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Split screen (computing)
Split screen is a display technique in computer graphics that consists of dividing graphics and/or text into non-overlapping adjacent parts, typically as two or four rectangular areas. This allows for the simultaneous presentation of (usually) related graphical and textual information on a computer display. TV sports adopted this presentation methodology in the 1960s for instant replay. Non-dynamic split screens differ from windowing systems in that the latter allowed overlapping and freely movable parts of the screen (the "windows") to present both related and unrelated application data to the user. In contrast, split-screen views are strictly limited to fixed positions. The split screen technique can also be used to run two instances of an application, potentially allowing another user to interact with the second instance. == In operating systems == Split screen modes are used by mobile operating systems to enable computer multitasking similar to the window interface present in desktop operating systems. Android supports split screen view of two apps natively on all devices, while certain devices, such as Samsung Galaxy Z TriFold, support three sumultaneous views. Split screen functionality is not supported on iOS, but a similar feature called Split View is present in iPadOS, first introduced in 2015 with the first generation of iPad Pro. == In video games == The split screen feature is commonly used in non-networked, also known as couch co-op, video games with multiplayer options. In its most easily understood form, a split screen for a multiplayer video game is an audiovisual output device (usually a standard television for video game consoles) where the display has been divided into 2-4 equally sized areas (depending on number of players) so that the players can explore different areas simultaneously without being close to each other. This has historically been remarkably popular on consoles, which until the 2000s did not have access to the Internet or any other network and is less common today with modern support for networked console-to-console multiplayer. In competitive split-screen games, it is customarily considered cheating to look at another player's screen section to gain an advantage. === History === Split screen gaming dates back to at least the 1970s, with games such Drag Race (1977) from Kee Games in the arcades being presented in this format. It has always been a common feature of two or more player home console and computer games too, with notable titles being Kikstart II for 8-bit systems, a number of 16-bit racing games (such as Lotus Esprit Turbo Challenge and Road Rash II), and action/strategy games (such as Toejam & Earl and Lemmings), all employing a vertical or horizontal screen split for two player games. Xenophobe is notable as a three-way split screen arcade title, although on home platforms it was reduced to one or two screens. The addition of four controller ports on home consoles also ushered in more four-way split screen games, with Mario Kart 64 and Goldeneye 007 on the Nintendo 64 being two well known examples. In arcades, machines tended to move towards having a whole screen for each player, or multiple connected machines, for multiplayer. On home machines, especially in the first and third person shooter genres, multiplayer is now more common over a network or the internet rather than locally with split screen. Starting from the late 2000s, the presence of split screen multiplayer has largely been declining due to the increasing prevalence of online multiplayer, though TechRadar reported a resurgence of split screen due to support from independent studios and increased interest from the players.
List of artificial intelligence projects
The following is a list of current and past, non-classified notable artificial intelligence projects. == Specialized projects == === Brain-inspired === Blue Brain Project, an attempt to create a synthetic brain by reverse-engineering the mammalian brain down to the molecular level. Google Brain, a deep learning project part of Google X attempting to have intelligence similar or equal to human-level. Human Brain Project, ten-year scientific research project, based on exascale supercomputers. === Cognitive architectures === 4CAPS, developed at Carnegie Mellon University under Marcel A. Just ACT-R, developed at Carnegie Mellon University under John R. Anderson. AIXI, Universal Artificial Intelligence developed by Marcus Hutter at IDSIA and ANU. CALO, a DARPA-funded, 25-institution effort to integrate many artificial intelligence approaches (natural language processing, speech recognition, machine vision, probabilistic logic, planning, reasoning, many forms of machine learning) into an AI assistant that learns to help manage your office environment. CHREST, developed under Fernand Gobet at Brunel University and Peter C. Lane at the University of Hertfordshire. CLARION, developed under Ron Sun at Rensselaer Polytechnic Institute and University of Missouri. CoJACK, an ACT-R inspired extension to the JACK multi-agent system that adds a cognitive architecture to the agents for eliciting more realistic (human-like) behaviors in virtual environments. Copycat, by Douglas Hofstadter and Melanie Mitchell at the Indiana University. DUAL, developed at the New Bulgarian University under Boicho Kokinov. FORR developed by Susan L. Epstein at The City University of New York. IDA and LIDA, implementing Global Workspace Theory, developed under Stan Franklin at the University of Memphis. OpenCog Prime, developed using the OpenCog Framework. Procedural Reasoning System (PRS), developed by Michael Georgeff and Amy L. Lansky at SRI International. Psi-Theory developed under Dietrich Dörner at the Otto-Friedrich University in Bamberg, Germany. Soar, developed under Allen Newell and John Laird at Carnegie Mellon University and the University of Michigan. Society of Mind and its successor The Emotion Machine proposed by Marvin Minsky. Subsumption architectures, developed e.g. by Rodney Brooks (though it could be argued whether they are cognitive). === Games === AlphaGo, software developed by Google that plays the Chinese board game Go. Chinook, a computer program that plays English draughts; the first to win the world champion title in the competition against humans. Deep Blue, a chess-playing computer developed by IBM which beat Garry Kasparov in 1997. Halite, an artificial intelligence programming competition created by Two Sigma in 2016. Libratus, a poker AI that beat world-class poker players in 2017, intended to be generalisable to other applications. The Matchbox Educable Noughts and Crosses Engine (sometimes called the Machine Educable Noughts and Crosses Engine or MENACE) was a mechanical computer made from 304 matchboxes designed and built by artificial intelligence researcher Donald Michie in 1961. Quick, Draw!, an online game developed by Google that challenges players to draw a picture of an object or idea and then uses a neural network to guess what the drawing is. The Samuel Checkers-playing Program (1959) was among the world's first successful self-learning programs, and as such a very early demonstration of the fundamental concept of artificial intelligence (AI). Stockfish AI, an open source chess engine currently ranked the highest in many computer chess rankings. TD-Gammon, a program that learned to play world-class backgammon partly by playing against itself (temporal difference learning with neural networks). === Internet activism === Serenata de Amor, project for the analysis of public expenditures and detect discrepancies. === Knowledge and reasoning === Alice (Microsoft), a project from Microsoft Research Lab aimed at improving decision-making in Economics Braina, an intelligent personal assistant application with a voice interface for Windows OS. Cyc, an attempt to assemble an ontology and database of everyday knowledge, enabling human-like reasoning. Eurisko, a language by Douglas Lenat for solving problems which consists of heuristics, including some for how to use and change its heuristics. Google Now, an intelligent personal assistant with a voice interface in Google's Android and Apple Inc.'s iOS, as well as Google Chrome web browser on personal computers. Holmes a new AI created by Wipro. Microsoft Cortana, an intelligent personal assistant with a voice interface in Microsoft's various Windows 10 editions. MindsDB, is an AI automation platform for building AI/ML powered features and applications. Mycin, an early medical expert system. Open Mind Common Sense, a project based at the MIT Media Lab to build a large common sense knowledge base from online contributions. Siri, an intelligent personal assistant and knowledge navigator with a voice-interface in Apple Inc.'s iOS and macOS. SNePS, simultaneously a logic-based, frame-based, and network-based knowledge representation, reasoning, and acting system. Viv (software), a new AI by the creators of Siri. Wolfram Alpha, an online service that answers queries by computing the answer from structured data. === Motion and manipulation === AIBO, the robot pet for the home, grew out of Sony's Computer Science Laboratory (CSL). Cog, a robot developed by MIT to study theories of cognitive science and artificial intelligence, now discontinued. === Music === Melomics, a bioinspired technology for music composition and synthesization of music, where computers develop their own style, rather than mimic musicians. === Natural language processing === AIML, an XML dialect for creating natural language software agents. Apache Lucene, a high-performance, full-featured text search engine library written entirely in Java. Apache OpenNLP, a machine learning based toolkit for the processing of natural language text. It supports the most common NLP tasks, such as tokenization, sentence segmentation, part-of-speech tagging, named entity extraction, chunking and parsing. Artificial Linguistic Internet Computer Entity (A.L.I.C.E.), a natural language processing chatterbot. ChatGPT, a chatbot built on top of OpenAI's GPT-3.5 and GPT-4 family of large language models. Claude, a family of large language models developed by Anthropic and launched in 2023. Claude LLMs achieved high coding scores in several recognized LLM benchmarks. Cleverbot, successor to Jabberwacky, now with 170m lines of conversation, Deep Context, fuzziness and parallel processing. Cleverbot learns from around 2 million user interactions per month. DeepSeek: Chinese chatbot funded by hedge fund High-Flyer. DBRX, 136 billion parameter open sourced large language model developed by Mosaic ML and Databricks. ELIZA, a famous 1966 computer program by Joseph Weizenbaum, which parodied person-centered therapy. FreeHAL, a self-learning conversation simulator (chatterbot) which uses semantic nets to organize its knowledge to imitate a very close human behavior within conversations. Gemini, a family of multimodal large language model developed by Google's DeepMind. Drives the Gemini chatbot, formerly known as Bard. GigaChat, a chatbot by Russian Sberbank. GPT-3, a 2020 language model developed by OpenAI that can produce text difficult to distinguish from that written by a human. Jabberwacky, a chatbot by Rollo Carpenter, aiming to simulate natural human chat. LaMDA, a family of conversational neural language models developed by Google. LLaMA, a 2023 language model family developed by Meta that includes 7, 13, 33 and 65 billion parameter models.[1] Mycroft, a free and open-source intelligent personal assistant that uses a natural language user interface. PARRY, another early chatterbot, written in 1972 by Kenneth Colby, attempting to simulate a paranoid schizophrenic. SHRDLU, an early natural language processing computer program developed by Terry Winograd at MIT from 1968 to 1970. SYSTRAN, a machine translation technology by the company of the same name, used by Yahoo!, AltaVista and Google, among others. === Speech recognition === CMU Sphinx, a group of speech recognition systems developed at Carnegie Mellon University. DeepSpeech, an open-source Speech-To-Text engine based on Baidu's deep speech research paper. Whisper, an open-source speech recognition system developed at OpenAI. === Speech synthesis === 15.ai, a real-time artificial intelligence text-to-speech tool developed by an anonymous researcher from MIT. Amazon Polly, a speech synthesis software by Amazon. Festival Speech Synthesis System, a general multi-lingual speech synthesis system developed at the Centre for Speech Technology Research (CSTR) at the University of Edinburgh. WaveNet, a deep neural network for generating raw audio. === Video === CapCut is a video editor tool, developed
Emotion recognition
Emotion recognition is the process of identifying human emotion. People vary widely in their accuracy at recognizing the emotions of others. Use of technology to help people with emotion recognition is a relatively nascent research area. Generally, the technology works best if it uses multiple modalities in context. To date, the most work has been conducted on automating the recognition of facial expressions from video, spoken expressions from audio, written expressions from text, and physiology as measured by wearables. == Human == Humans show a great deal of variability in their abilities to recognize emotion. A key point to keep in mind when learning about automated emotion recognition is that there are several sources of "ground truth", or truth about what the real emotion is. Suppose we are trying to recognize the emotions of Alex. One source is "what would most people say that Alex is feeling?" In this case, the 'truth' may not correspond to what Alex feels, but may correspond to what most people would say it looks like Alex feels. For example, Alex may actually feel sad, but he puts on a big smile and then most people say he looks happy. If an automated method achieves the same results as a group of observers it may be considered accurate, even if it does not actually measure what Alex truly feels. Another source of 'truth' is to ask Alex what he truly feels. This works if Alex has a good sense of his internal state, and wants to tell you what it is, and is capable of putting it accurately into words or a number. However, some people are alexithymic and do not have a good sense of their internal feelings, or they are not able to communicate them accurately with words and numbers. In general, getting to the truth of what emotion is actually present can take some work, can vary depending on the criteria that are selected, and will usually involve maintaining some level of uncertainty. == Automatic == Decades of scientific research have been conducted developing and evaluating methods for automated emotion recognition. There is now an extensive literature proposing and evaluating hundreds of different kinds of methods, leveraging techniques from multiple areas, such as signal processing, machine learning, computer vision, and speech processing. Different methodologies and techniques may be employed to interpret emotion such as Bayesian networks. , Gaussian Mixture models and Hidden Markov Models and deep neural networks. === Approaches === The accuracy of emotion recognition is usually improved when it combines the analysis of human expressions from multimodal forms such as texts, physiology, audio, or video. Different emotion types are detected through the integration of information from facial expressions, body movement and gestures, and speech. The technology is said to contribute in the emergence of the so-called emotional or emotive Internet. The existing approaches in emotion recognition to classify certain emotion types can be generally classified into three main categories: knowledge-based techniques, statistical methods, and hybrid approaches. ==== Knowledge-based techniques ==== Knowledge-based techniques (sometimes referred to as lexicon-based techniques), utilize domain knowledge and the semantic and syntactic characteristics of text and potentially spoken language in order to detect certain emotion types. In this approach, it is common to use knowledge-based resources during the emotion classification process such as WordNet, SenticNet, ConceptNet, and EmotiNet, to name a few. One of the advantages of this approach is the accessibility and economy brought about by the large availability of such knowledge-based resources. A limitation of this technique on the other hand, is its inability to handle concept nuances and complex linguistic rules. Knowledge-based techniques can be mainly classified into two categories: dictionary-based and corpus-based approaches. Dictionary-based approaches find opinion or emotion seed words in a dictionary and search for their synonyms and antonyms to expand the initial list of opinions or emotions. Corpus-based approaches on the other hand, start with a seed list of opinion or emotion words, and expand the database by finding other words with context-specific characteristics in a large corpus. While corpus-based approaches take into account context, their performance still vary in different domains since a word in one domain can have a different orientation in another domain. ==== Statistical methods ==== Statistical methods commonly involve the use of different supervised machine learning algorithms in which a large set of annotated data is fed into the algorithms for the system to learn and predict the appropriate emotion types. Machine learning algorithms generally provide more reasonable classification accuracy compared to other approaches, but one of the challenges in achieving good results in the classification process, is the need to have a sufficiently large training set. Some of the most commonly used machine learning algorithms include Support Vector Machines (SVM), Naive Bayes, and Maximum Entropy. Deep learning, which is under the unsupervised family of machine learning, is also widely employed in emotion recognition. Well-known deep learning algorithms include different architectures of Artificial Neural Network (ANN) such as Convolutional Neural Network (CNN), Long Short-term Memory (LSTM), and Extreme Learning Machine (ELM). The popularity of deep learning approaches in the domain of emotion recognition may be mainly attributed to its success in related applications such as in computer vision, speech recognition, and Natural Language Processing (NLP). ==== Hybrid approaches ==== Hybrid approaches in emotion recognition are essentially a combination of knowledge-based techniques and statistical methods, which exploit complementary characteristics from both techniques. Some of the works that have applied an ensemble of knowledge-driven linguistic elements and statistical methods include sentic computing and iFeel, both of which have adopted the concept-level knowledge-based resource SenticNet. The role of such knowledge-based resources in the implementation of hybrid approaches is highly important in the emotion classification process. Since hybrid techniques gain from the benefits offered by both knowledge-based and statistical approaches, they tend to have better classification performance as opposed to employing knowledge-based or statistical methods independently. A downside of using hybrid techniques however, is the computational complexity during the classification process. === Datasets === Data is an integral part of the existing approaches in emotion recognition and in most cases it is a challenge to obtain annotated data that is necessary to train machine learning algorithms. For the task of classifying different emotion types from multimodal sources in the form of texts, audio, videos or physiological signals, the following datasets are available: HUMAINE: provides natural clips with emotion words and context labels in multiple modalities Belfast database: provides clips with a wide range of emotions from TV programs and interview recordings SEMAINE: provides audiovisual recordings between a person and a virtual agent and contains emotion annotations such as angry, happy, fear, disgust, sadness, contempt, and amusement IEMOCAP: provides recordings of dyadic sessions between actors and contains emotion annotations such as happiness, anger, sadness, frustration, and neutral state eNTERFACE: provides audiovisual recordings of subjects from seven nationalities and contains emotion annotations such as happiness, anger, sadness, surprise, disgust, and fear DEAP: provides electroencephalography (EEG), electrocardiography (ECG), and face video recordings, as well as emotion annotations in terms of valence, arousal, and dominance of people watching film clips DREAMER: provides electroencephalography (EEG) and electrocardiography (ECG) recordings, as well as emotion annotations in terms of valence, dominance of people watching film clips MELD: is a multiparty conversational dataset where each utterance is labeled with emotion and sentiment. MELD provides conversations in video format and hence suitable for multimodal emotion recognition and sentiment analysis. MELD is useful for multimodal sentiment analysis and emotion recognition, dialogue systems and emotion recognition in conversations. MuSe: provides audiovisual recordings of natural interactions between a person and an object. It has discrete and continuous emotion annotations in terms of valence, arousal and trustworthiness as well as speech topics useful for multimodal sentiment analysis and emotion recognition. UIT-VSMEC: is a standard Vietnamese Social Media Emotion Corpus (UIT-VSMEC) with about 6,927 human-annotated sentences with six emotion labels, contributing to emotion recognition research in Vietnamese
Concordance (publishing)
A concordance is an alphabetical list of the principal words used in a book or body of work, listing every instance of each word with its immediate context. Historically, concordances have been compiled only for works of special importance, such as the Vedas, Bible, Qur'an or the works of Shakespeare, James Joyce or classical Latin and Greek authors, because of the time, difficulty, and expense involved in creating a concordance in the pre-computer era. A concordance is more than an index, with additional material such as commentary, definitions and topical cross-indexing which makes producing one a labor-intensive process even when assisted by computers. In the precomputing era, search technology was unavailable, and a concordance offered readers of long works such as the Bible something comparable to search results for every word that they would have been likely to search for. Today, the ability to combine the result of queries concerning multiple terms (such as searching for words near other words) has reduced interest in concordance publishing. In addition, mathematical techniques such as latent semantic indexing have been proposed as a means of automatically identifying linguistic information based on word context. A bilingual concordance is a concordance based on aligned parallel text. A topical concordance is a list of subjects that a book covers (usually The Bible), with the immediate context of the coverage of those subjects. Unlike a traditional concordance, the indexed word does not have to appear in the verse. The best-known topical concordance is Nave's Topical Bible. The first Bible concordance was compiled for the Vulgate Bible by Hugh of St Cher (d.1262), who employed 500 friars to assist him. In 1448, Rabbi Mordecai Nathan completed a concordance to the Hebrew Bible. It took him ten years. A concordance to the Greek New Testament was published in 1546 by Sixt Birck, and the Septuagint was done a by Conrad Kircher in 1602. The first concordance to the English Bible was published in 1550 by John Merbecke. According to Cruden, it did not employ the verse numbers devised by Robert Stephens in 1545, but "the pretty large concordance" of Mr Cotton did. Then followed Cruden's Concordance and Strong's Concordance. == Use in linguistics == Concordances are frequently used in linguistics, when studying a text. For example: comparing different usages of the same word analysing keywords analysing word frequencies finding and analysing phrases and idioms finding translations of subsentential elements, e.g. terminology, in bitexts and translation memories creating indexes and word lists (also useful for publishing) Concordancing techniques are widely used in national text corpora such as American National Corpus (ANC), British National Corpus (BNC), and Corpus of Contemporary American English (COCA) available on-line. Stand-alone applications that employ concordancing techniques are known as concordancers or more advanced corpus managers. Some of them have integrated part-of-speech taggers (POS taggers) and enable the user to create their own POS-annotated corpora to conduct various types of searches adopted in corpus linguistics. == Inversion == The reconstruction of the text of some of the Dead Sea Scrolls involved a concordance. Access to some of the scrolls was governed by a "secrecy rule" that allowed only the original International Team or their designates to view the original materials. After the death of Roland de Vaux in 1971, his successors repeatedly refused to even allow the publication of photographs to other scholars. This restriction was circumvented by Martin Abegg in 1991, who used a computer to "invert" a concordance of the missing documents made in the 1950s which had come into the hands of scholars outside of the International Team, to obtain an approximate reconstruction of the original text of 17 of the documents. This was soon followed by the release of the original text of the scrolls.
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
Sieve of Pritchard
In mathematics, the sieve of Pritchard is an algorithm for finding all prime numbers up to a specified bound. Like the ancient sieve of Eratosthenes, it has a simple conceptual basis in number theory. It is especially suited to quick hand computation for small bounds. Whereas the sieve of Eratosthenes marks off each non-prime for each of its prime factors, the sieve of Pritchard avoids considering almost all non-prime numbers by building progressively larger wheels, which represent the pattern of numbers not divisible by any of the primes processed thus far. It thereby achieves a better asymptotic complexity, and was the first sieve with a running time sublinear in the specified bound. Its asymptotic running-time has not been improved on, and it deletes fewer composites than any other known sieve. It was created in 1979 by Paul Pritchard. Since Pritchard has created a number of other sieve algorithms for finding prime numbers, the sieve of Pritchard is sometimes singled out by being called the wheel sieve (by Pritchard himself) or the dynamic wheel sieve. == Overview == A prime number is a natural number that has no natural number divisors other than the number 1 and itself. To find all the prime numbers less than or equal to a given integer N, a sieve algorithm examines a set of candidates in the range 2, 3, …, N, and eliminates those that are not prime, leaving the primes at the end. The sieve of Eratosthenes examines all of the range, first removing all multiples of the first prime 2, then of the next prime 3, and so on. The sieve of Pritchard instead examines a subset of the range consisting of numbers that occur on successive wheels, which represent the pattern of numbers left after each successive prime is processed by the sieve of Eratosthenes. For i > 0, the ith wheel Wi represents this pattern. It is the set of numbers between 1 and the product Pi = p1 · p2 ⋯ pi of the first i prime numbers that are not divisible by any of these prime numbers (and is said to have an associated length Pi). This is because adding Pi to a number does not change whether it is divisible by one of the first i prime numbers, since the remainder on division by any one of these primes is unchanged. So W1 = {1} with length P1 = 2 represents the pattern of odd numbers; W2 = {1,5} with length P2 = 6 represents the pattern of numbers not divisible by 2 or 3; etc. Wheels are so-called because Wi can be usefully visualized as a circle of circumference Pi with its members marked at their corresponding distances from an origin. Then rolling the wheel along the number line marks points corresponding to successive numbers not divisible by one of the first i prime numbers. The animation shows W2 being rolled up to 30. It is useful to define Wi → n for n > 0 to be the result of rolling Wi up to n. Then the animation generates W2 → 30 = {1,5,7,11,13,17,19,23,25,29}. Note that up to 52 − 1 = 24, this consists only of 1 and the primes between 5 and 25. The sieve of Pritchard is derived from the observation that this holds generally: for all i > 0, the values in Wi → (p2i+1 − 1) are 1 and the primes between pi+1 and p2i+1. It even holds for i = 0, where the wheel has length 1 and contains just 1 (representing all the natural numbers). So the sieve of Pritchard starts with the trivial wheel W0 and builds successive wheels until the square of the wheel's first member after 1 is at least N. Wheels grow very quickly, but only their values up to N are needed and generated. It remains to find a method for generating the next wheel. Note in the animation that W3 = {1,5,7,11,13,17,19,23,25,29} − {5 · 1 , 5 · 5} can be obtained by rolling W2 up to 30 and then removing 5 times each member of W2.This also holds generally: for all i ≥ 0, Wi+1 = (Wi → Pi+1) − {pi+1 · w | w ∈ Wi}. Rolling Wi past Pi just adds values to Wi, so the current wheel is first extended by getting each successive member starting with w = 1, adding Pi to it, and inserting the result in the set. Then the multiples of pi+1 are deleted. Care must be taken to avoid a number being deleted that itself needs to be multiplied by pi+1. The sieve of Pritchard as originally presented does so by first skipping past successive members until finding the maximum one needed, and then doing the deletions in reverse order by working back through the set. This is the method used in the first animation above. A simpler approach is just to gather the multiples of pi+1 in a list, and then delete them. Another approach is given by Gries and Misra. If the main loop terminates with a wheel whose length is less than N, it is extended up to N to generate the remaining primes. The algorithm, for finding all primes up to N, is therefore as follows: Start with a set W = {1} and length = 1 representing wheel 0, and prime p = 2. As long as p2 ≤ N, do the following: if length < N, then extend W by repeatedly getting successive members w of W starting with 1 and inserting length + w into W as long as it does not exceed p · length or N; increase length to the minimum of p · length and N. repeatedly delete p times each member of W by first finding the largest ≤ length and then working backwards. note the prime p, then set p to the next member of W after 1 (or 3 if p was 2). if length < N, then extend W to N by repeatedly getting successive members w of W starting with 1 and inserting length + w into W as long as it does not exceed N; On termination, the rest of the primes up to N are the members of W after 1. === Example === To find all the prime numbers less than or equal to 150, proceed as follows. Start with wheel 0 with length 1, representing all natural numbers 1, 2, 3...: 1 The first number after 1 for wheel 0 (when rolled) is 2; note it as a prime. Now form wheel 1 with length 2 × 1 = 2 by first extending wheel 0 up to 2 and then deleting 2 times each number in wheel 0, to get: 1 2 The first number after 1 for wheel 1 (when rolled) is 3; note it as a prime. Now form wheel 2 with length 3 × 2 = 6 by first extending wheel 1 up to 6 and then deleting 3 times each number in wheel 1, to get 1 2 3 5 The first number after 1 for wheel 2 is 5; note it as a prime. Now form wheel 3 with length 5 × 6 = 30 by first extending wheel 2 up to 30 and then deleting 5 times each number in wheel 2 (in reverse order), to get 1 2 3 5 7 11 13 17 19 23 25 29 The first number after 1 for wheel 3 is 7; note it as a prime. Now wheel 4 has length 7 × 30 = 210, so we only extend wheel 3 up to our limit 150. (No further extending will be done now that the limit has been reached.) We then delete 7 times each number in wheel 3 until we exceed our limit 150, to get the elements in wheel 4 up to 150: 1 2 3 5 7 11 13 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97 101 103 107 109 113 119 121 127 131 133 137 139 143 149 The first number after 1 for this partial wheel 4 is 11; note it as a prime. Since we have finished with rolling, we delete 11 times each number in the partial wheel 4 until we exceed our limit 150, to get the elements in wheel 5 up to 150: 1 2 3 5 7 11 13 17 19 23 25 29 31 37 41 43 47 49 53 59 61 67 71 73 77 79 83 89 91 97 101 103 107 109 113 119 121 127 131 133 137 139 143 149 The first number after 1 for this partial wheel 5 is 13. Since 13 squared is at least our limit 150, we stop. The remaining numbers (other than 1) are the rest of the primes up to our limit 150. Just 8 composite numbers are removed, once each. The rest of the numbers considered (other than 1) are prime. In comparison, the natural version of Eratosthenes sieve (stopping at the same point) removes composite numbers 184 times. == Pseudocode == The sieve of Pritchard can be expressed in pseudocode, as follows: algorithm Sieve of Pritchard is input: an integer N >= 2. output: the set of prime numbers in {1,2,...,N}. let W and Pr be sets of integer values, and all other variables integer values. k, W, length, p, Pr := 1, {1}, 2, 3, {2}; {invariant: p = pk+1 and W = Wk ∩ {\displaystyle \cap } {1,2,...,N} and length = minimum of Pk,N and Pr = the primes up to pk} while p2 <= N do if (length < N) then Extend W,length to minimum of plength,N; Delete multiples of p from W; Insert p into Pr; k, p := k+1, next(W, 1) if (length < N) then Extend W,length to N; return Pr ∪ {\displaystyle \cup } W - {1}; where next(W, w) is the next value in the ordered set W after w. procedure Extend W,length to n is {in: W = Wk and length = Pk and n > length} {out: W = Wk → {\displaystyle \rightarrow } n and length = n} integer w, x; w, x := 1, length+1; while x <= n do Insert x into W; w := next(W,w); x := length + w; length := n; procedure Delete multiples of p from W,length is integer w; w := p; while pw <= length do w := next(W,w); while w > 1 do w := prev(W,w); Remove pw from W; where prev(W, w) is the previous value in the ordered set W before w. The algorithm can be initialized with W0 instead of W1 at the minor complication of making next(W, 1) a special case when k = 0. This a