPO introduced its new Flagship publication the Green Technology Book. This digital-first publication aims to put innovation, technology and intellectual property at the forefront in the fight against climate change. The inaugural edition of this annual publication focused on available solutions for climate-change adaptation to reduce vulnerability as well as to increase resilience to the impacts of climate change. The book was created in cooperation with the Climate Technology Center and Network (CTCN) and the Egyptian Academy of Scientific Research and Technology (ASTR). It features 200 adaptation technologies, which are also available in the WIPO GREEN database of innovative technologies and needs. == Partners Network == WIPO GREEN partners are public or private institutions that wish to collaborate to advance WIPO GREEN’s mission. The network is aimed at helping the implementation and diffusion of green technology innovations around the world. Partners include government institutions, intergovernmental organizations, academia, and businesses – from small and medium-sized enterprises to Fortune 500 companies. As of 2022, WIPO GREEN has a network of over 146 partner organizations involved in green technology.

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  Algorithmic inference

  Algorithmic inference gathers new developments in the statistical inference methods made feasible by the powerful computing devices widely available to any data analyst. Cornerstones in this field are computational learning theory, granular computing, bioinformatics, and, long ago, structural probability (Fraser 1966). The main focus is on the algorithms which compute statistics rooting the study of a random phenomenon, along with the amount of data they must feed on to produce reliable results. This shifts the interest of mathematicians from the study of the distribution laws to the functional properties of the statistics, and the interest of computer scientists from the algorithms for processing data to the information they process. == The Fisher parametric inference problem == Concerning the identification of the parameters of a distribution law, the mature reader may recall lengthy disputes in the mid 20th century about the interpretation of their variability in terms of fiducial distribution (Fisher 1956), structural probabilities (Fraser 1966), priors/posteriors (Ramsey 1925), and so on. From an epistemology viewpoint, this entailed a companion dispute as to the nature of probability: is it a physical feature of phenomena to be described through random variables or a way of synthesizing data about a phenomenon? Opting for the latter, Fisher defines a fiducial distribution law of parameters of a given random variable that he deduces from a sample of its specifications. With this law he computes, for instance "the probability that μ (mean of a Gaussian variable – omeur note) is less than any assigned value, or the probability that it lies between any assigned values, or, in short, its probability distribution, in the light of the sample observed". == The classic solution == Fisher fought hard to defend the difference and superiority of his notion of parameter distribution in comparison to analogous notions, such as Bayes' posterior distribution, Fraser's constructive probability and Neyman's confidence intervals. For half a century, Neyman's confidence intervals won out for all practical purposes, crediting the phenomenological nature of probability. With this perspective, when you deal with a Gaussian variable, its mean μ is fixed by the physical features of the phenomenon you are observing, where the observations are random operators, hence the observed values are specifications of a random sample. Because of their randomness, you may compute from the sample specific intervals containing the fixed μ with a given probability that you denote confidence. === Example === Let X be a Gaussian variable with parameters μ {\displaystyle \mu } and σ 2 {\displaystyle \sigma ^{2}} and { X 1 , … , X m } {\displaystyle \{X_{1},\ldots ,X_{m}\}} a sample drawn from it. Working with statistics S μ = ∑ i = 1 m X i {\displaystyle S_{\mu }=\sum _{i=1}^{m}X_{i}} and S σ 2 = ∑ i = 1 m ( X i − X ¯ ) 2 , where X ¯ = S μ m {\displaystyle S_{\sigma ^{2}}=\sum _{i=1}^{m}(X_{i}-{\overline {X}})^{2},{\text{ where }}{\overline {X}}={\frac {S_{\mu }}{m}}} is the sample mean, we recognize that T = S μ − m μ S σ 2 m − 1 m = X ¯ − μ S σ 2 / ( m ( m − 1 ) ) {\displaystyle T={\frac {S_{\mu }-m\mu }{\sqrt {S_{\sigma ^{2}}}}}{\sqrt {\frac {m-1}{m}}}={\frac {{\overline {X}}-\mu }{\sqrt {S_{\sigma ^{2}}/(m(m-1))}}}} follows a Student's t distribution (Wilks 1962) with parameter (degrees of freedom) m − 1, so that f T ( t ) = Γ ( m / 2 ) Γ ( ( m − 1 ) / 2 ) 1 π ( m − 1 ) ( 1 + t 2 m − 1 ) m / 2 . {\displaystyle f_{T}(t)={\frac {\Gamma (m/2)}{\Gamma ((m-1)/2)}}{\frac {1}{\sqrt {\pi (m-1)}}}\left(1+{\frac {t^{2}}{m-1}}\right)^{m/2}.} Gauging T between two quantiles and inverting its expression as a function of μ {\displaystyle \mu } you obtain confidence intervals for μ {\displaystyle \mu } . With the sample specification: x = { 7.14 , 6.3 , 3.9 , 6.46 , 0.2 , 2.94 , 4.14 , 4.69 , 6.02 , 1.58 } {\displaystyle \mathbf {x} =\{7.14,6.3,3.9,6.46,0.2,2.94,4.14,4.69,6.02,1.58\}} having size m = 10, you compute the statistics s μ = 43.37 {\displaystyle s_{\mu }=43.37} and s σ 2 = 46.07 {\displaystyle s_{\sigma ^{2}}=46.07} , and obtain a 0.90 confidence interval for μ {\displaystyle \mu } with extremes (3.03, 5.65). == Inferring functions with the help of a computer == From a modeling perspective the entire dispute looks like a chicken-egg dilemma: either fixed data by first and probability distribution of their properties as a consequence, or fixed properties by first and probability distribution of the observed data as a corollary. The classic solution has one benefit and one drawback. The former was appreciated particularly back when people still did computations with sheet and pencil. Per se, the task of computing a Neyman confidence interval for the fixed parameter θ is hard: you do not know θ, but you look for disposing around it an interval with a possibly very low probability of failing. The analytical solution is allowed for a very limited number of theoretical cases. Vice versa a large variety of instances may be quickly solved in an approximate way via the central limit theorem in terms of confidence interval around a Gaussian distribution – that's the benefit. The drawback is that the central limit theorem is applicable when the sample size is sufficiently large. Therefore, it is less and less applicable with the sample involved in modern inference instances. The fault is not in the sample size on its own part. Rather, this size is not sufficiently large because of the complexity of the inference problem. With the availability of large computing facilities, scientists refocused from isolated parameters inference to complex functions inference, i.e. re sets of highly nested parameters identifying functions. In these cases we speak about learning of functions (in terms for instance of regression, neuro-fuzzy system or computational learning) on the basis of highly informative samples. A first effect of having a complex structure linking data is the reduction of the number of sample degrees of freedom, i.e. the burning of a part of sample points, so that the effective sample size to be considered in the central limit theorem is too small. Focusing on the sample size ensuring a limited learning error with a given confidence level, the consequence is that the lower bound on this size grows with complexity indices such as VC dimension or detail of a class to which the function we want to learn belongs. === Example === A sample of 1,000 independent bits is enough to ensure an absolute error of at most 0.081 on the estimation of the parameter p of the underlying Bernoulli variable with a confidence of at least 0.99. The same size cannot guarantee a threshold less than 0.088 with the same confidence 0.99 when the error is identified with the probability that a 20-year-old man living in New York does not fit the ranges of height, weight and waistline observed on 1,000 Big Apple inhabitants. The accuracy shortage occurs because both the VC dimension and the detail of the class of parallelepipeds, among which the one observed from the 1,000 inhabitants' ranges falls, are equal to 6. == The general inversion problem solving the Fisher question == With insufficiently large samples, the approach: fixed sample – random properties suggests inference procedures in three steps: === Definition === For a random variable and a sample drawn from it a compatible distribution is a distribution having the same sampling mechanism M X = ( Z , g θ ) {\displaystyle {\mathcal {M}}_{X}=(Z,g_{\boldsymbol {\theta }})} of X with a value θ {\displaystyle {\boldsymbol {\theta }}} of the random parameter Θ {\displaystyle \mathbf {\Theta } } derived from a master equation rooted on a well-behaved statistic s. === Example === You may find the distribution law of the Pareto parameters A and K as an implementation example of the population bootstrap method as in the figure on the left. Implementing the twisting argument method, you get the distribution law F M ( μ ) {\displaystyle F_{M}(\mu )} of the mean M of a Gaussian variable X on the basis of the statistic s M = ∑ i = 1 m x i {\textstyle s_{M}=\sum _{i=1}^{m}x_{i}} when Σ 2 {\displaystyle \Sigma ^{2}} is known to be equal to σ 2 {\displaystyle \sigma ^{2}} (Apolloni, Malchiodi & Gaito 2006). Its expression is: F M ( μ ) = Φ ( m μ − s M σ m ) , {\displaystyle F_{M}(\mu )=\Phi {\left({\frac {m\mu -s_{M}}{\sigma {\sqrt {m}}}}\right)},} shown in the figure on the right, where Φ {\displaystyle \Phi } is the cumulative distribution function of a standard normal distribution. Computing a confidence interval for M given its distribution function is straightforward: we need only find two quantiles (for instance δ / 2 {\displaystyle \delta /2} and 1 − δ / 2 {\displaystyle 1-\delta /2} quantiles in case we are interested in a confidence interval of level δ symmetric in the tail's probabilities) as indicated on the left in the diagram showing the behavior of

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  Dataset shift

  Dataset shift is a phenomenon in machine learning and statistics in which the joint distribution of input variables and target labels is different in the training phase and the deployment or test phase (i.e., P t r a i n ( X , Y ) ≠ P t e s t ( X , Y ) {\displaystyle P_{train}(X,Y)\neq P_{test}(X,Y)} ). This happens when the statistical properties of data used to train a model are no longer representative of the data encountered in real-world use, often resulting in degraded predictive performance and diminished generalization ability. Dataset shift is a generic term for a number of particular types of distributional change. Covariate shift is when the distribution of the input features changes, but the conditional relationship between inputs and outputs remains constant . Prior probability shift (or label shift) happens when the distribution of target labels changes, but the conditional distribution of inputs given labels stays the same. Concept shift (also known as concept drift) is the change of the conditional relationship between inputs and outputs that renders previously learned patterns invalid over time. A key challenge for deploying machine learning systems is dataset shift, in particular in dynamic environments where the data distributions change over time. Detecting and mitigating such shifts is an active area of research, e.g., drift detection, domain adaptation, continual learning.

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  Multimodal representation learning

  Multimodal representation learning is a subfield of representation learning focused on integrating and interpreting information from different modalities, such as text, images, audio, or video, by projecting them into a shared latent space. This allows for semantically similar content across modalities to be mapped to nearby points within that space, facilitating a unified understanding of diverse data types. By automatically learning meaningful features from each modality and capturing their inter-modal relationships, multimodal representation learning enables a unified representation that enhances performance in cross-media analysis tasks such as video classification, event detection, and sentiment analysis. It also supports cross-modal retrieval and translation, including image captioning, video description, and text-to-image synthesis. == Motivation == The primary motivations for multimodal representation learning arise from the inherent nature of real-world data and the limitations of unimodal approaches. Since multimodal data offers complementary and supplementary information about an object or event from different perspectives, it is more informative than relying on a single modality. A key motivation is to narrow the heterogeneity gap that exists between different modalities by projecting their features into a shared semantic subspace. This allows semantically similar content across modalities to be represented by similar vectors, facilitating the understanding of relationships and correlations between them. Multimodal representation learning aims to leverage the unique information provided by each modality to achieve a more comprehensive and accurate understanding of concepts. These unified representations are crucial for improving performance in various cross-media analysis tasks such as video classification, event detection, and sentiment analysis. They also enable cross-modal retrieval, allowing users to search and retrieve content across different modalities. Additionally, it facilitates cross-modal translation, where information can be converted from one modality to another, as seen in applications like image captioning and text-to-image synthesis. The abundance of ubiquitous multimodal data in real-world applications, including understudied areas like healthcare, finance, and human-computer interaction (HCI), further motivates the development of effective multimodal representation learning techniques. == Approaches and methods == === Canonical-correlation analysis based methods === Canonical-correlation analysis (CCA) was first introduced in 1936 by Harold Hotelling and is a fundamental approach for multimodal learning. CCA aims to find linear relationships between two sets of variables. Given two data matrices X ∈ R n × p {\displaystyle X\in \mathbb {R} ^{n\times p}} and Y ∈ R n × q {\displaystyle Y\in \mathbb {R} ^{n\times q}} representing different modalities, CCA finds projection vectors w x ∈ R p {\displaystyle w_{x}\in \mathbb {R} ^{p}} and w y ∈ R q {\displaystyle w_{y}\in \mathbb {R} ^{q}} that maximizes the correlation between the projected variables: ρ = max w x , w y w x ⊤ Σ x y w y w x ⊤ Σ x x w x w y ⊤ Σ y y w y {\displaystyle \rho =\max _{w_{x},w_{y}}{\frac {w_{x}^{\top }\Sigma _{xy}w_{y}}{{\sqrt {w_{x}^{\top }\Sigma _{xx}w_{x}}}{\sqrt {w_{y}^{\top }\Sigma _{yy}w_{y}}}}}} such that Σ x x {\displaystyle \Sigma _{xx}} and Σ y y {\displaystyle \Sigma _{yy}} are the within-modality covariance matrices, and Σ x y {\displaystyle \Sigma _{xy}} is the between-modality covariance matrix. However, standard CCA is limited by its linearity, which led to the development of nonlinear extensions, such as kernel CCA and deep CCA. ==== Kernel CCA ==== Kernel canonical correlation analysis (KCCA) extends traditional CCA to capture nonlinear relationships between modalities by implicitly mapping the data into high dimensional feature spaces using kernel functions. Given kernel functions K x {\displaystyle K_{x}} and K y {\displaystyle K_{y}} with corresponding Gram matrices K x ∈ R n × n {\displaystyle K_{x}\in \mathbb {R} ^{n\times n}} and K y ∈ R n × n {\displaystyle K_{y}\in \mathbb {R} ^{n\times n}} , KCCA seeks coefficients α {\displaystyle \alpha } and β {\displaystyle \beta } that maximize: ρ = max α , β α ⊤ K x K y β α ⊤ K x 2 α β ⊤ K y 2 β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{\top }K_{x}Ky\beta }{{\sqrt {\alpha ^{\top }K_{x}^{2}\alpha }}{\sqrt {\beta ^{\top }K_{y}^{2}\beta }}}}} To prevent overfitting, regularization terms are typically added, resulting in: ρ = max α , β α T K x K y β α T ( K x 2 + λ x K x ) α β T ( K y 2 + λ y K y ) β {\displaystyle \rho =\max _{\alpha ,\beta }{\frac {\alpha ^{T}K_{x}K_{y}\beta }{{\sqrt {\alpha ^{T}\left(K_{x}^{2}+\lambda _{x}K_{x}\right)\alpha }}{\sqrt {\;\beta ^{T}\left(K_{y}^{2}+\lambda _{y}K_{y}\right)\beta }}}}} where λ x {\displaystyle \lambda _{x}} and λ y {\displaystyle \lambda _{y}} are regularization parameters. KCCA has proven effective for tasks such as cross-modal retrieval and semantic analysis, though it faces computational challenges with large datasets due to its O ( n 2 ) {\displaystyle O(n^{2})} memory requirement for sorting kernel matrices. KCCA was proposed independently by several researchers. ==== Deep CCA ==== Deep canonical correlation analysis (DCCA), introduced in 2013, employs neural networks to learn nonlinear transformations for maximizing the correlation between modalities. DCCA uses separate neural networks f x {\displaystyle f_{x}} and f y {\displaystyle f_{y}} for each modality to transform the original data before applying CCA: max W x , W y , θ x , θ y corr ⁡ ( f x ( X ; θ x ) , f y ( Y ; θ y ) ) {\displaystyle \max _{W_{x},W_{y},\theta _{x},\theta _{y}}\operatorname {corr} \left(f_{x}(X;\theta _{x}),f_{y}(Y;\theta _{y})\right)} where θ x {\displaystyle \theta _{x}} and θ y {\displaystyle \theta _{y}} represent the parameters of the neural networks, and W x {\displaystyle W_{x}} and W y {\displaystyle W_{y}} are the CCA projection matrices. The correlation objective is computed as: corr ⁡ ( H x , H y ) = tr ⁡ ( T − 1 / 2 H x T H y S − 1 / 2 ) {\displaystyle \operatorname {corr} (H_{x},H_{y})=\operatorname {tr} \left(T^{-1/2}H_{x}^{T}H_{y}S^{-1/2}\right)} where H x = f x ( X ) {\displaystyle H_{x}=f_{x}(X)} and H y = f y ( Y ) {\displaystyle H_{y}=f_{y}(Y)} are the network outputs, T = H x T H x + r x I {\displaystyle T=H_{x}^{T}H_{x}+r_{x}I} , S = H y T H y + r y I {\displaystyle S=H_{y}^{T}H_{y}+r_{y}I} and r x , r y {\displaystyle r_{x},r_{y}} are the regularization parameters. DCCA overcomes the limitations of linear CCA and kernel CCA by learning complex nonlinear relationships while maintaining computational efficiency for large datasets through mini-batch optimization. === Graph-based methods === Graph-based approaches for multimodal representation learning leverage graph structure to model relationships between entities across different modalities. These methods typically represent each modality as a graph and then learn embedding that preserve cross-modal similarities, enabling more effective joint representation of heterogeneous data. One such method is cross-modal graph neural networks (CMGNNs) that extend traditional graph neural networks (GNNs) to handle data from multiple modalities by constructing graphs that capture both intra-modal and inter-modal relationships. These networks model interactions across modalities by representing them as nodes and their relationships as edges. Other graph-based methods include Probabilistic Graphical Models (PGMs) such as deep belief networks (DBN) and deep Boltzmann machines (DBM). These models can learn a joint representation across modalities, for instance, a multimodal DBN achieves this by adding a shared restricted Boltzmann Machine (RBM) hidden layer on top of modality-specific DBNs. Additionally, the structure of data in some domains like Human-Computer Interaction (HCI), such as the view hierarchy of app screens, can potentially be modeled using graph-like structures. The field of graph representation learning is also relevant, with ongoing progress in developing evaluation benchmarks. === Diffusion maps === Another set of methods relevant to multimodal representation learning are based on diffusion maps and their extensions to handle multiple modalities. ==== Multi-view diffusion maps ==== Multi-view diffusion maps address the challenge of achieving multi-view dimensionality reduction by effectively utilizing the availability of multiple views to extract a coherent low-dimensional representation of the data. The core idea is to exploit both the intrinsic relations within each view and the mutual relations between the different views, defining a cross-view model where a random walk process implicitly hops between objects in different views. A multi-view kernel matrix is constructed by combining these relations, defining a cross-view diffusion process and associ

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  BERT (language model)

  Bidirectional encoder representations from transformers (BERT) is a language model introduced in October 2018 by researchers at Google. It learns to represent text as a sequence of vectors using self-supervised learning. It uses the encoder-only transformer architecture. BERT dramatically improved the state of the art for large language models. As of 2020, BERT is a ubiquitous baseline in natural language processing (NLP) experiments. BERT is trained by masked token prediction and next sentence prediction. With this training, BERT learns contextual, latent representations of tokens in their context, similar to ELMo and GPT-2. It found applications for many natural language processing tasks, such as coreference resolution and polysemy resolution. It improved on ELMo and spawned the study of "BERTology", which attempts to interpret what is learned by BERT. BERT was originally implemented in the English language at two model sizes, BERTBASE (110 million parameters) and BERTLARGE (340 million parameters). Both were trained on the Toronto BookCorpus (800M words) and English Wikipedia (2,500M words). The weights were released on GitHub. On March 11, 2020, 24 smaller models were released, the smallest being BERTTINY with just 4 million parameters. == Architecture == BERT is an "encoder-only" transformer architecture. At a high level, BERT consists of 4 modules: Tokenizer: This module converts a piece of English text into a sequence of integers ("tokens"). Embedding: This module converts the sequence of tokens into an array of real-valued vectors representing the tokens. It represents the conversion of discrete token types into a lower-dimensional Euclidean space. Encoder: a stack of Transformer blocks with self-attention, but without causal masking. Task head: This module converts the final representation vectors into one-shot encoded tokens again by producing a predicted probability distribution over the token types. It can be viewed as a simple decoder, decoding the latent representation into token types, or as an "un-embedding layer". The task head is necessary for pre-training, but it is often unnecessary for so-called "downstream tasks," such as question answering or sentiment classification. Instead, one removes the task head and replaces it with a newly initialized module suited for the task, and finetune the new module. The latent vector representation of the model is directly fed into this new module, allowing for sample-efficient transfer learning. === Embedding === This section describes the embedding used by BERTBASE. The other one, BERTLARGE, is similar, just larger. The tokenizer of BERT is WordPiece, which is a sub-word strategy like byte-pair encoding. Its vocabulary size is 30,000, and any token not appearing in its vocabulary is replaced by [UNK] ("unknown"). The first layer is the embedding layer, which contains three components: token type embeddings, position embeddings, and segment type embeddings. Token type: The token type is a standard embedding layer, translating a one-hot vector into a dense vector based on its token type. Position: The position embeddings are based on a token's position in the sequence. BERT uses absolute position embeddings, where each position in a sequence is mapped to a real-valued vector. Each dimension of the vector consists of a sinusoidal function that takes the position in the sequence as input. Segment type: Using a vocabulary of just 0 or 1, this embedding layer produces a dense vector based on whether the token belongs to the first or second text segment in that input. In other words, type-1 tokens are all tokens that appear after the [SEP] special token. All prior tokens are type-0. The three embedding vectors are added together representing the initial token representation as a function of these three pieces of information. After embedding, the vector representation is normalized using a LayerNorm operation, outputting a 768-dimensional vector for each input token. After this, the representation vectors are passed forward through 12 Transformer encoder blocks, and are decoded back to 30,000-dimensional vocabulary space using a basic affine transformation layer. === Architectural family === The encoder stack of BERT has 2 free parameters: L {\displaystyle L} , the number of layers, and H {\displaystyle H} , the hidden size. There are always H / 64 {\displaystyle H/64} self-attention heads, and the feed-forward/filter size is always 4 H {\displaystyle 4H} . By varying these two numbers, one obtains an entire family of BERT models. For BERT: the feed-forward size and filter size are synonymous. Both of them denote the number of dimensions in the middle layer of the feed-forward network. the hidden size and embedding size are synonymous. Both of them denote the number of real numbers used to represent a token. The notation for encoder stack is written as L/H. For example, BERTBASE is written as 12L/768H, BERTLARGE as 24L/1024H, and BERTTINY as 2L/128H. == Training == === Pre-training === BERT was pre-trained simultaneously on two tasks: Masked language modeling (MLM): In this task, BERT ingests a sequence of words, where one word may be randomly changed ("masked"), and BERT tries to predict the original words that had been changed. For example, in the sentence "The cat sat on the [MASK]," BERT would need to predict "mat." This helps BERT learn bidirectional context, meaning it understands the relationships between words not just from left to right or right to left but from both directions at the same time. Next sentence prediction (NSP): In this task, BERT is trained to predict whether one sentence logically follows another. For example, given two sentences, "The cat sat on the mat" and "It was a sunny day", BERT has to decide if the second sentence is a valid continuation of the first one. This helps BERT understand relationships between sentences, which is important for tasks like question answering or document classification. ==== Masked language modeling ==== In masked language modeling, 15% of tokens would be randomly selected for masked-prediction task, and the training objective was to predict the masked token given its context. In more detail, the selected token is: replaced with a [MASK] token with probability 80%, replaced with a random word token with probability 10%, not replaced with probability 10%. The reason not all selected tokens are masked is to avoid the dataset shift problem. The dataset shift problem arises when the distribution of inputs seen during training differs significantly from the distribution encountered during inference. A trained BERT model might be applied to word representation (like Word2Vec), where it would be run over sentences not containing any [MASK] tokens. It is later found that more diverse training objectives are generally better. As an illustrative example, consider the sentence "my dog is cute". It would first be divided into tokens like "my1 dog2 is3 cute4". Then a random token in the sentence would be picked. Let it be the 4th one "cute4". Next, there would be three possibilities: with probability 80%, the chosen token is masked, resulting in "my1 dog2 is3 [MASK]4"; with probability 10%, the chosen token is replaced by a uniformly sampled random token, such as "happy", resulting in "my1 dog2 is3 happy4"; with probability 10%, nothing is done, resulting in "my1 dog2 is3 cute4". After processing the input text, the model's 4th output vector is passed to its decoder layer, which outputs a probability distribution over its 30,000-dimensional vocabulary space. ==== Next sentence prediction ==== Given two sentences, the model predicts if they appear sequentially in the training corpus, outputting either [IsNext] or [NotNext]. During training, the algorithm sometimes samples two sentences from a single continuous span in the training corpus, while at other times, it samples two sentences from two discontinuous spans. The first sentence starts with a special token, [CLS] (for "classify"). The two sentences are separated by another special token, [SEP] (for "separate"). After processing the two sentences, the final vector for the [CLS] token is passed to a linear layer for binary classification into [IsNext] and [NotNext]. For example: Given "[CLS] my dog is cute [SEP] he likes playing [SEP]", the model should predict [IsNext]. Given "[CLS] my dog is cute [SEP] how do magnets work [SEP]", the model should predict [NotNext]. === Fine-tuning === BERT is meant as a general pretrained model for various applications in natural language processing. That is, after pre-training, BERT can be fine-tuned with fewer resources on smaller datasets to optimize its performance on specific tasks such as natural language inference and text classification, and sequence-to-sequence-based language generation tasks such as question answering and conversational response generation. The original BERT paper published results demonstrating that a small amount of fine

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  Learning to rank

  Learning to rank (LTR) or machine-learned ranking (MLR) is the application of machine learning, often supervised, semi-supervised or reinforcement learning, in the construction of ranking models for information retrieval and recommender systems. Training data may, for example, consist of lists of items with some partial order specified between items in each list. This order is typically induced by giving a numerical or ordinal score or a binary judgment (e.g. "relevant" or "not relevant") for each item. The goal of constructing the ranking model is to rank new, unseen lists in a similar way to rankings in the training data. == Applications == === In information retrieval === Ranking is a central part of many information retrieval problems, such as document retrieval, collaborative filtering, sentiment analysis, and online advertising. A possible architecture of a machine-learned search engine is shown in the accompanying figure. Training data consists of queries and documents matching them together with the relevance degree of each match. It may be prepared manually by human assessors (or raters, as Google calls them), who check results for some queries and determine relevance of each result. It is not feasible to check the relevance of all documents, and so typically a technique called pooling is used — only the top few documents, retrieved by some existing ranking models are checked. This technique may introduce selection bias. Alternatively, training data may be derived automatically by analyzing clickthrough logs (i.e. search results which got clicks from users), query chains, or such search engines' features as Google's (since-replaced) SearchWiki. Clickthrough logs can be biased by the tendency of users to click on the top search results on the assumption that they are already well-ranked. Training data is used by a learning algorithm to produce a ranking model which computes the relevance of documents for actual queries. Typically, users expect a search query to complete in a short time (such as a few hundred milliseconds for web search), which makes it impossible to evaluate a complex ranking model on each document in the corpus, and so a two-phase scheme is used. First, a small number of potentially relevant documents are identified using simpler retrieval models which permit fast query evaluation, such as the vector space model, Boolean model, weighted AND, or BM25. This phase is called top- k {\displaystyle k} document retrieval and many heuristics were proposed in the literature to accelerate it, such as using a document's static quality score and tiered indexes. In the second phase, a more accurate but computationally expensive machine-learned model is used to re-rank these documents. === In other areas === Learning to rank algorithms have been applied in areas other than information retrieval: In machine translation for ranking a set of hypothesized translations; In computational biology for ranking candidate 3-D structures in protein structure prediction problems; In recommender systems for identifying a ranked list of related news articles to recommend to a user after he or she has read a current news article. == Feature vectors == For the convenience of MLR algorithms, query-document pairs are usually represented by numerical vectors, which are called feature vectors. Such an approach is sometimes called bag of features and is analogous to the bag of words model and vector space model used in information retrieval for representation of documents. Components of such vectors are called features, factors or ranking signals. They may be divided into three groups (features from document retrieval are shown as examples): Query-independent or static features — those features, which depend only on the document, but not on the query. For example, PageRank or document's length. Such features can be precomputed in off-line mode during indexing. They may be used to compute document's static quality score (or static rank), which is often used to speed up search query evaluation. Query-dependent or dynamic features — those features, which depend both on the contents of the document and the query, such as TF-IDF score or other non-machine-learned ranking functions. Query-level features or query features, which depend only on the query. For example, the number of words in a query. Some examples of features, which were used in the well-known LETOR dataset: TF, TF-IDF, BM25, and language modeling scores of document's zones (title, body, anchors text, URL) for a given query; Lengths and IDF sums of document's zones; Document's PageRank, HITS ranks and their variants. Selecting and designing good features is an important area in machine learning, which is called feature engineering. == Evaluation measures == There are several measures (metrics) which are commonly used to judge how well an algorithm is doing on training data and to compare the performance of different MLR algorithms. Often a learning-to-rank problem is reformulated as an optimization problem with respect to one of these metrics. Examples of ranking quality measures: Mean average precision (MAP); DCG and NDCG; Precision@n, NDCG@n, where "@n" denotes that the metrics are evaluated only on top n documents; Mean reciprocal rank; Kendall's tau; Spearman's rho. DCG and its normalized variant NDCG are usually preferred in academic research when multiple levels of relevance are used. Other metrics such as MAP, MRR and precision, are defined only for binary judgments. Recently, there have been proposed several new evaluation metrics which claim to model user's satisfaction with search results better than the DCG metric: Expected reciprocal rank (ERR); Yandex's pfound. Both of these metrics are based on the assumption that the user is more likely to stop looking at search results after examining a more relevant document, than after a less relevant document. == Approaches == Learning to Rank approaches are often categorized using one of three approaches: pointwise (where individual documents are ranked), pairwise (where pairs of documents are ranked into a relative order), and listwise (where an entire list of documents are ordered). Tie-Yan Liu of Microsoft Research Asia has analyzed existing algorithms for learning to rank problems in his book Learning to Rank for Information Retrieval. He categorized them into three groups by their input spaces, output spaces, hypothesis spaces (the core function of the model) and loss functions: the pointwise, pairwise, and listwise approach. In practice, listwise approaches often outperform pairwise approaches and pointwise approaches. This statement was further supported by a large scale experiment on the performance of different learning-to-rank methods on a large collection of benchmark data sets. In this section, without further notice, x {\displaystyle x} denotes an object to be evaluated, for example, a document or an image, f ( x ) {\displaystyle f(x)} denotes a single-value hypothesis, h ( ⋅ ) {\displaystyle h(\cdot )} denotes a bi-variate or multi-variate function and L ( ⋅ ) {\displaystyle L(\cdot )} denotes the loss function. === Pointwise approach === In this case, it is assumed that each query-document pair in the training data has a numerical or ordinal score. Then the learning-to-rank problem can be approximated by a regression problem — given a single query-document pair, predict its score. Formally speaking, the pointwise approach aims at learning a function f ( x ) {\displaystyle f(x)} predicting the real-value or ordinal score of a document x {\displaystyle x} using the loss function L ( f ; x j , y j ) {\displaystyle L(f;x_{j},y_{j})} . A number of existing supervised machine learning algorithms can be readily used for this purpose. Ordinal regression and classification algorithms can also be used in pointwise approach when they are used to predict the score of a single query-document pair, and it takes a small, finite number of values. === Pairwise approach === In this case, the learning-to-rank problem is approximated by a classification problem — learning a binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} that can tell which document is better in a given pair of documents. The classifier shall take two documents as its input and the goal is to minimize a loss function L ( h ; x u , x v , y u , v ) {\displaystyle L(h;x_{u},x_{v},y_{u,v})} . The loss function typically reflects the number and magnitude of inversions in the induced ranking. In many cases, the binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} is implemented with a scoring function f ( x ) {\displaystyle f(x)} . As an example, RankNet adapts a probability model and defines h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} as the estimated probability of the document x u {\displaystyle x_{u}} has higher quality than x v {\displaystyle x_{v}} : P u , v ( f ) = CDF ( f ( x u ) − f ( x v ) ) , {\displaystyle P_{u,v}(f)={\text{CDF}

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  Sample complexity

  The sample complexity of a machine learning algorithm represents the number of training-samples that it needs in order to successfully learn a target function. More precisely, the sample complexity is the number of training-samples that we need to supply to the algorithm, so that the function returned by the algorithm is within an arbitrarily small error of the best possible function, with probability arbitrarily close to 1. There are two variants of sample complexity: The weak variant fixes a particular input-output distribution; The strong variant takes the worst-case sample complexity over all input-output distributions. The No free lunch theorem, discussed below, proves that, in general, the strong sample complexity is infinite, i.e. that there is no algorithm that can learn the globally-optimal target function using a finite number of training samples. However, if we are only interested in a particular class of target functions (e.g., only linear functions) then the sample complexity is finite, and it depends linearly on the VC dimension on the class of target functions. == Definition == Let X {\displaystyle X} be a space which we call the input space, and Y {\displaystyle Y} be a space which we call the output space, and let Z {\displaystyle Z} denote the product X × Y {\displaystyle X\times Y} . For example, in the setting of binary classification, X {\displaystyle X} is typically a finite-dimensional vector space and Y {\displaystyle Y} is the set { − 1 , 1 } {\displaystyle \{-1,1\}} . Fix a hypothesis space H {\displaystyle {\mathcal {H}}} of functions h : X → Y {\displaystyle h\colon X\to Y} . A learning algorithm over H {\displaystyle {\mathcal {H}}} is a computable map from Z {\displaystyle Z} to H {\displaystyle {\mathcal {H}}} . In other words, it is an algorithm that takes as input a finite sequence of training samples and outputs a function from X {\displaystyle X} to Y {\displaystyle Y} . Typical learning algorithms include empirical risk minimization, without or with Tikhonov regularization. Fix a loss function L : Y × Y → R ≥ 0 {\displaystyle {\mathcal {L}}\colon Y\times Y\to \mathbb {R} _{\geq 0}} , for example, the square loss L ( y , y ′ ) = ( y − y ′ ) 2 {\displaystyle {\mathcal {L}}(y,y')=(y-y')^{2}} , where h ( x ) = y ′ {\displaystyle h(x)=y'} . For a given distribution ρ {\displaystyle \rho } on X × Y {\displaystyle X\times Y} , the expected risk of a hypothesis (a function) h ∈ H {\displaystyle h\in {\mathcal {H}}} is E ( h ) := E ρ [ L ( h ( x ) , y ) ] = ∫ X × Y L ( h ( x ) , y ) d ρ ( x , y ) {\displaystyle {\mathcal {E}}(h):=\mathbb {E} _{\rho }[{\mathcal {L}}(h(x),y)]=\int _{X\times Y}{\mathcal {L}}(h(x),y)\,d\rho (x,y)} In our setting, we have h = A ( S n ) {\displaystyle h={\mathcal {A}}(S_{n})} , where A {\displaystyle {\mathcal {A}}} is a learning algorithm and S n = ( ( x 1 , y 1 ) , … , ( x n , y n ) ) ∼ ρ n {\displaystyle S_{n}=((x_{1},y_{1}),\ldots ,(x_{n},y_{n}))\sim \rho ^{n}} is a sequence of vectors which are all drawn independently from ρ {\displaystyle \rho } . Define the optimal risk E H ∗ = inf h ∈ H E ( h ) . {\displaystyle {\mathcal {E}}_{\mathcal {H}}^{}={\underset {h\in {\mathcal {H}}}{\inf }}{\mathcal {E}}(h).} Set h n = A ( S n ) {\displaystyle h_{n}={\mathcal {A}}(S_{n})} , for each sample size n {\displaystyle n} . h n {\displaystyle h_{n}} is a random variable and depends on the random variable S n {\displaystyle S_{n}} , which is drawn from the distribution ρ n {\displaystyle \rho ^{n}} . The algorithm A {\displaystyle {\mathcal {A}}} is called consistent if E ( h n ) {\displaystyle {\mathcal {E}}(h_{n})} probabilistically converges to E H ∗ {\displaystyle {\mathcal {E}}_{\mathcal {H}}^{}} . In other words, for all ϵ , δ > 0 {\displaystyle \epsilon ,\delta >0} , there exists a positive integer N {\displaystyle N} , such that, for all sample sizes n ≥ N {\displaystyle n\geq N} , we have Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] < δ . {\displaystyle \Pr _{\rho ^{n}}[{\mathcal {E}}(h_{n})-{\mathcal {E}}_{\mathcal {H}}^{}\geq \varepsilon ]<\delta .} The sample complexity of A {\displaystyle {\mathcal {A}}} is then the minimum N {\displaystyle N} for which this holds, as a function of ρ , ϵ {\displaystyle \rho ,\epsilon } , and δ {\displaystyle \delta } . We write the sample complexity as N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} to emphasize that this value of N {\displaystyle N} depends on ρ , ϵ {\displaystyle \rho ,\epsilon } , and δ {\displaystyle \delta } . If A {\displaystyle {\mathcal {A}}} is not consistent, then we set N ( ρ , ϵ , δ ) = ∞ {\displaystyle N(\rho ,\epsilon ,\delta )=\infty } . If there exists an algorithm for which N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is finite, then we say that the hypothesis space H {\displaystyle {\mathcal {H}}} is learnable. In others words, the sample complexity N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} defines the rate of consistency of the algorithm: given a desired accuracy ϵ {\displaystyle \epsilon } and confidence δ {\displaystyle \delta } , one needs to sample N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} data points to guarantee that the risk of the output function is within ϵ {\displaystyle \epsilon } of the best possible, with probability at least 1 − δ {\displaystyle 1-\delta } . In probably approximately correct (PAC) learning, one is concerned with whether the sample complexity is polynomial, that is, whether N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is bounded by a polynomial in 1 / ϵ {\displaystyle 1/\epsilon } and 1 / δ {\displaystyle 1/\delta } . If N ( ρ , ϵ , δ ) {\displaystyle N(\rho ,\epsilon ,\delta )} is polynomial for some learning algorithm, then one says that the hypothesis space H {\displaystyle {\mathcal {H}}} is PAC-learnable. This is a stronger notion than being learnable. == Unrestricted hypothesis space: infinite sample complexity == One can ask whether there exists a learning algorithm so that the sample complexity is finite in the strong sense, that is, there is a bound on the number of samples needed so that the algorithm can learn any distribution over the input-output space with a specified target error. More formally, one asks whether there exists a learning algorithm A {\displaystyle {\mathcal {A}}} , such that, for all ϵ , δ > 0 {\displaystyle \epsilon ,\delta >0} , there exists a positive integer N {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} , we have sup ρ ( Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] ) < δ , {\displaystyle \sup _{\rho }\left(\Pr _{\rho ^{n}}[{\mathcal {E}}(h_{n})-{\mathcal {E}}_{\mathcal {H}}^{}\geq \varepsilon ]\right)<\delta ,} where h n = A ( S n ) {\displaystyle h_{n}={\mathcal {A}}(S_{n})} , with S n = ( ( x 1 , y 1 ) , … , ( x n , y n ) ) ∼ ρ n {\displaystyle S_{n}=((x_{1},y_{1}),\ldots ,(x_{n},y_{n}))\sim \rho ^{n}} as above. The No Free Lunch Theorem says that without restrictions on the hypothesis space H {\displaystyle {\mathcal {H}}} , this is not the case, i.e., there always exist "bad" distributions for which the sample complexity is arbitrarily large. Thus, in order to make statements about the rate of convergence of the quantity sup ρ ( Pr ρ n [ E ( h n ) − E H ∗ ≥ ε ] ) , {\displaystyle \sup _{\rho }\left(\Pr _{\rho ^{n}}[{\mathcal {E}}(h_{n})-{\mathcal {E}}_{\mathcal {H}}^{}\geq \varepsilon ]\right),} one must either constrain the space of probability distributions ρ {\displaystyle \rho } , e.g. via a parametric approach, or constrain the space of hypotheses H {\displaystyle {\mathcal {H}}} , as in distribution-free approaches. == Restricted hypothesis space: finite sample-complexity == The latter approach leads to concepts such as VC dimension and Rademacher complexity which control the complexity of the space H {\displaystyle {\mathcal {H}}} . A smaller hypothesis space introduces more bias into the inference process, meaning that E H ∗ {\displaystyle {\mathcal {E}}_{\mathcal {H}}^{}} may be greater than the best possible risk in a larger space. However, by restricting the complexity of the hypothesis space it becomes possible for an algorithm to produce more uniformly consistent functions. This trade-off leads to the concept of regularization. It is a theorem from VC theory that the following three statements are equivalent for a hypothesis space H {\displaystyle {\mathcal {H}}} : H {\displaystyle {\mathcal {H}}} is PAC-learnable. The VC dimension of H {\displaystyle {\mathcal {H}}} is finite. H {\displaystyle {\mathcal {H}}} is a uniform Glivenko-Cantelli class. This gives a way to prove that certain hypothesis spaces are PAC learnable, and by extension, learnable. === An example of a PAC-learnable hypothesis space === X = R d , Y = { − 1 , 1 } {\displaystyle X=\mathbb {R} ^{d},Y=\{-1,1\}} , and let H {\displaystyle {\mathcal {H}}} be the space of affine functions on X {\displaystyle X} , that is, functions of the form x ↦ ⟨ w , x ⟩ + b {\displaystyle x\mapsto \langl

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