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  Machine unlearning

  Machine unlearning is a branch of machine learning focused on removing specific undesired element, such as private data, wrong or manipulated training data, outdated information, copyrighted material, harmful content, dangerous abilities, or misinformation, without needing to rebuild models from the ground up. Large language models, like the ones powering ChatGPT, may be asked not just to remove specific elements but also to unlearn a "concept," "fact," or "knowledge," which aren't easily linked to specific examples. New terms such as "model editing," "concept editing," and "knowledge unlearning" have emerged to describe this process. == History == Early research efforts were largely motivated by Article 17 of the GDPR, the European Union's privacy regulation commonly known as the "right to be forgotten" (RTBF), introduced in 2014. The GDPR did not anticipate that the development of large language models would make data erasure a complex task. This issue has since led to research on "machine unlearning," with a growing focus on removing copyrighted material, harmful content, dangerous capabilities, and misinformation. Just as early experiences in humans shape later ones, some concepts are more fundamental and harder to unlearn. A piece of knowledge may be so deeply embedded in the model's knowledge graph that unlearning it could cause internal contradictions, requiring adjustments to other parts of the graph to resolve them. Researchers have now also started studying unlearning in the context of removing incorrect or adversarially manipulated training data such as systematically biased labels or poisoning attacks. == Motivations == At present, machine unlearning is motivated by a growing range of concerns that extend well beyond the field's original focus on data privacy. A widely used taxonomy in the literature distinguishes two high-level categories of motivation. Access revocation covers cases where a data subject or rights holder requests the removal of data they own or control. This is most commonly associated with RTBF established by the European Union's General Data Protection Regulation (GDPR) and analogous legislation such as the California Consumer Privacy Act (CCPA). These regulations grant individuals the legal right to request erasure of their personal data from any system that has processed it, including models that were trained on it. Access revocation also encompasses the removal of copyrighted or pay-walled content that was incorporated into training corpora without the necessary licenses, a concern that has become prominent with the widespread use of largely web-scraped pre-training datasets. Model correction covers cases where the model exhibits undesirable behavior arising from the training data, regardless of any individual's request. This includes: Removal of toxic, biased, or unsafe outputs introduced by harmful content in the training set Correction of stale or factually incorrect associations, such as outdated knowledge encoded in a deployed model Removal of dangerous capabilities, such as detailed knowledge of the synthesis of chemical or biological agents Correction of the influence of data poisoning or adversarial attacks that have corrupted model behavior This second category has been formalized as corrective machine unlearning, which frames unlearning as a post-training mechanism for repairing the effects of bad or harmful training data. It is closely related to the AI safety literature, where data filtering alone has been found insufficient to prevent hazardous knowledge from being encoded in model weights, motivating unlearning as a complementary risk mitigation strategy. A further distinction has been drawn in the literature between removal {eliminating the influence of specific training data on model parameters) and suppression (preventing the model from generating specific outputs regardless of how that knowledge is encoded). These two goals are not equivalent: removing training data does not guarantee meaningful output suppression, and suppressing outputs does not constitute removal of the underlying training data's influence. == SISA Training == SISA is a training strategy consisting of four mechanisms designed to make machine unlearning more efficient by structuring how models are trained and updated. Its goal is to allow a system to remove the influence of specific data points without retraining an entire model from scratch. By reorganizing training data and workflows, SISA reduces the computational burden of unlearning requests. Sharding divides the training dataset into multiple disjoint subsets, or shards. Each shard is used to train a separate model instance. This ensures that a single data point affects only one shard, so unlearning it requires updating only the corresponding shard rather than the full model. Isolation refers to training each shard independently, with nothing shared across shards during the training process. This separation prevents cross-contamination between shards, ensuring that forgetting data in one shard does not require adjustments to any others. Slicing breaks the data within each shard into sequential slices and stores model states after each slice is trained on. When an unlearning request targets a piece of data, the system can roll back to the checkpoint before the point was seen and retrain only from that slice forward. This reduces retraining time even within a shard. Aggregation occurs at inference, when the model is queried. It combines the outputs of each shard to determine the output of the overall model. This is often through majority voting or averaging. This allows SISA-trained systems to behave like a single model despite being composed of multiple shard-level models. Together, these mechanisms enable machine learning systems to forget specific data points with far lower computational cost than full retraining. The trade-off is that sharding and slicing can lead to reduced model accuracy, worse generalization, and increased storage requirements for the intermediate checkpoints. This can be tolerable based on the needs of the individual or organization to comply with "right to be forgotten" or efficiently recover from backdoor attacks. == Algorithms == Machine unlearning algorithms are broadly categorized into exact and approximate methods, reflecting a fundamental trade-off between formal guarantees and computational tractability. === Exact Unlearning === Exact unlearning methods produce a model that is statistically indistinguishable from one retrained from scratch on the dataset with the forget data removed. The canonical framework for exact unlearning is SISA Training (Sharded, Isolated, Sliced, and Aggregated), introduced by Bourtoule et al. (2021). SISA partitions the training dataset into disjoint shards and trains a separate sub-model on each. At inference time, predictions are aggregated across sub-models. When an unlearning request is received, only the sub-model corresponding to the shard containing the target data requires retraining, reducing computational overhead proportionally to the number of shards. Exact methods provide the strongest guarantees but become prohibitively expensive for large pre-trained neural networks and are generally limited to settings where training can be structured in advance. === Approximate Unlearning === Approximate unlearning methods seek to produce a model whose behavior is sufficiently close to an exactly unlearned model without the cost of full retraining. These methods dominate practical applications. Common approaches include: Gradient Ascent: The model is fine-tuned by maximizing the loss on the forget set, directly degrading its performance on targeted data. This is the most direct approach but risks destabilizing performance on retained data. Random Labelling: The model is fine-tuned on the forget set using randomly shuffled labels, confusing its associations with the targeted data while producing a less aggressive weight shift than pure gradient ascent. Gradient Difference: Combines gradient ascent on the forget set with simultaneous gradient descent on the retain set, using the retain objective as a regularizer to preserve general model utility. KL Divergence Regularization: Minimizes the KL divergence between the outputs of the unlearned model and the original model on the retain set, anchoring behavior on data the model should remember. Weight Pruning and Fine-tuning: Parameters with the smallest L1-norm are pruned — targeting weights most weakly associated with general knowledge and potentially most associated with the forget set — followed by fine-tuning on the retain set to restore utility. Layer Reset and Fine-tuning: The first or last k layers are re-initialized to random weights and the model is subsequently fine-tuned on the retain set. This is a coarse but computationally simple approach. Selective Synaptic Dampening: Uses influence functions to estimate the effect of individual trainin

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  SmarterChild

  SmarterChild was a chatbot available on AOL Instant Messenger and Windows Live Messenger (previously MSN Messenger) networks. == History == SmarterChild was an apparently intelligent agent or "bot" developed by ActiveBuddy, Inc., with offices in New York and Sunnyvale. It was widely distributed across global instant messaging networks. SmarterChild became very popular, attracting over 30 million Instant Messenger "buddies" on AIM (AOL), MSN and Yahoo Messenger over the course of its lifetime. Founded in 2000, ActiveBuddy was the brainchild of Robert Hoffer and Timothy Kay, who later brought seasoned advertising executive Peter Levitan on board as CEO. The concept for conversational instant messaging bots came from the founder's vision to add natural language comprehension functionality to the increasingly popular AIM instant messaging application. The original implementation took shape as a demo that Kay programmed in Perl in his Los Altos garage to connect a single buddy name, "ActiveBuddy", to look up stock symbols, and later allow AIM users to play Colossal Cave Adventure, a word-based adventure game, and MIT's Boris Katz Start Question Answering System but quickly grew to include a wide range of database applications the company called 'knowledge domains' including instant access to news, weather, stock information, movie times, yellow pages listings, and detailed sports data, as well as a variety of tools (personal assistant, calculators, translator, etc.). None of the individual domains which the company had named “stocksBuddy”, “sportsBuddy”, etc. ever launched publicly. When Stephen Klein came on board as COO — and eventually CEO — he insisted that all of the disparate test “buddies” be launched together with the company’s highly-developed colloquial chat domain. He suggested using “SmarterChild”, a username coined by Tim Kay which Tim was using to test various things. The bundled domains were launched publicly as SmarterChild (on AIM initially) in June 2001. SmarterChild provided information wrapped in fun and quirky conversation. The company generated no revenue from SmarterChild, but used it as a demonstration of the power of what Klein called “conversational computing”. The company subsequently marketed Automated Service Agents—delivering immediate answers to customer service inquiries—-to large corporations, like Comcast, Cingular, TimeWarner Cable, etc. SmarterChild's popularity spawned targeted marketing-oriented bots for Radiohead, Austin Powers, Intel, Keebler, The Sporting News and others. ActiveBuddy co-founders, Kay and Hoffer, as co-inventors, were issued two controversial U.S. patents in 2002. ActiveBuddy changed its name to Colloquis (briefly Conversagent) and targeted development of consumer-facing enterprise customer service agents, which the company marketed as Automated Service Agents. Microsoft acquired Colloquis in October 2006 and proceeded to de-commission SmarterChild and kill off the Automated Service Agent business as well. Robert Hoffer, ActiveBuddy co-founder, licensed the technology from Microsoft after Microsoft abandoned the Colloquis technology.

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  Contextual image classification

  Contextual image classification, a topic of pattern recognition in computer vision, is an approach of classification based on contextual information in images. "Contextual" means this approach is focusing on the relationship of the nearby pixels, which is also called neighbourhood. The goal of this approach is to classify the images by using the contextual information. == Introduction == Similar as processing language, a single word may have multiple meanings unless the context is provided, and the patterns within the sentences are the only informative segments we care about. For images, the principle is same. Find out the patterns and associate proper meanings to them. As the image illustrated below, if only a small portion of the image is shown, it is very difficult to tell what the image is about. Even try another portion of the image, it is still difficult to classify the image. However, if we increase the contextual of the image, then it makes more sense to recognize. As the full images shows below, almost everyone can classify it easily. During the procedure of segmentation, the methods which do not use the contextual information are sensitive to noise and variations, thus the result of segmentation will contain a great deal of misclassified regions, and often these regions are small (e.g., one pixel). Compared to other techniques, this approach is robust to noise and substantial variations for it takes the continuity of the segments into account. Several methods of this approach will be described below. == Applications == === Functioning as a post-processing filter to a labelled image === This approach is very effective against small regions caused by noise. And these small regions are usually formed by few pixels or one pixel. The most probable label is assigned to these regions. However, there is a drawback of this method. The small regions also can be formed by correct regions rather than noise, and in this case the method is actually making the classification worse. This approach is widely used in remote sensing applications. === Improving the post-processing classification === This is a two-stage classification process: For each pixel, label the pixel and form a new feature vector for it. Use the new feature vector and combine the contextual information to assign the final label to the === Merging the pixels in earlier stages === Instead of using single pixels, the neighbour pixels can be merged into homogeneous regions benefiting from contextual information. And provide these regions to classifier. === Acquiring pixel feature from neighbourhood === The original spectral data can be enriched by adding the contextual information carried by the neighbour pixels, or even replaced in some occasions. This kind of pre-processing methods are widely used in textured image recognition. The typical approaches include mean values, variances, texture description, etc. === Combining spectral and spatial information === The classifier uses the grey level and pixel neighbourhood (contextual information) to assign labels to pixels. In such case the information is a combination of spectral and spatial information. === Powered by the Bayes minimum error classifier === Contextual classification of image data is based on the Bayes minimum error classifier (also known as a naive Bayes classifier). Present the pixel: A pixel is denoted as x 0 {\displaystyle x_{0}} . The neighbourhood of each pixel x 0 {\displaystyle x_{0}} is a vector and denoted as N ( x 0 ) {\displaystyle N(x_{0})} . The values in the neighbourhood vector is denoted as f ( x i ) {\displaystyle f(x_{i})} . Each pixel is presented by the vector ξ = ( f ( x 0 ) , f ( x 1 ) , … , f ( x k ) ) {\displaystyle \xi =\left(f(x_{0}),f(x_{1}),\ldots ,f(x_{k})\right)} x i ∈ N ( x 0 ) ; i = 1 , … , k {\displaystyle x_{i}\in N(x_{0});\quad i=1,\ldots ,k} The labels (classification) of pixels in the neighbourhood N ( x 0 ) {\displaystyle N(x_{0})} are presented as a vector η = ( θ 0 , θ 1 , … , θ k ) {\displaystyle \eta =\left(\theta _{0},\theta _{1},\ldots ,\theta _{k}\right)} θ i ∈ { ω 0 , ω 1 , … , ω k } {\displaystyle \theta _{i}\in \left\{\omega _{0},\omega _{1},\ldots ,\omega _{k}\right\}} ω s {\displaystyle \omega _{s}} here denotes the assigned class. A vector presents the labels in the neighbourhood N ( x 0 ) {\displaystyle N(x_{0})} without the pixel x 0 {\displaystyle x_{0}} η ^ = ( θ 1 , θ 2 , … , θ k ) {\displaystyle {\hat {\eta }}=\left(\theta _{1},\theta _{2},\ldots ,\theta _{k}\right)} The neighbourhood: Size of the neighbourhood. There is no limitation of the size, but it is considered to be relatively small for each pixel x 0 {\displaystyle x_{0}} . A reasonable size of neighbourhood would be 3 × 3 {\displaystyle 3\times 3} of 4-connectivity or 8-connectivity ( x 0 {\displaystyle x_{0}} is marked as red and placed in the centre). The calculation: Apply the minimum error classification on a pixel x 0 {\displaystyle x_{0}} , if the probability of a class ω r {\displaystyle \omega _{r}} being presenting the pixel x 0 {\displaystyle x_{0}} is the highest among all, then assign ω r {\displaystyle \omega _{r}} as its class. θ 0 = ω r if P ( ω r ∣ f ( x 0 ) ) = max s = 1 , 2 , … , R P ( ω s ∣ f ( x 0 ) ) {\displaystyle \theta _{0}=\omega _{r}\quad {\text{ if }}\quad P(\omega _{r}\mid f(x_{0}))=\max _{s=1,2,\ldots ,R}P(\omega _{s}\mid f(x_{0}))} The contextual classification rule is described as below, it uses the feature vector x 1 {\displaystyle x_{1}} rather than x 0 {\displaystyle x_{0}} . θ 0 = ω r if P ( ω r ∣ ξ ) = max s = 1 , 2 , … , R P ( ω s ∣ ξ ) {\displaystyle \theta _{0}=\omega _{r}\quad {\text{ if }}\quad P(\omega _{r}\mid \xi )=\max _{s=1,2,\ldots ,R}P(\omega _{s}\mid \xi )} Use the Bayes formula to calculate the posteriori probability P ( ω s ∣ ξ ) {\displaystyle P(\omega _{s}\mid \xi )} P ( ω s ∣ ξ ) = p ( ξ ∣ ω s ) P ( ω s ) p ( ξ ) {\displaystyle P(\omega _{s}\mid \xi )={\frac {p(\xi \mid \omega _{s})P(\omega _{s})}{p\left(\xi \right)}}} The number of vectors is the same as the number of pixels in the image. For the classifier uses a vector corresponding to each pixel x i {\displaystyle x_{i}} , and the vector is generated from the pixel's neighbourhood. The basic steps of contextual image classification: Calculate the feature vector ξ {\displaystyle \xi } for each pixel. Calculate the parameters of probability distribution p ( ξ ∣ ω s ) {\displaystyle p(\xi \mid \omega _{s})} and P ( ω s ) {\displaystyle P(\omega _{s})} Calculate the posterior probabilities P ( ω r ∣ ξ ) {\displaystyle P(\omega _{r}\mid \xi )} and all labels θ 0 {\displaystyle \theta _{0}} . Get the image classification result. == Algorithms == === Template matching === The template matching is a "brute force" implementation of this approach. The concept is first create a set of templates, and then look for small parts in the image match with a template. This method is computationally high and inefficient. It keeps an entire templates list during the whole process and the number of combinations is extremely high. For a m × n {\displaystyle m\times n} pixel image, there could be a maximum of 2 m × n {\displaystyle 2^{m\times n}} combinations, which leads to high computation. This method is a top down method and often called table look-up or dictionary look-up. === Lower-order Markov chain === The Markov chain also can be applied in pattern recognition. The pixels in an image can be recognised as a set of random variables, then use the lower order Markov chain to find the relationship among the pixels. The image is treated as a virtual line, and the method uses conditional probability. === Hilbert space-filling curves === The Hilbert curve runs in a unique pattern through the whole image, it traverses every pixel without visiting any of them twice and keeps a continuous curve. It is fast and efficient. === Markov meshes === The lower-order Markov chain and Hilbert space-filling curves mentioned above are treating the image as a line structure. The Markov meshes however will take the two dimensional information into account. === Dependency tree === The dependency tree is a method using tree dependency to approximate probability distributions.

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  Gallery software

  Gallery software is software that helps the user publish or share photos, pictures, videos or other digital media. Most galleries are located on Web servers, where users are allowed to register and publish their pictures. Gallery software usually features automatic image resizing, allows digital media be categorized into sets, and allows comments. == Types == Early digital media publishing and sharing was done with imageboards. The boards are by topics, sometimes called "chan". Each discussion in a "chan" are started with a piece of digital media, and follow-up discussions can contain another piece too. Software works in this way: Futallaby, Danbooru. Traditionally, galleries are managed. An administrator maintains a set of or hierarchy of albums. The users can upload their digital media in one of the existing albums defined by an administrator, or create their own albums. The users with sufficient permission can re-categorise the digital media others uploaded. Often, the site's administrator can define which album the users are allowed to categorise their media into, or delete other user's content. Examples are open source galleries Coppermine, Gallery Project. There are decentralised gallery software that does not have an administrator for managing contents. Pinterest, Flickr and DeviantArt has been successful with this model. Open source gallery software MediaGoblin works in this way. Each user can create their own "collections", to categorise theirs or other users' media. However users cannot put media into other user's collections. Each user's category is separate. There is no centralised theme or hierarchy for the media.

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  Agent verification

  Agent verification is activity to gain assurances that purposeful artificial constructs act in accordance with their specifications. While primitive forms of inorganic agents have been used in manufacturing for centuries, the study of artificial agents did not begin until the mid 20th century. Foundational work on such agents was closely bound with the emergence of artificial intelligence as an academic discipline. Early agents deployed for industrial control systems and in computing were often controlled by quite simple logic however, not involving artificial intelligence as such. When deployed as part of a multi-agent system, even such simple agents could require special agent orientated testing methods, as their collective behaviour was challenging to verify with traditional testing techniques. Difficulties in providing assurances that agents will not behave in dangerous ways became more prevalent after the introduction of LLM agents, especially after the rapid acceleration of their deployment in 2025. The verification of agent behaviour can be conducted by formal or informal methods. Informal verification requires less mathematical skill. But when agents are part of systems where errors have significant risks — such as danger to human life, environmental damage or major financial loss — formal verification is preferred. Both regulators and system designers themselves like formal verification as it provides a high degree of mathematical certainty. It is not however always possible to formally test all aspects of an agent based system's behaviour, especially where newer LLM based agents are concerned, due in part to their high degree of autonomy. Accordingly, agent verification for low impact deployments might be carried out only with informal methods, while for high impact deployments, it may be performed with a mix of formal and informal techniques. == Terminology == In academia, the term agent verification is often defined to mean activity concerned with gaining assurance that the agent behaves in accordance with its specification - whether by processes such as testing or simulation. 'Verification' is typically contrasted with 'validation', the latter meaning activity concerned with checking that the specification itself meets user or real world needs. Such definitions are not universally adhered to however - for example, in some workplaces and documents, the words 'verification' and 'validation' can be used synonymously. Efforts to gain confidence in Agents have intensified sharply since 2025 due to the rapid roll out of LLM agents; different terms are sometimes used in the commercial sector. Here the term 'agent verification' can be used in the same sense as it is in academia, but sometimes the same activity can be covered by more ambiguous and wider ranging terms such as 'Agent governance' , 'Agent observability' or 'AI agent policing'. == History == === Classical agents === The theoretical underpinnings for artificial (inorganic) agents emerged in the mid 20th century, with establishment of cybernetics and artificial intelligence. Oliver Selfridge's 1958 Pandemonium - A Paradigm for Learning paper was an important early theoretical contribution in establishing agent oriented architecture. Practical implementations of agents for real world applications began to become widespread in the 1990s, after the introduction of the belief–desire–intention software model (BDI), and agent-oriented programming. Pure digital agents were deployed in computer infrastructure for purposes such as monitoring, while agents connected to real-world sensors and actuators were increasingly used in industrial control systems. While the concept of artificial agents was interwoven with early artificial intelligence studies right from the start, early agents lacked general purpose reasoning capabilities, often only having simple if then logic. Even a device as simple as a thermostat, which has a sensor and a means of acting, can be considered a proto agent in this sense. Verifying the behaviours of a simple single agent system is not generally especially difficult, but it can be a different matter when several simple agents coexist in the same system. Craig Reynolds's work on boids showed that relatively complex, "intelligent" behaviour can emerge from a number of such simple agents working together in a Multi-agent system (MAS). By the 1990s, even the behaviour of a single agent system could sometimes be quite complex; in accordance with the Belief–desire–intention software model, agents could have believes that might evolve over time. Agents were increasingly introduced that were controlled by quite large decision tree models, which had new vulnerabilities to adversarial attack. It was becoming increasingly apparent that traditional software verification methods had limitations for testing such agents, or even for the more primitive type of agents when they were deployed as part of a MAS. It was the use of agents for industrial control systems, sometimes associated with robotics, that lent urgency to the practice of agent verification. Informal testing might be acceptable for digital agents used say to monitor whether each of an organisation's computers are properly licensed. But with an increasing potential for faulty agents to result in a failure that might cause a large fire to break out at a chemical manufacturing plant, a botched medical operation, or even a crashed aircraft, the need to develop reliable means of verifying behaviour of such agents was considered urgent. The Foundation for Intelligent Physical Agents was established in 1996. From the late 90s, a growing number of industry and university based scientists began working on the problem, with researchers publishing papers on the verification of both single and multi agent systems. Much of this work showed how formal verification techniques like model checking could be used to gain a high level of assurance that agent based systems would conform with their specification. A 2018 systematic review covering 231 studies found that model checking was the most common technique for agent verification, with theorem proving the second most commonly used formal verification method. In the first two decades of the 20th century, agents run by AI became more common, with Siri and Alexa being well known examples. But such agents still lacked general reasoning capabilities and did not pose new pressing problems for agent verification. === General purpose reasoning agents === The advent of LLMs created huge potential for further use of artificial agents, as agents based on them could have general purpose cognitive abilities. Agents run by LLMs (and occasionally non-LLM foundation models) have similar vulnerability to adversarial attack as those run by decision tree models. The wider scope of actions for LLM agents has created new challenges for their verification, over and above those present for classical agents. For example, the LLM's neural network endows it with infinite domains, an especial challenge for traditional formal verification techniques. Academics began to study the problems involved in verifying LLM agents from 2018. Deployment of such agents began to accelerate in late 2023 after OpenAI's "function-calling" API was made available, and especially after Anthropic's late 2024 introduction of Model Context Protocol (MCP), a standardised way for LLM agents to gain contextual awareness, and to act on the world by calling various external tools. The rapid rollout of LLM agents following MCP's release has seen the task of agent verification receive increased attention within academia, and also from the private sector. In 2024 and 2025 several startups focusing on LLM agent verification have been founded in both Europe and the US to meet growing demand. == Approaches == === Formal verification === Formal verification involves proving the correctness of some or all aspects of a system using mathematical methods. Such methods can range from manual formal proof, to verification assisted with automated theorem provers like Isabelle. For agent verification, model checking is by far the most frequently used formal verification method; for pre-LLM models it was often complemented with techniques using computation tree logic. Another common method is theorem proving. Formal verification provides a higher degree of confidence than informal methods, but it is not always used, even when it is possible. Sometimes a person or organisation developing software agents won't have the necessary skills, or may not see it as worth the effort if the agent(s) will not have the ability to cause much harm even if they malfunction. When agents are deployed in systems where errors could have serious consequences, the ability of formal verification methods to provide mathematical certainty tends to be strongly preferred by both regulators and designers themselves. But even for high impact systems, formal verificatio

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  Weak supervision

  Weak supervision (also known as semi-supervised learning) is a paradigm in machine learning, the relevance and notability of which increased with the advent of large language models due to the large amount of data required to train them. It is characterized by using a combination of a small amount of human-labeled data (exclusively used in more expensive and time-consuming supervised learning paradigm), followed by a large amount of unlabeled data (used exclusively in unsupervised learning paradigm). In other words, the desired output values are provided only for a subset of the training data. The remaining data is unlabeled or imprecisely labeled. Intuitively, it can be seen as an exam and labeled data as sample problems that the teacher solves for the class as an aid in solving another set of problems. In the transductive setting, these unsolved problems act as exam questions. In the inductive setting, they become practice problems of the sort that will make up the exam. == Problem == The acquisition of labeled data for a learning problem often requires a skilled human agent (e.g. to transcribe an audio segment) or a physical experiment (e.g. determining the 3D structure of a protein or determining whether there is oil at a particular location). The cost associated with the labeling process thus may render large, fully labeled training sets infeasible, whereas acquisition of unlabeled data is relatively inexpensive. In such situations, semi-supervised learning can be of great practical value. Semi-supervised learning is also of theoretical interest in machine learning and as a model for human learning. == Technique == More formally, semi-supervised learning assumes a set of l {\displaystyle l} independently identically distributed examples x 1 , … , x l ∈ X {\displaystyle x_{1},\dots ,x_{l}\in X} with corresponding labels y 1 , … , y l ∈ Y {\displaystyle y_{1},\dots ,y_{l}\in Y} and u {\displaystyle u} unlabeled examples x l + 1 , … , x l + u ∈ X {\displaystyle x_{l+1},\dots ,x_{l+u}\in X} are processed. Semi-supervised learning combines this information to surpass the classification performance that can be obtained either by discarding the unlabeled data and doing supervised learning or by discarding the labels and doing unsupervised learning. Semi-supervised learning may refer to either transductive learning or inductive learning. The goal of transductive learning is to infer the correct labels for the given unlabeled data x l + 1 , … , x l + u {\displaystyle x_{l+1},\dots ,x_{l+u}} only. The goal of inductive learning is to infer the correct mapping from X {\displaystyle X} to Y {\displaystyle Y} . It is unnecessary (and, according to Vapnik's principle, imprudent) to perform transductive learning by way of inferring a classification rule over the entire input space; however, in practice, algorithms formally designed for transduction or induction are often used interchangeably. == Assumptions == In order to make any use of unlabeled data, some relationship to the underlying distribution of data must exist. Semi-supervised learning algorithms make use of at least one of the following assumptions: === Continuity / smoothness assumption === Points that are close to each other are more likely to share a label. This is also generally assumed in supervised learning and yields a preference for geometrically simple decision boundaries. In the case of semi-supervised learning, the smoothness assumption additionally yields a preference for decision boundaries in low-density regions, so few points are close to each other but in different classes. === Cluster assumption === The data tend to form discrete clusters, and points in the same cluster are more likely to share a label (although data that shares a label may spread across multiple clusters). This is a special case of the smoothness assumption and gives rise to feature learning with clustering algorithms. === Manifold assumption === The data lie approximately on a manifold of much lower dimension than the input space. In this case learning the manifold using both the labeled and unlabeled data can avoid the curse of dimensionality. Then learning can proceed using distances and densities defined on the manifold. The manifold assumption is practical when high-dimensional data are generated by some process that may be hard to model directly, but which has only a few degrees of freedom. For instance, human voice is controlled by a few vocal folds, and images of various facial expressions are controlled by a few muscles. In these cases, it is better to consider distances and smoothness in the natural space of the generating problem, rather than in the space of all possible acoustic waves or images, respectively. == History == The heuristic approach of self-training (also known as self-learning or self-labeling) is historically the oldest approach to semi-supervised learning, with examples of applications starting in the 1960s. The transductive learning framework was formally introduced by Vladimir Vapnik in the 1970s. Interest in inductive learning using generative models also began in the 1970s. A probably approximately correct learning bound for semi-supervised learning of a Gaussian mixture was demonstrated by Ratsaby and Venkatesh in 1995. == Methods == === Generative models === Generative approaches to statistical learning first seek to estimate p ( x | y ) {\displaystyle p(x|y)} , the distribution of data points belonging to each class. The probability p ( y | x ) {\displaystyle p(y|x)} that a given point x {\displaystyle x} has label y {\displaystyle y} is then proportional to p ( x | y ) p ( y ) {\displaystyle p(x|y)p(y)} by Bayes' rule. Semi-supervised learning with generative models can be viewed either as an extension of supervised learning (classification plus information about p ( x ) {\displaystyle p(x)} ) or as an extension of unsupervised learning (clustering plus some labels). Generative models assume that the distributions take some particular form p ( x | y , θ ) {\displaystyle p(x|y,\theta )} parameterized by the vector θ {\displaystyle \theta } . If these assumptions are incorrect, the unlabeled data may actually decrease the accuracy of the solution relative to what would have been obtained from labeled data alone. However, if the assumptions are correct, then the unlabeled data necessarily improves performance. The unlabeled data are distributed according to a mixture of individual-class distributions. In order to learn the mixture distribution from the unlabeled data, it must be identifiable, that is, different parameters must yield different summed distributions. Gaussian mixture distributions are identifiable and commonly used for generative models. The parameterized joint distribution can be written as p ( x , y | θ ) = p ( y | θ ) p ( x | y , θ ) {\displaystyle p(x,y|\theta )=p(y|\theta )p(x|y,\theta )} by using the chain rule. Each parameter vector θ {\displaystyle \theta } is associated with a decision function f θ ( x ) = argmax y p ( y | x , θ ) {\displaystyle f_{\theta }(x)={\underset {y}{\operatorname {argmax} }}\ p(y|x,\theta )} . The parameter is then chosen based on fit to both the labeled and unlabeled data, weighted by λ {\displaystyle \lambda } : argmax Θ ( log ⁡ p ( { x i , y i } i = 1 l | θ ) + λ log ⁡ p ( { x i } i = l + 1 l + u | θ ) ) {\displaystyle {\underset {\Theta }{\operatorname {argmax} }}\left(\log p(\{x_{i},y_{i}\}_{i=1}^{l}|\theta )+\lambda \log p(\{x_{i}\}_{i=l+1}^{l+u}|\theta )\right)} === Low-density separation === Another major class of methods attempts to place boundaries in regions with few data points (labeled or unlabeled). One of the most commonly used algorithms is the transductive support vector machine, or TSVM (which, despite its name, may be used for inductive learning as well). Whereas support vector machines for supervised learning seek a decision boundary with maximal margin over the labeled data, the goal of TSVM is a labeling of the unlabeled data such that the decision boundary has maximal margin over all of the data. In addition to the standard hinge loss ( 1 − y f ( x ) ) + {\displaystyle (1-yf(x))_{+}} for labeled data, a loss function ( 1 − | f ( x ) | ) + {\displaystyle (1-|f(x)|)_{+}} is introduced over the unlabeled data by letting y = sign ⁡ f ( x ) {\displaystyle y=\operatorname {sign} {f(x)}} . TSVM then selects f ∗ ( x ) = h ∗ ( x ) + b {\displaystyle f^{}(x)=h^{}(x)+b} from a reproducing kernel Hilbert space H {\displaystyle {\mathcal {H}}} by minimizing the regularized empirical risk: f ∗ = argmin f ( ∑ i = 1 l ( 1 − y i f ( x i ) ) + + λ 1 ‖ h ‖ H 2 + λ 2 ∑ i = l + 1 l + u ( 1 − | f ( x i ) | ) + ) {\displaystyle f^{}={\underset {f}{\operatorname {argmin} }}\left(\displaystyle \sum _{i=1}^{l}(1-y_{i}f(x_{i}))_{+}+\lambda _{1}\|h\|_{\mathcal {H}}^{2}+\lambda _{2}\sum _{i=l+1}^{l+u}(1-|f(x_{i})|)_{+}\right)} An exact solution is intractable due to the non-convex term ( 1 − | f ( x ) | ) + {\displayst

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  Legendre moment

  In mathematics, Legendre moments are a type of image moment and are achieved by using the Legendre polynomial. Legendre moments are used in areas of image processing including: pattern and object recognition, image indexing, line fitting, feature extraction, edge detection, and texture analysis. Legendre moments have been studied as a means to reduce image moment calculation complexity by limiting the amount of information redundancy through approximation. == Legendre moments == Source: With order of m + n, and object intensity function f(x,y): L m n = ( 2 m + 1 ) ( 2 n + 1 ) 4 ∫ − 1 1 ∫ − 1 1 P m ( x ) P n ( y ) f ( x , y ) d x d y {\displaystyle L_{mn}={\frac {(2m+1)(2n+1)}{4}}\int \limits _{-1}^{1}\int \limits _{-1}^{1}P_{m}(x)P_{n}(y)f(x,y)\,dx\,dy} where m,n = 1, 2, 3, ...∞ with the nth-order Legendre polynomials being: P n ( x ) = ∑ k = 0 n a k , n x k = ( − 1 ) n 2 n n ! ( d d x ) [ ( 1 − x 2 ) n ] {\displaystyle P_{n}(x)=\sum _{k=0}^{n}a_{k,n}x^{k}={\frac {(-1)^{n}}{2^{n}n!}}\left({\frac {d}{dx}}\right)[(1-x^{2})^{n}]} which can also be written: P n ( x ) = ∑ k = 0 D ( n ) ( − 1 ) k ( 2 n − 2 k ) ! 2 n k ! ( n − k ) ! ( n − 2 k ) ! x n − 2 k = ( 2 n ) ! 2 n ( n ! ) 2 x n − ( 2 n − 2 ) ! 2 n 1 ! ( n − 1 ) ! ( n − 2 ) ! x n − 2 + ⋯ {\displaystyle {\begin{aligned}P_{n}(x)&=\sum _{k=0}^{D(n)}(-1)^{k}{\frac {(2n-2k)!}{2^{n}k!(n-k)!(n-2k)!}}x^{n-2k}\\[5pt]&={\frac {(2n)!}{2^{n}(n!)^{2}}}x^{n}-{\frac {(2n-2)!}{2^{n}1!(n-1)!(n-2)!}}x^{n-2}+\cdots \end{aligned}}} where D(n) = floor(n/2). The set of Legendre polynomials {Pn(x)} form an orthogonal set on the interval [−1,1]: ∫ − 1 1 P n ( x ) P m ( x ) d x = 2 2 n + 1 δ n m {\displaystyle \int _{-1}^{1}P_{n}(x)P_{m}(x)\,dx={\frac {2}{2n+1}}\delta _{nm}} A recurrence relation can be used to compute the Legendre polynomial: ( n + 1 ) P n + 1 ( x ) − ( 2 n + 1 ) x P n ( x ) + n P n − 1 ( x ) = 0 {\displaystyle (n+1)P_{n+1}(x)-(2n+1)xP_{n}(x)+nP_{n-1}(x)=0} f(x,y) can be written as an infinite series expansion in terms of Legendre polynomials [−1 ≤ x,y ≤ 1.]: f ( x , y ) = ∑ m = 0 ∞ ∑ n = 0 ∞ λ m n P m ( x ) P n ( y ) {\displaystyle f(x,y)=\sum _{m=0}^{\infty }\sum _{n=0}^{\infty }\lambda _{mn}P_{m}(x)P_{n}(y)}

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  Physics-informed neural networks

  In machine learning, physics-informed neural networks (PINNs), also referred to as theory-trained neural networks (TTNs), are a type of universal function approximator that can embed the knowledge of any physical laws that govern a given data-set in the learning process, and can be described by partial differential equations (PDEs). Low data availability for some biological and engineering problems limit the robustness of conventional machine learning models used for these applications. The prior knowledge of general physical laws acts in the training of neural networks (NNs) as a regularization agent that limits the space of admissible solutions, increasing the generalizability of the function approximation. This way, embedding this prior information into a neural network results in enhancing the information content of the available data, facilitating the learning algorithm to capture the right solution and to generalize well even with a low amount of training examples. Because they process continuous spatial and time coordinates and output continuous PDE solutions, they can be categorized as neural fields. == Function approximation == Most of the physical laws that govern the dynamics of a system can be described by partial differential equations. For example, the Navier–Stokes equations are a set of partial differential equations derived from the conservation laws (i.e., conservation of mass, momentum, and energy) that govern fluid mechanics. The solution of the Navier–Stokes equations with appropriate initial and boundary conditions allows the quantification of flow dynamics in a precisely defined geometry. However, these equations cannot be solved exactly and therefore numerical methods must be used (such as finite differences, finite elements and finite volumes). In this setting, these governing equations must be solved while accounting for prior assumptions, linearization, and adequate time and space discretization. Recently, solving the governing partial differential equations of physical phenomena using deep learning has emerged as a new field of scientific machine learning (SciML), leveraging the universal approximation theorem and high expressivity of neural networks. In general, deep neural networks could approximate any high-dimensional function given that sufficient training data are supplied. However, such networks do not consider the physical characteristics underlying the problem, and the level of approximation accuracy provided by them is still heavily dependent on careful specifications of the problem geometry as well as the initial and boundary conditions. Without this preliminary information, the solution is not unique and may lose physical correctness. To remedy this, Physics-Informed Neural Networks (PINNs) leverage governing physical equations in neural network training. Namely, PINNs are designed to be trained to satisfy the given training data as well as the imposed governing equations. In this fashion, a neural network can be guided with training datasets that do not necessarily need to be large or complete. An accurate solution of partial differential equations can potentially be found without knowing the boundary conditions. Therefore, with some knowledge about the physical characteristics of the problem and some form of training data (even sparse and incomplete), PINNs may be used for finding an optimal solution with high fidelity. PINNs can be applied to a wide range of problems in computational science, and are a pioneering technology leading to the development of new classes of numerical solvers for PDEs. PINNs can be thought of as a mesh-free alternative to traditional approaches (e.g., CFD for fluid dynamics), and new data-driven approaches for model inversion and system identification. Notably, a trained PINN network can be used to predict values on simulation grids of different resolutions without needing to be retrained. Additionally, the derivatives used in the partial differential equations can be computed using automatic differentiation (AD), which is assessed to be superior to numerical or symbolic differentiation. == Modeling and computation == A general nonlinear partial differential equation can be written as: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} where u ( t , x ) {\displaystyle u(t,x)} denotes the solution, N [ ⋅ ; λ ] {\displaystyle {\mathcal {N}}[\cdot ;\lambda ]} is a nonlinear operator parameterized by λ {\displaystyle \lambda } , and Ω {\displaystyle \Omega } is a subset of R D {\displaystyle \mathbb {R} ^{D}} . This general form of governing equations summarizes a wide range of problems in mathematical physics, such as conservative laws, diffusion process, advection-diffusion systems, and kinetic equations. Given noisy measurements of a generic dynamic system described by the equation above, PINNs can be designed to solve two classes of problems: data-driven solutions of partial differential equations data-driven discovery of partial differential equations === Data-driven solution of partial differential equations === The data-driven solution of PDE computes the hidden state u ( t , x ) {\displaystyle u(t,x)} of the system given boundary data and/or measurements z {\displaystyle z} , and fixed model parameters λ {\displaystyle \lambda } . We solve: u t + N [ u ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u]=0,\quad x\in \Omega ,\quad t\in [0,T]} . by defining the residual f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ] {\displaystyle f:=u_{t}+{\mathcal {N}}[u]} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network. This network can be differentiated using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} is the error between the PINN u ( t , x ) {\displaystyle u(t,x)} and the set of boundary conditions and measured data on the set of points Γ {\displaystyle \Gamma } where the boundary conditions and data are defined. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the mean-squared error of the residual function. This second term encourages the PINN to learn the structural information expressed by the PDE during the training process. This approach has been used to yield computationally efficient physics-informed surrogate models with applications in the forecasting of physical processes, model predictive control, multi-physics and multi-scale modeling, and simulation. It has been shown to converge to the solution of the PDE. === Data-driven discovery of partial differential equations === Given noisy and incomplete measurements z {\displaystyle z} of the state of the system, the data-driven discovery of PDEs results in computing the unknown state u ( t , x ) {\displaystyle u(t,x)} and learning model parameters λ {\displaystyle \lambda } that best describe the observed data: u t + N [ u ; λ ] = 0 , x ∈ Ω , t ∈ [ 0 , T ] {\displaystyle u_{t}+{\mathcal {N}}[u;\lambda ]=0,\quad x\in \Omega ,\quad t\in [0,T]} By defining f ( t , x ) {\displaystyle f(t,x)} as: f := u t + N [ u ; λ ] = 0 {\displaystyle f:=u_{t}+{\mathcal {N}}[u;\lambda ]=0} , and approximating u ( t , x ) {\displaystyle u(t,x)} by a deep neural network, f ( t , x ) {\displaystyle f(t,x)} results in a PINN. This network can be derived using automatic differentiation. The parameters of u ( t , x ) {\displaystyle u(t,x)} and f ( t , x ) {\displaystyle f(t,x)} , together with the parameter λ {\displaystyle \lambda } of the differential operator can be then learned by minimizing the following loss function L tot {\displaystyle L_{\text{tot}}} : L tot = L u + L f {\displaystyle L_{\text{tot}}=L_{u}+L_{f}} where: L u = ‖ u − z ‖ Γ {\displaystyle L_{u}=\Vert u-z\Vert _{\Gamma }} , with u {\displaystyle u} and z {\displaystyle z} state solutions and measurements at sparse location Γ {\displaystyle \Gamma } , respectively. L f = ‖ f ‖ Γ {\displaystyle L_{f}=\Vert f\Vert _{\Gamma }} is the residual function. This second term requires the structured information represented by the partial differential equations to be satisfied in the training process. This strategy allows for discovering dynamic models described by nonlinear PDEs assembling computationally efficient and fully differentiable surrogate models that may find application in predictive forecasting, control, and data assimilation. == Extensions and applications == === For piece-wise function approximation === PINNs are unable to approximate PDEs that have strong non-linearity or sharp gradients (such as those that commonly occur in practical fluid flow problems). Piecewise approximation has been an old practic

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  Geometric primitive

  In vector computer graphics, CAD systems, and geographic information systems, a geometric primitive (or prim) is the simplest (i.e. 'atomic' or irreducible) geometric shape that the system can handle (draw, store). Sometimes the subroutines that draw the corresponding objects are called "geometric primitives" as well. The most "primitive" primitives are point and straight line segments, which were all that early vector graphics systems had. In constructive solid geometry, primitives are simple geometric shapes such as a cube, cylinder, sphere, cone, pyramid, torus. Modern 2D computer graphics systems may operate with primitives which are curves (segments of straight lines, circles and more complicated curves), as well as shapes (boxes, arbitrary polygons, circles). A common set of two-dimensional primitives includes lines, points, and polygons, although some people prefer to consider triangles primitives, because every polygon can be constructed from triangles (polygon triangulation). All other graphic elements are built up from these primitives. In three dimensions, triangles or polygons positioned in three-dimensional space can be used as primitives to model more complex 3D forms. In some cases, curves (such as Bézier curves, circles, etc.) may be considered primitives; in other cases, curves are complex forms created from many straight, primitive shapes. == Common primitives == The set of geometric primitives is based on the dimension of the region being represented: Point (0-dimensional), a single location with no height, width, or depth. Line or curve (1-dimensional), having length but no width, although a linear feature may curve through a higher-dimensional space. Planar surface or curved surface (2-dimensional), having length and width. Volumetric region or solid (3-dimensional), having length, width, and depth. In GIS, the terrain surface is often spoken of colloquially as "2 1/2 dimensional," because only the upper surface needs to be represented. Thus, elevation can be conceptualized as a scalar field property or function of two-dimensional space, affording it a number of data modeling efficiencies over true 3-dimensional objects. A shape of any of these dimensions greater than zero consists of an infinite number of distinct points. Because digital systems are finite, only a sample set of the points in a shape can be stored. Thus, vector data structures typically represent geometric primitives using a strategic sample, organized in structures that facilitate the software interpolating the remainder of the shape at the time of analysis or display, using the algorithms of Computational geometry. A Point is a single coordinate in a Cartesian coordinate system. Some data models allow for Multipoint features consisting of several disconnected points. A Polygonal chain or Polyline is an ordered list of points (termed vertices in this context). The software is expected to interpolate the intervening shape of the line between adjacent points in the list as a parametric curve, most commonly a straight line, but other types of curves are frequently available, including circular arcs, cubic splines, and Bézier curves. Some of these curves require additional points to be defined that are not on the line itself, but are used for parametric control. A Polygon is a polyline that closes at its endpoints, representing the boundary of a two-dimensional region. The software is expected to use this boundary to partition 2-dimensional space into an interior and exterior. Some data models allow for a single feature to consist of multiple polylines, which could collectively connect to form a single closed boundary, could represent a set of disjoint regions (e.g., the state of Hawaii), or could represent a region with holes (e.g., a lake with an island). A Parametric shape is a standardized two-dimensional or three-dimensional shape defined by a minimal set of parameters, such as an ellipse defined by two points at its foci, or three points at its center, vertex, and co-vertex. A Polyhedron or Polygon mesh is a set of polygon faces in three-dimensional space that are connected at their edges to completely enclose a volumetric region. In some applications, closure may not be required or may be implied, such as modeling terrain. The software is expected to use this surface to partition 3-dimensional space into an interior and exterior. A triangle mesh is a subtype of polyhedron in which all faces must be triangles, the only polygon that will always be planar, including the Triangulated irregular network (TIN) commonly used in GIS. A parametric mesh represents a three-dimensional surface by a connected set of parametric functions, similar to a spline or Bézier curve in two dimensions. The most common structure is the Non-uniform rational B-spline (NURBS), supported by most CAD and animation software. == Application in GIS == A wide variety of vector data structures and formats have been developed during the history of Geographic information systems, but they share a fundamental basis of storing a core set of geometric primitives to represent the location and extent of geographic phenomena. Locations of points are almost always measured within a standard Earth-based coordinate system, whether the spherical Geographic coordinate system (latitude/longitude), or a planar coordinate system, such as the Universal Transverse Mercator. They also share the need to store a set of attributes of each geographic feature alongside its shape; traditionally, this has been accomplished using the data models, data formats, and even software of relational databases. Early vector formats, such as POLYVRT, the ARC/INFO Coverage, and the Esri shapefile support a basic set of geometric primitives: points, polylines, and polygons, only in two dimensional space and the latter two with only straight line interpolation. TIN data structures for representing terrain surfaces as triangle meshes were also added. Since the mid 1990s, new formats have been developed that extend the range of available primitives, generally standardized by the Open Geospatial Consortium's Simple Features specification. Common geometric primitive extensions include: three-dimensional coordinates for points, lines, and polygons; a fourth "dimension" to represent a measured attribute or time; curved segments in lines and polygons; text annotation as a form of geometry; and polygon meshes for three-dimensional objects. Frequently, a representation of the shape of a real-world phenomenon may have a different (usually lower) dimension than the phenomenon being represented. For example, a city (a two-dimensional region) may be represented as a point, or a road (a three-dimensional volume of material) may be represented as a line. This dimensional generalization correlates with tendencies in spatial cognition. For example, asking the distance between two cities presumes a conceptual model of the cities as points, while giving directions involving travel "up," "down," or "along" a road imply a one-dimensional conceptual model. This is frequently done for purposes of data efficiency, visual simplicity, or cognitive efficiency, and is acceptable if the distinction between the representation and the represented is understood, but can cause confusion if information users assume that the digital shape is a perfect representation of reality (i.e., believing that roads really are lines). == In 3D modelling == In CAD software or 3D modelling, the interface may present the user with the ability to create primitives which may be further modified by edits. For example, in the practice of box modelling the user will start with a cuboid, then use extrusion and other operations to create the model. In this use the primitive is just a convenient starting point, rather than the fundamental unit of modelling. A 3D package may also include a list of extended primitives which are more complex shapes that come with the package. For example, a teapot is listed as a primitive in 3D Studio Max. == In graphics hardware == Various graphics accelerators exist with hardware acceleration for rendering specific primitives such as lines or triangles, frequently with texture mapping and shaders. Modern 3D accelerators typically accept sequences of triangles as triangle strips.

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  Core FTP

  Core FTP LE is a freeware secure FTP client for Windows, developed by CoreFTP.com. Features include FTP, SSL/TLS, SFTP via SSH, and HTTP/HTTPS support. Secure FTP clients encrypt account information and data transferred across the internet, protecting data from being seen, or sniffed across networks. Core FTP is a traditional FTP client with local files displayed on the left, remote files on the right. Core FTP Server is a secure FTP server for Windows, developed by CoreFTP.com, starting in 2010. == Licensing == CoreFTP LE is free for personal, educational, non-profit, and business use.

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  Artificial intelligence in fraud detection

  Artificial intelligence is used by many different businesses and organizations. It is widely used in the financial sector, especially by accounting firms, to help detect fraud. In 2022, PricewaterhouseCoopers reported that fraud has impacted 46% of all businesses in the world. The shift from working in person to working from home has brought increased access to data. According to an FTC (Federal Trade Commission) study from 2022, customers reported fraud of approximately $5.8 billion in 2021, an increase of 70% from the year before. The majority of these scams were imposter scams and online shopping frauds. Furthermore, artificial intelligence plays a crucial role in developing advanced algorithms and machine learning models that enhance fraud detection systems, enabling businesses to stay ahead of evolving fraudulent tactics in an increasingly digital landscape. == Tools == === Expert systems === Expert systems were first designed in the 1970s as an expansion into artificial intelligence technologies. Their design is based on the premise of decreasing potential user error in decision-making and emulating mental reasoning used by experts in a particular field. They differentiate themselves from traditional linear reasoning models by separating identified points in data and processing them individually at the same time. Though, these systems do not rely purely on machine-learned intelligence. Information regarding rules, practices, and procedures in the form of "if-then" statements are implemented into the programming of the system. Users interact with the system by feeding information into the system either through direct entry or import of external data. An inference system compares the information provided by the user with corresponding rules that are believed to specifically apply to the situation. Using this information and the corresponding rules will be used to create a solution to the user's query. Expert systems will generally not operate properly when the common procedures for a specified situation are ambiguous due to the need for well-defined rules. Implementation of expert systems in accounting procedures is feasible in areas where professional judgment is required. Situations where expert systems are applicable include investigations into transactions that involve potential fraudulent entries, instances of going concern, and the evaluation of risk in the planning stages of an audit. === Continuous auditing === Continuous auditing is a set of processes that assess various aspects of information gathered in an audit to classify areas of risk and potential weaknesses in financial Internal controls at a more frequent rate than traditional methods. Instead of analyzing recorded transactions and journal entries periodically, continuous auditing focuses on interpreting the character of these actions more frequently. The frequency of these processes being undertaken as well as highlighting areas of importance is up to the discretion of their implementer, who commonly makes such decisions based on the level of risk in the accounts being evaluated and the goals of implementing the system. Performance of these processes can occur as frequently as being nearly instantaneous with an entry being posted. The processes involved with analyzing financial data in continuous auditing can include the creation of spreadsheets to allow for interactive information gathering, calculation of financial ratios for comparison with previously created models, and detection of errors in entered figures. A primary goal of this practice is to allow for quicker and easier detection of instances of faulty controls, errors, and instances of fraud. === Machine learning and deep learning === The ability of machine learning and deep learning to swiftly and effectively sort through vast volumes of data in the forms of various documents relevant to companies and documents being audited makes them applicable to the domains of audit and fraud detection. Examples of this include recognizing key language in contracts, identifying levels of risk of fraud in transactions, and assessing journal entries for misstatement. == Applications == === 'Big 4' Accounting Firms === Deloitte created an Al-enabled document-reviewing system in 2014. The system automates the method of reviewing and extracting relevant information from different business documents. Deloitte claims that this innovation has made a difference by reducing time spent going through lawful contract documents, invoices, money-related articulations, and board minutes by up to 50%. Working with IBM's Watson, Deloitte is developing cognitive-technology-enhanced commerce arrangements for its clients. LeasePoint is fueled by IBM TRIRIGA (this product evolved into IBM Maximo Real Estate and Facilities) and uses Deloitte's industrial information to create an end-to-end leasing portfolio. Automated Cognitive Resource Assessment employs IBM's Maximo innovation to progress the proficiency of asset inspection. Ernst and Young (EY) connected Al to the investigation of lease contracts. EY (Australia) has also received Al-enabled auditing technology. Collaborating with H20.ai, PwC developed an Al-enabled framework (GL.ai) capable of analyzing reports and preparing reports. PwC claims to have made a significant investment in normal dialect processing (NLP), an Al-enabled innovation to process unstructured information efficiently. KPMG built a portfolio of Al instruments, called KPMG Ignite, to upgrade trade decisions and forms. Working with Microsoft and IBM Watson, KPMG is creating instruments to coordinate Al, data analytics, Cognitive Technologies, and RPA. == Advantages == === Efficiency === The process of auditing an entity in an attempt to detect fraudulent activity requires the repeating of investigatory processes until an error or misstatement may be identified. Under traditional methods, these processes would be carried out by a human being. Proponents of artificial intelligence in fraud detection have stated that these traditional methods are inefficient and can be more quickly accomplished with the aid of an intelligent computing system. A survey of 400 chief executive officers created by KPMG in 2016 found that approximately 58% believed that artificial intelligence would play a key role in making audits more efficient in the future. === Data interpretation === Higher levels of fraud detection entail the use of professional judgement to interpret data. Supporters of artificial intelligence being used in financial audits have claimed that increased risks from instances of higher data interpretation can be minimized through such technologies. One necessary element of an audit of financial statements that requires professional judgement is the implementation of thresholds for materiality. Materiality entails the distinction between errors and transactions in financial statements that would impact decisions made by users of those financial statements. The threshold for materiality in an audit is set by the auditor based on various factors. Artificial intelligence has been used to interpret data and suggest materiality thresholds to be implemented through the use of expert systems. === Decreased costs === Those in favor of using artificial intelligence to complete investigations of fraud have stated that such technologies decrease the amount of time required to complete tasks that are repetitive. The claim further states that such efficiencies allow for lowered resource requirements, which can then be further spent on tasks that have not been fully automated. The audit firm Ernst & Young has posited these claims by declaring that their deep learning systems have been used to reduce time spent on administrative tasks by analyzing relevant audit documents. According to the firm, this has allowed their employees to focus more on judgement and analysis. == Disadvantages == === Job Displacement === The inescapable reception of computer based intelligence and robotization advancements might prompt critical work relocation across different enterprises. As artificial intelligence frameworks become more equipped for performing undertakings customarily completed by people, there is a worry that specific work jobs could become out of date, prompting joblessness and financial imbalance. === Initial investment requirement === Along with a knowledge of coding and building systems through computer programs, we are seeing the advantages of these systems, but since they are so new, they require a large investment to start building such a system. Any firm that is planning on implementing an AI system to detect fraud must hire a team of data scientists, along with upgrading their cloud system and data storage. The system must be consistently monitored and updated to be the most efficient form of itself, otherwise the likelihood of fraud being involved in those transactions increases. If one does not initially invest in such a syst

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  Sentence embedding

  In natural language processing, a sentence embedding is a representation of a sentence as a vector of numbers which encodes meaningful semantic information. State of the art embeddings are based on the learned hidden layer representation of dedicated sentence transformer models. BERT pioneered an approach involving the use of a dedicated [CLS] token prepended to the beginning of each sentence inputted into the model; the final hidden state vector of this token encodes information about the sentence and can be fine-tuned for use in sentence classification tasks. In practice however, BERT's sentence embedding with the [CLS] token achieves poor performance, often worse than simply averaging non-contextual word embeddings. SBERT later achieved superior sentence embedding performance by fine tuning BERT's [CLS] token embeddings through the usage of a siamese neural network architecture on the SNLI dataset. Other approaches are loosely based on the idea of distributional semantics applied to sentences. Skip-Thought trains an encoder-decoder structure for the task of neighboring sentences predictions; this has been shown to achieve worse performance than approaches such as InferSent or SBERT. An alternative direction is to aggregate word embeddings, such as those returned by Word2vec, into sentence embeddings. The most straightforward approach is to simply compute the average of word vectors, known as continuous bag-of-words (CBOW). However, more elaborate solutions based on word vector quantization have also been proposed. One such approach is the vector of locally aggregated word embeddings (VLAWE), which demonstrated performance improvements in downstream text classification tasks. == Applications == In recent years, sentence embedding has seen a growing level of interest due to its applications in natural language queryable knowledge bases through the usage of vector indexing for semantic search. LangChain for instance utilizes sentence transformers for purposes of indexing documents. In particular, an indexing is generated by generating embeddings for chunks of documents and storing (document chunk, embedding) tuples. Then given a query in natural language, the embedding for the query can be generated. A top k similarity search algorithm is then used between the query embedding and the document chunk embeddings to retrieve the most relevant document chunks as context information for question answering tasks. This approach is also known formally as retrieval-augmented generation. Though not as predominant as BERTScore, sentence embeddings are commonly used for sentence similarity evaluation which sees common use for the task of optimizing a Large language model's generation parameters is often performed via comparing candidate sentences against reference sentences. By using the cosine-similarity of the sentence embeddings of candidate and reference sentences as the evaluation function, a grid-search algorithm can be utilized to automate hyperparameter optimization. == Evaluation == A way of testing sentence encodings is to apply them on Sentences Involving Compositional Knowledge (SICK) corpus for both entailment (SICK-E) and relatedness (SICK-R). In the best results are obtained using a BiLSTM network trained on the Stanford Natural Language Inference (SNLI) Corpus. The Pearson correlation coefficient for SICK-R is 0.885 and the result for SICK-E is 86.3. A slight improvement over previous scores is presented in: SICK-R: 0.888 and SICK-E: 87.8 using a concatenation of bidirectional Gated recurrent unit.

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  Elastix (image registration)

  Elastix is an image registration toolbox built upon the Insight Segmentation and Registration Toolkit (ITK). It is entirely open-source and provides a wide range of algorithms employed in image registration problems. Its components are designed to be modular to ease a fast and reliable creation of various registration pipelines tailored for case-specific applications. It was first developed by Stefan Klein and Marius Staring under the supervision of Josien P.W. Pluim at Image Sciences Institute (ISI). Its first version was command-line based, allowing the final user to employ scripts to automatically process big data-sets and deploy multiple registration pipelines with few lines of code. Nowadays, to further widen its audience, a version called SimpleElastix is also available, developed by Kasper Marstal, which allows the integration of elastix with high level languages, such as Python, Java, and R. == Image registration fundamentals == Image registration is a well-known technique in digital image processing that searches for the geometric transformation that, applied to a moving image, obtains a one-to-one map with a target image. Generally, the images acquired from different sensors (multimodal), time instants (multitemporal), and points of view (multiview) should be correctly aligned to proceed with further processing and feature extraction. Even though there are a plethora of different approaches to image registration, the majority is composed of the same macro building blocks, namely the transformation, the interpolator, the metric, and the optimizer. Registering two or more images can be framed as an optimization problem that requires multiple iterations to converge to the best solution. Starting from an initial transformation computed from the image moments the optimization process searches for the best transformation parameters based on the value of the selected similarity metric. The figure on the right shows the high-level representation of the registration of two images, where the reference remains constant during the entire process, while the moving one will be transformed according to the transformation parameters. In other words, the registration ends when the similarity metric, which is a mathematical function with a certain number of parameters to be optimized, reaches the optimal value which is highly dependent on the specific application. == Main building blocks == Following the structure of the image registration workflow, the elastix toolbox proposes a modular solution that implements for each of the building blocks different algorithms, highly employed in medical image registration, and helps the final users to build their specific pipeline by selecting the most suitable algorithm for each of the main building blocks. Each block is easily configurable both by selecting pre-defined initialization values or by trying multiple sets of parameters and then choosing the most performing one. The registration is performed on images, and the elastix toolbox supports all the data formats supported by ITK, ranging from JPEG and PNG to medical standard formats such as DICOM and NIFTI. It also stores physical pixel spacing, the origin and the relative position to an external world reference system, when provided in the metadata, to facilitate the registration process, especially in medical field applications. === Transformation === The transformation is an essential building block, since it defines the allowable transformations. In image registration, the main distinction can be done between parallel-to-parallel and parallel-to-non parallel (deformable) line mapping transformations. In the elastix toolbox, the final users can select one transformation or compose more transformations either through addition or via composition. Below are reported the different transformation models in order of increasing flexibility, along with the corresponding elastix class names between brackets. Translation (TranslationTransform) allows only translations Rigid (EulerTransform) expands the translation adding rotations and the object is seen as a rigid body Similarity (SimilarityTransform) expands the rigid transformation by introducing isotropic scaling Affine (AffineTransform) expands the rigid transformation allowing both scaling and shear B-splines (BSplineTransform) is a deformable transformation usually preceded by a rigid or affine one Thin-plate splines (SplineKernelTransform) is a deformable transformation belonging to the class of kernel-based transformations that is a composition of and affine and a non-rigid part === Metric === The similarity metric is the mathematical function whose parameters should be optimized to reach the desired registration, and, during the process, it is computed multiple times. Below are reported the available metrics computed employing the reference and the transformed images and the corresponding elastix class names between brackets. Mean squared difference (AdvancedMeanSquares) to be used for mono-modal applications Normalized correlation coefficient (AdvancedNormalizedCorrelation) to be used for images that have an intensity linear relationship Mutual information (AdvancedMattesMutualInformation) to be used for both mono- and multi-modal applications and optimized to reach better performance compared to the normalized version Normalized mutual information (NormalizedMutualInformation) for both mono- and multi-modal applications Kappa statistic (AdvancedKappaStatistic) to be used only for binary images === Sampler === For the computation of the similarity metrics, it is not always necessary to consider all the voxels and, sometimes, it can be useful to use only a fraction of the voxels of the images, i.e. to reduce the execution time for big input images. Below are reported the available criteria for selecting a fraction of the voxels for the similarity metric computation and the corresponding elastix class names between brackets. Full (Full) to employ all the voxels Grid (Grid) to employ a regular grid defined by the user to downsample the image Random (Random) to randomly select a percentage of voxels defined by the users (all voxels have equal probability to be selected) Random coordinate (RandomCoordinate) like the random criterion, but in this case also off-grid positions can be selected to simplify the optimization process === Interpolator === After the application of the transformation, it may occur that the voxels used for the similarity metric computation are at non-voxel positions, so intensity interpolation should be performed to ensure the correctness of the computed values. Below are reported the implemented interpolators and the corresponding elastix class names between brackets. Nearest neighbor (NearestNeighborInterpolator) exploits little resources, but gives low quality results Linear (LinearInterpolator) is sufficient in general applications N-th order B-spline (BSplineInterpolator) can be used to increase the order N, increasing quality and computation time. N=0 and N=1 indicate the nearest neighbor and linear cases respectively. === Optimizer === The optimizer defines the strategy employed for searching the best transformation parameter to reach the correct registration, and it is commonly an iterative strategy. Below are reported some of the implemented optimization strategies. Gradient descent Robbins-Monro, similar to the gradient descent, but employing an approximation of the cost function derivatives A wider range of optimizers is also available, such as Quasi-Newton or evolutionary strategies. === Other features === The elastix software also offers other features that can be employed to speed up the registration procedure and to provide more advanced algorithms to the end-users. Some examples are the introduction of blur and Gaussian pyramid to reduce data complexity, and multi-image and multi-metric framework to deal with more complex applications. == Applications == Elastix has applications mainly in the medical field, where image registration is fundamental to get comprehensive information regarding the analysed anatomical region. It is widely employed in image-guided surgery, tumour monitoring, and treatment assessment. For example, in radiotherapy planning, image registration allows to correctly deliver the treatment and evaluate the obtained results. Thanks to the wide range of implemented algorithms, the use of the elastix software allows physicians and researchers to test different registration pipelines from the simplest to more complex ones, and to save the best one as a configuration file. This file and the fact that the software is completely open-source makes it easy to reproduce the work, that can help supporting the open science paradigm, and allows fast reuse on different patients data. In image-guided surgery, registration time and accuracy are critical points, considering that, during the registration, the patient is on the operating table, and the imag

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  Transderivational search

  Transderivational search (often abbreviated to TDS) is a psychological and cybernetics term, meaning when a search is being conducted for a fuzzy match across a broad field. In computing the equivalent function can be performed using content-addressable memory. Unlike usual searches, which look for literal (i.e. exact, logical, or regular expression) matches, a transderivational search is a search for a possible meaning or possible match as part of communication, and without which an incoming communication cannot be made any sense of whatsoever. It is thus an integral part of processing language, and of attaching meaning to communication. In NLP (Neuro-linguistic programming), a transderivational search (Bandler and Grinder, 1976) is essentially the process of searching back through one's stored memories and mental representations to find the personal reference experiences from which a current understanding or mental map has been derived. By the end of 1976, Grinder and Bandler had combined Satir’s and Perls’ language patterns and Erickson’s hypnotic language and use of metaphor with anchoring to create new processes that they called collapsing anchors, trans-derivational search, changing personal history, and reframing. A psychological example of TDS is in Ericksonian hypnotherapy, where vague suggestions are used that the patient must process intensely in order to find their own meanings, thus ensuring that the practitioner does not intrude his own beliefs into the subject's inner world. == TDS in human communication and processing == Because TDS is a compelling, automatic and unconscious state of internal focus and processing (i.e. a type of everyday trance state), and often a state of internal lack of certainty, or openness to finding an answer (since something is being checked out at that moment), it can be utilized or interrupted, in order to create, or deepen, trance. TDS is a fundamental part of human language and cognitive processing. Arguably, every word or utterance a person hears, for example, and everything they see or feel and take note of, results in a very brief trance while TDS is carried out to establish a contextual meaning for it. === Examples === Leading statements: "And those thoughts you had yesterday..." the human mind cannot process hearing this phrase, without at some level searching internally for some thoughts or other that it had yesterday, to make the subject of the sentence. "The many colors that fruit can be" likewise starts the human mind considering even if briefly, different fruit sorted by color. "You did it again, didn't you!" This everyday manipulative use of TDS usually sends the recipient looking internally for some "it" they may have done for which blame is being fairly given. Regardless of whether such a matter can be identified, guilt or anger may result. "There has been pain, hasn't there" the mind of a patient suffering an illness will find it very hard or impossible to hear or answer this sentence without conducting internal searches to verify whether this is true or not, or to find an example if so. "You'd forgotten something [or: some part of your body], hadn't you?" the mind usually checks through the various things, or parts of the body, on hearing this, seeing if each in turn has been forgotten. Textual ambiguity: "Do you remember line dancing on the steps?" Without sufficient context, some statements may trigger TDS in order to resolve inherent ambiguity in the interpretation of a posed question. Do I remember a bygone fad called "line dancing on the steps"? Do I remember personally engaging in dancing in the past? Do I remember my routine practice dancing by focusing on the steps of the dance? Do I tend to forget about dancing when I am standing on steps? "Penny-wise and pound the table dance to the beat of a different drummer". The mixing of cliché and stock phrases may trigger TDS in order to reconcile the discrepancies between expected and actual utterances in sequence. Although TDS is often associated with spoken language, it can be induced in any perceptual system. Thus Milton Erickson's "hypnotic handshake" is a technique that leaves the other person performing TDS in search of meaning to a deliberately ambiguous use of touch.

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  Eigenmoments

  EigenMoments is a set of orthogonal, noise robust, invariant to rotation, scaling and translation and distribution sensitive moments. Their application can be found in signal processing and computer vision as descriptors of the signal or image. The descriptors can later be used for classification purposes. It is obtained by performing orthogonalization, via eigen analysis on geometric moments. == Framework summary == EigenMoments are computed by performing eigen analysis on the moment space of an image by maximizing signal-to-noise ratio in the feature space in form of Rayleigh quotient. This approach has several benefits in Image processing applications: Dependency of moments in the moment space on the distribution of the images being transformed, ensures decorrelation of the final feature space after eigen analysis on the moment space. The ability of EigenMoments to take into account distribution of the image makes it more versatile and adaptable for different genres. Generated moment kernels are orthogonal and therefore analysis on the moment space becomes easier. Transformation with orthogonal moment kernels into moment space is analogous to projection of the image onto a number of orthogonal axes. Nosiy components can be removed. This makes EigenMoments robust for classification applications. Optimal information compaction can be obtained and therefore a few number of moments are needed to characterize the images. == Problem formulation == Assume that a signal vector s ∈ R n {\displaystyle s\in {\mathcal {R}}^{n}} is taken from a certain distribution having correlation C ∈ R n × n {\displaystyle C\in {\mathcal {R}}^{n\times n}} , i.e. C = E [ s s T ] {\displaystyle C=E[ss^{T}]} where E[.] denotes expected value. Dimension of signal space, n, is often too large to be useful for practical application such as pattern classification, we need to transform the signal space into a space with lower dimensionality. This is performed by a two-step linear transformation: q = W T X T s , {\displaystyle q=W^{T}X^{T}s,} where q = [ q 1 , . . . , q n ] T ∈ R k {\displaystyle q=[q_{1},...,q_{n}]^{T}\in {\mathcal {R}}^{k}} is the transformed signal, X = [ x 1 , . . . , x n ] T ∈ R n × m {\displaystyle X=[x_{1},...,x_{n}]^{T}\in {\mathcal {R}}^{n\times m}} a fixed transformation matrix which transforms the signal into the moment space, and W = [ w 1 , . . . , w n ] T ∈ R m × k {\displaystyle W=[w_{1},...,w_{n}]^{T}\in {\mathcal {R}}^{m\times k}} the transformation matrix which we are going to determine by maximizing the SNR of the feature space resided by q {\displaystyle q} . For the case of Geometric Moments, X would be the monomials. If m = k = n {\displaystyle m=k=n} , a full rank transformation would result, however usually we have m ≤ n {\displaystyle m\leq n} and k ≤ m {\displaystyle k\leq m} . This is specially the case when n {\displaystyle n} is of high dimensions. Finding W {\displaystyle W} that maximizes the SNR of the feature space: S N R t r a n s f o r m = w T X T C X w w T X T N X w , {\displaystyle SNR_{transform}={\frac {w^{T}X^{T}CXw}{w^{T}X^{T}NXw}},} where N is the correlation matrix of the noise signal. The problem can thus be formulated as w 1 , . . . , w k = a r g m a x w w T X T C X w w T X T N X w {\displaystyle {w_{1},...,w_{k}}=argmax_{w}{\frac {w^{T}X^{T}CXw}{w^{T}X^{T}NXw}}} subject to constraints: w i T X T N X w j = δ i j , {\displaystyle w_{i}^{T}X^{T}NXw_{j}=\delta _{ij},} where δ i j {\displaystyle \delta _{ij}} is the Kronecker delta. It can be observed that this maximization is Rayleigh quotient by letting A = X T C X {\displaystyle A=X^{T}CX} and B = X T N X {\displaystyle B=X^{T}NX} and therefore can be written as: w 1 , . . . , w k = a r g m a x x w T A w w T B w {\displaystyle {w_{1},...,w_{k}}={\underset {x}{\operatorname {arg\,max} }}{\frac {w^{T}Aw}{w^{T}Bw}}} , w i T B w j = δ i j {\displaystyle w_{i}^{T}Bw_{j}=\delta _{ij}} === Rayleigh quotient === Optimization of Rayleigh quotient has the form: max w R ( w ) = max w w T A w w T B w {\displaystyle \max _{w}R(w)=\max _{w}{\frac {w^{T}Aw}{w^{T}Bw}}} and A {\displaystyle A} and B {\displaystyle B} , both are symmetric and B {\displaystyle B} is positive definite and therefore invertible. Scaling w {\displaystyle w} does not change the value of the object function and hence and additional scalar constraint w T B w = 1 {\displaystyle w^{T}Bw=1} can be imposed on w {\displaystyle w} and no solution would be lost when the objective function is optimized. This constraint optimization problem can be solved using Lagrangian multiplier: max w w T A w {\displaystyle \max _{w}{w^{T}Aw}} subject to w T B w = 1 {\displaystyle {w^{T}Bw}=1} max w L ( w ) = max w ( w T A w − λ w T B w ) {\displaystyle \max _{w}{\mathcal {L}}(w)=\max _{w}(w{T}Aw-\lambda w^{T}Bw)} equating first derivative to zero and we will have: A w = λ B w {\displaystyle Aw=\lambda Bw} which is an instance of Generalized Eigenvalue Problem (GEP). The GEP has the form: A w = λ B w {\displaystyle Aw=\lambda Bw} for any pair ( w , λ ) {\displaystyle (w,\lambda )} that is a solution to above equation, w {\displaystyle w} is called a generalized eigenvector and λ {\displaystyle \lambda } is called a generalized eigenvalue. Finding w {\displaystyle w} and λ {\displaystyle \lambda } that satisfies this equations would produce the result which optimizes Rayleigh quotient. One way of maximizing Rayleigh quotient is through solving the Generalized Eigen Problem. Dimension reduction can be performed by simply choosing the first components w i {\displaystyle w_{i}} , i = 1 , . . . , k {\displaystyle i=1,...,k} , with the highest values for R ( w ) {\displaystyle R(w)} out of the m {\displaystyle m} components, and discard the rest. Interpretation of this transformation is rotating and scaling the moment space, transforming it into a feature space with maximized SNR and therefore, the first k {\displaystyle k} components are the components with highest k {\displaystyle k} SNR values. The other method to look at this solution is to use the concept of simultaneous diagonalization instead of Generalized Eigen Problem. === Simultaneous diagonalization === Let A = X T C X {\displaystyle A=X^{T}CX} and B = X T N X {\displaystyle B=X^{T}NX} as mentioned earlier. We can write W {\displaystyle W} as two separate transformation matrices: W = W 1 W 2 . {\displaystyle W=W_{1}W_{2}.} W 1 {\displaystyle W_{1}} can be found by first diagonalize B: P T B P = D B {\displaystyle P^{T}BP=D_{B}} . Where D B {\displaystyle D_{B}} is a diagonal matrix sorted in increasing order. Since B {\displaystyle B} is positive definite, thus D B > 0 {\displaystyle D_{B}>0} . We can discard those eigenvalues that large and retain those close to 0, since this means the energy of the noise is close to 0 in this space, at this stage it is also possible to discard those eigenvectors that have large eigenvalues. Let P ^ {\displaystyle {\hat {P}}} be the first k {\displaystyle k} columns of P {\displaystyle P} , now P T ^ B P ^ = D B ^ {\displaystyle {\hat {P^{T}}}B{\hat {P}}={\hat {D_{B}}}} where D B ^ {\displaystyle {\hat {D_{B}}}} is the k × k {\displaystyle k\times k} principal submatrix of D B {\displaystyle D_{B}} . Let W 1 = P ^ D B ^ − 1 / 2 {\displaystyle W_{1}={\hat {P}}{\hat {D_{B}}}^{-1/2}} and hence: W 1 T B W 1 = ( P ^ D B ^ − 1 / 2 ) T B ( P ^ D B ^ − 1 / 2 ) = I {\displaystyle W_{1}^{T}BW_{1}=({\hat {P}}{\hat {D_{B}}}^{-1/2})^{T}B({\hat {P}}{\hat {D_{B}}}^{-1/2})=I} . W 1 {\displaystyle W_{1}} whiten B {\displaystyle B} and reduces the dimensionality from m {\displaystyle m} to k {\displaystyle k} . The transformed space resided by q ′ = W 1 T X T s {\displaystyle q'=W_{1}^{T}X^{T}s} is called the noise space. Then, we diagonalize W 1 T A W 1 {\displaystyle W_{1}^{T}AW_{1}} : W 2 T W 1 T A W 1 W 2 = D A {\displaystyle W_{2}^{T}W_{1}^{T}AW_{1}W_{2}=D_{A}} , where W 2 T W 2 = I {\displaystyle W_{2}^{T}W_{2}=I} . D A {\displaystyle D_{A}} is the matrix with eigenvalues of W 1 T A W 1 {\displaystyle W_{1}^{T}AW_{1}} on its diagonal. We may retain all the eigenvalues and their corresponding eigenvectors since most of the noise are already discarded in previous step. Finally the transformation is given by: W = W 1 W 2 {\displaystyle W=W_{1}W_{2}} where W {\displaystyle W} diagonalizes both the numerator and denominator of the SNR, W T A W = D A {\displaystyle W^{T}AW=D_{A}} , W T B W = I {\displaystyle W^{T}BW=I} and the transformation of signal s {\displaystyle s} is defined as q = W T X T s = W 2 T W 1 T X T s {\displaystyle q=W^{T}X^{T}s=W_{2}^{T}W_{1}^{T}X^{T}s} . === Information loss === To find the information loss when we discard some of the eigenvalues and eigenvectors we can perform following analysis: η = 1 − t r a c e ( W 1 T A W 1 ) t r a c e ( D B − 1 / 2 P T A P D B − 1 / 2 ) = 1 − t r a c e ( D B ^ − 1 / 2 P ^ T A P ^ D B ^ − 1 / 2 ) t r a c e ( D B − 1 / 2 P T A P D B − 1 / 2 ) {\displaystyle {\begin{array}{lll}\eta &=&

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