AI Chat Bots Like Character AI

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  Image destriping

  Image destriping is the process of removing stripes or streaks from images and videos without disrupting the original image/video. These artifacts plague a range of fields in scientific imaging including atomic force microscopy, light sheet fluorescence microscopy, and planetary satellite imaging. The most common image processing techniques to reduce stripe artifacts is with Fourier filtering. Unfortunately, filtering methods risk altering or suppressing useful image data. Methods developed for multiple-sensor imaging systems in planetary satellites use statistical-based methods to match signal distribution across multiple sensors. More recently, a new class of approaches leverage compressed sensing, to regularize an optimization problem, and recover stripe free images. In many cases, these destriped images have little to no artifacts, even at low signal to noise ratios.

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  Multilinear principal component analysis

  Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac, or by multilinear tensor models, such as multilinear principal component analysis (MPCA) or multilinear (tensor) independent component analysis (MICA). In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between multilinear data models that employed 2nd order statistics versus higher order statistics to compute a set of independent components for each mode, such as Multilinear ICA Multilinear PCA may be applied to compute the causal factors of data formation, or as signal processing tool on data tensors whose individual observation have either been vectorized, or whose observations are treated as a collection of column/row observations, an "observation as a matrix", and concatenated into a data tensor. The latter approach is suitable for compression and reducing redundancy in the rows, columns and fibers that are unrelated to the causal factors of data formation. Vasilescu and Terzopoulos in their paper "TensorFaces" introduced the M-mode SVD algorithm which are algorithms misidentified in the literature as the HOSVD or the Tucker which employ the power method or gradient descent, respectively. Vasilescu and Terzopoulos framed the data analysis, recognition and synthesis problems as multilinear tensor problems. Data is viewed as the compositional consequence of several causal factors, that are well suited for multi-modal tensor factor analysis. The power of the tensor framework was showcased by analyzing human motion joint angles, facial images or textures in the following papers: Human Motion Signatures (CVPR 2001, ICPR 2002), face recognition – TensorFaces, (ECCV 2002, CVPR 2003, etc.) and computer graphics – TensorTextures (Siggraph 2004). == The algorithm == The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent. == Feature selection == MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition while a semi-supervised MPCA feature selection is employed in visualization tasks. == Extensions == Various extension of MPCA: Robust MPCA (RMPCA) Multi-Tensor Factorization, that also finds the number of components automatically (MTF)

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  Sammon mapping

  Sammon mapping or Sammon projection is an algorithm that maps a high-dimensional space to a space of lower dimensionality (see multidimensional scaling) by trying to preserve the structure of inter-point distances in high-dimensional space in the lower-dimension projection. It is particularly suited for use in exploratory data analysis. The method was proposed by John W. Sammon in 1969. It is considered a non-linear approach as the mapping cannot be represented as a linear combination of the original variables as possible in techniques such as principal component analysis, which also makes it more difficult to use for classification applications. Denote the distance between ith and jth objects in the original space by d i j ∗ {\displaystyle \scriptstyle d_{ij}^{}} , and the distance between their projections by d i j {\displaystyle \scriptstyle d_{ij}^{}} . Sammon's mapping aims to minimize the following error function, which is often referred to as Sammon's stress or Sammon's error: E = 1 ∑ i < j d i j ∗ ∑ i < j ( d i j ∗ − d i j ) 2 d i j ∗ . {\displaystyle E={\frac {1}{\sum \limits _{i Read more →

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  Recursive neural network

  A recursive neural network is a kind of deep neural network created by applying the same set of weights recursively over a structured input, to produce a structured prediction over variable-size input structures, or a scalar prediction on it, by traversing a given structure in topological order. These networks were first introduced to learn distributed representations of structure (such as logical terms), but have been successful in multiple applications, for instance in learning sequence and tree structures in natural language processing (mainly continuous representations of phrases and sentences based on word embeddings). == Architectures == === Basic === In the simplest architecture, nodes are combined into parents using a weight matrix (which is shared across the whole network) and a non-linearity such as the tanh {\displaystyle \tanh } hyperbolic function. If c 1 {\displaystyle c_{1}} and c 2 {\displaystyle c_{2}} are n {\displaystyle n} -dimensional vector representations of nodes, their parent will also be an n {\displaystyle n} -dimensional vector, defined as: p 1 , 2 = tanh ⁡ ( W [ c 1 ; c 2 ] ) {\displaystyle p_{1,2}=\tanh(W[c_{1};c_{2}])} where W {\displaystyle W} is a learned n × 2 n {\displaystyle n\times 2n} weight matrix. This architecture, with a few improvements, has been used for successfully parsing natural scenes, syntactic parsing of natural language sentences, and recursive autoencoding and generative modeling of 3D shape structures in the form of cuboid abstractions. === Recursive cascade correlation (RecCC) === RecCC is a constructive neural network approach to deal with tree domains with pioneering applications to chemistry and extension to directed acyclic graphs. === Unsupervised RNN === A framework for unsupervised RNN has been introduced in 2004. === Tensor === Recursive neural tensor networks use a single tensor-based composition function for all nodes in the tree. == Training == === Stochastic gradient descent === Typically, stochastic gradient descent (SGD) is used to train the network. The gradient is computed using backpropagation through structure (BPTS), a variant of backpropagation through time used for recurrent neural networks. == Properties == The universal approximation capability of RNNs over trees has been proved in literature. == Related models == === Recurrent neural networks === Recurrent neural networks are recursive artificial neural networks with a certain structure: that of a linear chain. Whereas recursive neural networks operate on any hierarchical structure, combining child representations into parent representations, recurrent neural networks operate on the linear progression of time, combining the previous time step and a hidden representation into the representation for the current time step. === Tree Echo State Networks === An efficient approach to implement recursive neural networks is given by the Tree Echo State Network within the reservoir computing paradigm. === Extension to graphs === Extensions to graphs include graph neural network (GNN), Neural Network for Graphs (NN4G), and more recently convolutional neural networks for graphs.

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  Medical data breach

  Medical data, including patients' identity information, health status, disease diagnosis and treatment, and biogenetic information, not only involve patients' privacy but also have a special sensitivity and important value, which may bring physical and mental distress and property loss to patients and even negatively affect social stability and national security once leaked. However, the development and application of medical AI must rely on a large amount of medical data for algorithm training, and the larger and more diverse the amount of data, the more accurate the results of its analysis and prediction will be. However, the application of big data technologies such as data collection, analysis and processing, cloud storage, and information sharing has increased the risk of data leakage. In the United States, the rate of such breaches has increased over time, with 176 million records breached by the end of 2017. By 2024, the U.S. Department of Health and Human Services reported 725 large healthcare data breaches affecting approximately 275 million individual records in a single year, marking a significant escalation in both the frequency and scale of incidents. == Black market for health data == In February 2015 an NPR report claimed that organized crime networks had ways of selling health data in the black market. In 2015 a Beazley employee estimated that medical records could sell on the black market for US$40-50. == How data is lost == Theft, data loss, hacking, and unauthorized account access are ways in which medical data breaches happen. Among reported breaches of medical information in the United States networked information systems accounted for the largest number of records breached. There are many data breaches happening in the US health care system, among business associates of the health care providers that continuously gain access to patients' data. == List of data breaches == In February 2024, a ransomware attack on Change Healthcare, a subsidiary of UnitedHealth Group, compromised the protected health information of approximately 100 million individuals, making it the largest healthcare data breach in United States history. The attack disrupted claims processing for healthcare providers nationwide for several weeks. In May 2024, MediSecure suffered a cyberattack involving ransomware in Australia. In May 2021, the Health Service Executive in the Republic of Ireland was the victim of a cyberattack involving ransomware, in the Health Service Executive cyberattack, with admission records and test results present in a sample of the data reviewed by the Financial Times. In October 2018, the Centers for Medicare and Medicaid Services in the US reported that around 75,000 individual records had been affected by a data breach that took place through the ACA Agent and Broker Portal. In 2018, Social Indicators Research published the scientific evidence of 173,398,820 (over 173 million) individuals affected in USA from October 2008 (when the data were collected) to September 2017 (when the statistical analysis took place). In 2015, Anthem Inc. lost data for 37 million people in the Anthem medical data breach In 2014 4.5 million people using Complete Health Systems had their data stolen In 2013-14 1 million people using Montana Department of Public Health and Human Services had their data stolen In 2013 4 million people using Advocate Health and Hospitals Corporation had their data stolen In 2011 4.9 million users of Tricare services had their data stolen due to an employee error by Science Applications International Corporation In 2011 1.9 million people using Health Net had their data stolen In 2011 1 million people using Nemours Foundation had their data stolen In 2010 6800 people using New York-Presbyterian Hospital and Columbia University Medical Center had their data breached. In response, those organizations agreed to pay the United States Department of Health and Human Services a US$4.8 million dollar fine. In 2009 1 million people using BlueCross BlueShield of Tennessee had their data stolen == Regulation == In the United States, the Health Insurance Portability and Accountability Act and Health Information Technology for Economic and Clinical Health Act require companies to report data breaches to affected individuals and the federal government. Under the HIPAA Breach Notification Rule, covered entities must notify affected individuals without unreasonable delay and no later than 60 days after discovering a breach of unsecured protected health information. Breaches affecting 500 or more individuals must also be reported to the HHS Secretary and to prominent media outlets serving the affected state or jurisdiction within the same timeframe; HHS publicly lists these larger breaches on its breach portal, commonly known as the "wall of shame." Breaches affecting fewer than 500 individuals are reported to HHS annually, no later than 60 days after the end of the calendar year in which they were discovered. Health Information Privacy Health Insurance Portability and Accountability Act of 1996 (HIPAA). - 45 CFR Parts 160 and 164, Standards for Privacy of Individually Identifiable Health Information and Security Standards for the Protection of Electronic Protected Health Information. HIPAA includes provisions designed to save health care businesses money by encouraging electronic transactions, as well as regulations to protect the security and confidentiality of patient information. The Privacy Rule became effective April 14, 2001, and most covered entities (health plans, health care clearinghouses, and health care providers that conduct certain financial and administrative transactions electronically) had until April 2003 to comply. This security provision became effective April 21, 2003. The Health Insurance Portability and Accountability Act (HIPAA) is the baseline set of federal regulations governing medical information. It does three things: i. i. i.Establish a structure for how personal health information is disclosed and establish the rights of individuals with respect to health information; ii.Specify security standards for the retention and transmission of electronic patient information; iii.Need a common format and data structure for the electronic exchange of health information. California-Specific Laws California’s medical privacy laws, primarily the Confidentiality of Medical Information Act (CMIA), the data breach sections of the Civil Code, and sections of the Health and Safety Code, provide HIPAA-like protections, although the terminology is different. HIPAA establishes a federal "minimum standard" that applies where there are gaps in California law, and HIPAA also specifies that stricter state laws will override or supersede HIPAA. California's health care privacy laws apply to providers who provide personal health records (PHR), while HIPAA only applies when the provider providing the PHR is a business associate of a covered entity. Federal law does not grant individuals the right to file a lawsuit in the event of a data breach (only the Attorney General can file a lawsuit), but California law does. This means that California law sets a higher standard for medical privacy, and that individuals in California enjoy stronger legal protections and more ways to hold entities that violate their medical privacy accountable. In the UK, the legal framework for how patient data is cared for and processed is the Data Protection Act 2018 (DPA), which incorporates the EU General Data Protection Regulation (GDPR) into law, and the common law duty of confidentiality (CLDC). The data protection legislation requires that the collection and processing of personal data be fair, lawful and transparent. This means that the collection and processing of data as defined by data protection legislation must always have a valid lawful basis and must also meet the requirements of the CLDC. In the China, Article 18 of the "National Health Care Big Data Standards, Security and Services Management Measures (for Trial Implementation)" (National Health Planning and Development (2018) No. 23) promulgated by the National Health Care Commission in 2018 states, "The responsible unit shall adopt measures such as data classification, important data backup, and encryption authentication to guarantee the security of health care big data." However, the scope and definition of important data are not covered. Although the "Information Security Technology-Healthcare Data Security Guide" (the "Guide") issued by the National Standardization Committee also proposes that important data should be evaluated and approved in accordance with the regulations, there is likewise no definition of the connotation and definition of important data.

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  Winner-take-all (computing)

  Winner-take-all is a computational principle applied in computational models of neural networks by which neurons compete with each other for activation. In the classical form, only the neuron with the highest activation stays active while all other neurons shut down; however, other variations allow more than one neuron to be active, for example the soft winner take-all, by which a power function is applied to the neurons. == Neural networks == In the theory of artificial neural networks, winner-take-all networks are a case of competitive learning in recurrent neural networks. Output nodes in the network mutually inhibit each other, while simultaneously activating themselves through reflexive connections. After some time, only one node in the output layer will be active, namely the one corresponding to the strongest input. Thus the network uses nonlinear inhibition to pick out the largest of a set of inputs. Winner-take-all is a general computational primitive that can be implemented using different types of neural network models, including both continuous-time and spiking networks. Winner-take-all networks are commonly used in computational models of the brain, particularly for distributed decision-making or action selection in the cortex. Important examples include hierarchical models of vision, and models of selective attention and recognition. They are also common in artificial neural networks and neuromorphic analog VLSI circuits. It has been formally proven that the winner-take-all operation is computationally powerful compared to other nonlinear operations, such as thresholding. In many practical cases, there is not only one single neuron which becomes active but there are exactly k neurons which become active for a fixed number k. This principle is referred to as k-winners-take-all. === Example algorithm === Consider a single linear neuron, with inputs x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} . Each input has weight w i {\displaystyle w_{i}} , and the output of the neuron is ∑ i w i x i {\displaystyle \sum _{i}w_{i}x_{i}} . In the Instar learning rule, on each input vector, the weight vectors are modified according to Δ w i = η ( x i − w i ) {\displaystyle \Delta w_{i}=\eta (x_{i}-w_{i})} where η {\displaystyle \eta } is the learning rate. This rule is unsupervised, since we need just the input vector, not a reference output. Now, consider multiple linear neurons y 1 , … , y m {\displaystyle y_{1},\dots ,y_{m}} . The output of each satisfies y i = ∑ j w i j x j {\displaystyle y_{i}=\sum _{j}w_{ij}x_{j}} . In the winner-take-all algorithm, the weights are modified as follows. Given an input vector x {\displaystyle x} , each output is computed. The neuron with the largest output is selected, and the weights going into that neuron are modified according to the Instar learning rule. All other weights remain unchanged. The k-winners-take-all rule is similar, except that the Instar learning rule is applied to the weights going into the k neurons with the largest outputs. == Circuit example == A simple, but popular CMOS winner-take-all circuit is shown on the right. This circuit was originally proposed by Lazzaro et al. (1989) using MOS transistors biased to operate in the weak-inversion or subthreshold regime. In the particular case shown there are only two inputs (IIN,1 and IIN,2), but the circuit can be easily extended to multiple inputs in a straightforward way. It operates on continuous-time input signals (currents) in parallel, using only two transistors per input. In addition, the bias current IBIAS is set by a single global transistor that is common to all the inputs. The largest of the input currents sets the common potential VC. As a result, the corresponding output carries almost all the bias current, while the other outputs have currents that are close to zero. Thus, the circuit selects the larger of the two input currents, i.e., if IIN,1 > IIN,2, we get IOUT,1 = IBIAS and IOUT,2 = 0. Similarly, if IIN,2 > IIN,1, we get IOUT,1 = 0 and IOUT,2 = IBIAS. A SPICE-based DC simulation of the CMOS winner-take-all circuit in the two-input case is shown on the right. As shown in the top subplot, the input IIN,1 was fixed at 6nA, while IIN,2 was linearly increased from 0 to 10nA. The bottom subplot shows the two output currents. As expected, the output corresponding to the larger of the two inputs carries the entire bias current (10nA in this case), forcing the other output current nearly to zero. == Other uses == In stereo matching algorithms, following the taxonomy proposed by Scharstein and Szelliski, winner-take-all is a local method for disparity computation. Adopting a winner-take-all strategy, the disparity associated with the minimum or maximum cost value is selected at each pixel. It is axiomatic that in the electronic commerce market, early dominant players such as AOL or Yahoo! get most of the rewards. By 1998, one study found the top 5% of all web sites garnered more than 74% of all traffic. The winner-take-all hypothesis in economics suggests that once a technology or a firm gets ahead, it will do better and better over time, whereas lagging technology and firms will fall further behind. See First-mover advantage.

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  Effective fitness

  In natural evolution and artificial evolution (e.g. artificial life and evolutionary computation) the fitness (or performance or objective measure) of a schema is rescaled to give its effective fitness which takes into account crossover and mutation. Effective fitness is used in Evolutionary Computation to understand population dynamics. While a biological fitness function only looks at reproductive success, an effective fitness function tries to encompass things that are needed to be fulfilled for survival on population level. In homogeneous populations, reproductive fitness and effective fitness are equal. When a population moves away from homogeneity a higher effective fitness is reached for the recessive genotype. This advantage will decrease while the population moves toward an equilibrium. The deviation from this equilibrium displays how close the population is to achieving a steady state. When this equilibrium is reached, the maximum effective fitness of the population is achieved. Problem solving with evolutionary computation is realized with a cost function. If cost functions are applied to swarm optimization they are called a fitness function. Strategies like reinforcement learning and NEAT neuroevolution are creating a fitness landscape which describes the reproductive success of cellular automata. The effective fitness function models the number of fit offspring and is used in calculations that include evolutionary processes, such as mutation and crossover, important on the population level. The effective fitness model is superior to its predecessor, the standard reproductive fitness model. It advances in the qualitatively and quantitatively understanding of evolutionary concepts like bloat, self-adaptation, and evolutionary robustness. While reproductive fitness only looks at pure selection, effective fitness describes the flow of a population and natural selection by taking genetic operators into account. A normal fitness function fits to a problem, while an effective fitness function is an assumption if the objective was reached. The difference is important for designing fitness functions with algorithms like novelty search in which the objective of the agents is unknown. In the case of bacteria effective fitness could include production of toxins and rate of mutation of different plasmids, which are mostly stochastically determined == Applications == When evolutionary equations of the studied population dynamics are available, one can algorithmically compute the effective fitness of a given population. Though the perfect effective fitness model is yet to be found, it is already known to be a good framework to the better understanding of the moving of the genotype-phenotype map, population dynamics, and the flow on fitness landscapes. Models using a combination of Darwinian fitness functions and effective functions are better at predicting population trends. Effective models could be used to determine therapeutic outcomes of disease treatment. Other models could determine effective protein engineering and works towards finding novel or heightened biochemistry.

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  Rule-based machine learning

  Rule-based machine learning (RBML) is a term in computer science intended to encompass any machine learning method that identifies, learns, or evolves 'rules' to store, manipulate or apply. The defining characteristic of a rule-based machine learner is the identification and utilization of a set of relational rules that collectively represent the knowledge captured by the system. Rule-based machine learning approaches include learning classifier systems, association rule learning, artificial immune systems, and any other method that relies on a set of rules, each covering contextual knowledge. While rule-based machine learning is conceptually a type of rule-based system, it is distinct from traditional rule-based systems, which are often hand-crafted, and other rule-based decision makers. This is because rule-based machine learning applies some form of learning algorithm such as Rough sets theory to identify and minimise the set of features and to automatically identify useful rules, rather than a human needing to apply prior domain knowledge to manually construct rules and curate a rule set. == Rules == Rules typically take the form of an '{IF:THEN} expression', (e.g. {IF 'condition' THEN 'result'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign}). An individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Therefore rule-based machine learning methods typically comprise a set of rules, or knowledge base, that collectively make up the prediction model usually known as decision algorithm. Rules can also be interpreted in various ways depending on the domain knowledge, data types(discrete or continuous) and in combinations. == RIPPER == Repeated incremental pruning to produce error reduction (RIPPER) is a propositional rule learner proposed by William W. Cohen as an optimized version of IREP.

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  Soterml

  SoTerML (Soil and Terrain Markup Language) is a XML-based markup language for storing and exchanging soil and terrain related data. SoTerML development is being done within The e-SoTer Platform. GEOSS plans a global Earth Observation System and, within this framework, the e-SOTER project addresses the felt need for a global soil and terrain database. The Centre for Geospatial Science (Currently Nottingham Geospatial Institute) at the University of Nottingham has initiated the development since January 2009. Further development and maintenance is currently handled in National Soil Resources Institute (NSRI) at Cranfield University, UK. The role of CGS is within the development of the e-SOTER dissemination platform, which is based on INSPIRE principles. The SoTerML development included: 1. Development of a data dictionary for nomenclatures and various data sources (data and metadata). 2. Development of an exchange format/procedures from the World Reference Base 2006.

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  BrownBoost

  BrownBoost is a boosting algorithm that may be robust to noisy datasets. BrownBoost is an adaptive version of the boost by majority algorithm. As is the case for all boosting algorithms, BrownBoost is used in conjunction with other machine learning methods. BrownBoost was introduced by Yoav Freund in 2001. == Motivation == AdaBoost performs well on a variety of datasets; however, it can be shown that AdaBoost does not perform well on noisy data sets. This is a result of AdaBoost's focus on examples that are repeatedly misclassified. In contrast, BrownBoost effectively "gives up" on examples that are repeatedly misclassified. The core assumption of BrownBoost is that noisy examples will be repeatedly mislabeled by the weak hypotheses and non-noisy examples will be correctly labeled frequently enough to not be "given up on." Thus only noisy examples will be "given up on," whereas non-noisy examples will contribute to the final classifier. In turn, if the final classifier is learned from the non-noisy examples, the generalization error of the final classifier may be much better than if learned from noisy and non-noisy examples. The user of the algorithm can set the amount of error to be tolerated in the training set. Thus, if the training set is noisy (say 10% of all examples are assumed to be mislabeled), the booster can be told to accept a 10% error rate. Since the noisy examples may be ignored, only the true examples will contribute to the learning process. == Algorithm description == BrownBoost uses a non-convex potential loss function, thus it does not fit into the AdaBoost framework. The non-convex optimization provides a method to avoid overfitting noisy data sets. However, in contrast to boosting algorithms that analytically minimize a convex loss function (e.g. AdaBoost and LogitBoost), BrownBoost solves a system of two equations and two unknowns using standard numerical methods. The only parameter of BrownBoost ( c {\displaystyle c} in the algorithm) is the "time" the algorithm runs. The theory of BrownBoost states that each hypothesis takes a variable amount of time ( t {\displaystyle t} in the algorithm) which is directly related to the weight given to the hypothesis α {\displaystyle \alpha } . The time parameter in BrownBoost is analogous to the number of iterations T {\displaystyle T} in AdaBoost. A larger value of c {\displaystyle c} means that BrownBoost will treat the data as if it were less noisy and therefore will give up on fewer examples. Conversely, a smaller value of c {\displaystyle c} means that BrownBoost will treat the data as more noisy and give up on more examples. During each iteration of the algorithm, a hypothesis is selected with some advantage over random guessing. The weight of this hypothesis α {\displaystyle \alpha } and the "amount of time passed" t {\displaystyle t} during the iteration are simultaneously solved in a system of two non-linear equations ( 1. uncorrelated hypothesis w.r.t example weights and 2. hold the potential constant) with two unknowns (weight of hypothesis α {\displaystyle \alpha } and time passed t {\displaystyle t} ). This can be solved by bisection (as implemented in the JBoost software package) or Newton's method (as described in the original paper by Freund). Once these equations are solved, the margins of each example ( r i ( x j ) {\displaystyle r_{i}(x_{j})} in the algorithm) and the amount of time remaining s {\displaystyle s} are updated appropriately. This process is repeated until there is no time remaining. The initial potential is defined to be 1 m ∑ j = 1 m 1 − erf ( c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}({\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Since a constraint of each iteration is that the potential be held constant, the final potential is 1 m ∑ j = 1 m 1 − erf ( r i ( x j ) / c ) = 1 − erf ( c ) {\displaystyle {\frac {1}{m}}\sum _{j=1}^{m}1-{\mbox{erf}}(r_{i}(x_{j})/{\sqrt {c}})=1-{\mbox{erf}}({\sqrt {c}})} . Thus the final error is likely to be near 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} . However, the final potential function is not the 0–1 loss error function. For the final error to be exactly 1 − erf ( c ) {\displaystyle 1-{\mbox{erf}}({\sqrt {c}})} , the variance of the loss function must decrease linearly w.r.t. time to form the 0–1 loss function at the end of boosting iterations. This is not yet discussed in the literature and is not in the definition of the algorithm below. The final classifier is a linear combination of weak hypotheses and is evaluated in the same manner as most other boosting algorithms. == BrownBoost learning algorithm definition == Input: m {\displaystyle m} training examples ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x j ∈ X , y j ∈ Y = { − 1 , + 1 } {\displaystyle x_{j}\in X,\,y_{j}\in Y=\{-1,+1\}} The parameter c {\displaystyle c} Initialise: s = c {\displaystyle s=c} . (The value of s {\displaystyle s} is the amount of time remaining in the game) r i ( x j ) = 0 {\displaystyle r_{i}(x_{j})=0} ∀ j {\displaystyle \forall j} . The value of r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin at iteration i {\displaystyle i} for example x j {\displaystyle x_{j}} . While s > 0 {\displaystyle s>0} : Set the weights of each example: W i ( x j ) = e − ( r i ( x j ) + s ) 2 c {\displaystyle W_{i}(x_{j})=e^{-{\frac {(r_{i}(x_{j})+s)^{2}}{c}}}} , where r i ( x j ) {\displaystyle r_{i}(x_{j})} is the margin of example x j {\displaystyle x_{j}} Find a classifier h i : X → { − 1 , + 1 } {\displaystyle h_{i}:X\to \{-1,+1\}} such that ∑ j W i ( x j ) h i ( x j ) y j > 0 {\displaystyle \sum _{j}W_{i}(x_{j})h_{i}(x_{j})y_{j}>0} Find values α , t {\displaystyle \alpha ,t} that satisfy the equation: ∑ j h i ( x j ) y j e − ( r i ( x j ) + α h i ( x j ) y j + s − t ) 2 c = 0 {\displaystyle \sum _{j}h_{i}(x_{j})y_{j}e^{-{\frac {(r_{i}(x_{j})+\alpha h_{i}(x_{j})y_{j}+s-t)^{2}}{c}}}=0} . (Note this is similar to the condition E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} set forth by Schapire and Singer. In this setting, we are numerically finding the W i + 1 = exp ⁡ ( ⋯ ⋯ ) {\displaystyle W_{i+1}=\exp \left({\frac {\cdots }{\cdots }}\right)} such that E W i + 1 [ h i ( x j ) y j ] = 0 {\displaystyle E_{W_{i+1}}[h_{i}(x_{j})y_{j}]=0} .) This update is subject to the constraint ∑ ( Φ ( r i ( x j ) + α h ( x j ) y j + s − t ) − Φ ( r i ( x j ) + s ) ) = 0 {\displaystyle \sum \left(\Phi \left(r_{i}(x_{j})+\alpha h(x_{j})y_{j}+s-t\right)-\Phi \left(r_{i}(x_{j})+s\right)\right)=0} , where Φ ( z ) = 1 − erf ( z / c ) {\displaystyle \Phi (z)=1-{\mbox{erf}}(z/{\sqrt {c}})} is the potential loss for a point with margin r i ( x j ) {\displaystyle r_{i}(x_{j})} Update the margins for each example: r i + 1 ( x j ) = r i ( x j ) + α h ( x j ) y j {\displaystyle r_{i+1}(x_{j})=r_{i}(x_{j})+\alpha h(x_{j})y_{j}} Update the time remaining: s = s − t {\displaystyle s=s-t} Output: H ( x ) = sign ( ∑ i α i h i ( x ) ) {\displaystyle H(x)={\textrm {sign}}\left(\sum _{i}\alpha _{i}h_{i}(x)\right)} == Empirical results == In preliminary experimental results with noisy datasets, BrownBoost outperformed AdaBoost's generalization error; however, LogitBoost performed as well as BrownBoost. An implementation of BrownBoost can be found in the open source software JBoost.

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  Hinge loss

  In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs). For an intended output t = ±1 and a classifier score y, the hinge loss of the prediction y is defined as ℓ ( y ) = max ( 0 , 1 − t ⋅ y ) {\displaystyle \ell (y)=\max(0,1-t\cdot y)} Note that y {\displaystyle y} should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, y = w ⋅ x + b {\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b} , where ( w , b ) {\displaystyle (\mathbf {w} ,b)} are the parameters of the hyperplane and x {\displaystyle \mathbf {x} } is the input variable(s). When t and y have the same sign (meaning y predicts the right class) and | y | ≥ 1 {\displaystyle |y|\geq 1} , the hinge loss ℓ ( y ) = 0 {\displaystyle \ell (y)=0} . When they have opposite signs, ℓ ( y ) {\displaystyle \ell (y)} increases linearly with y, and similarly if | y | < 1 {\displaystyle |y|<1} , even if it has the same sign (correct prediction, but not by enough margin). The Hinge loss is not a proper scoring rule. == Extensions == While binary SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion, it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed. For example, Crammer and Singer defined it for a linear classifier as ℓ ( y ) = max ( 0 , 1 + max y ≠ t w y x − w t x ) {\displaystyle \ell (y)=\max(0,1+\max _{y\neq t}\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} , where t {\displaystyle t} is the target label, w t {\displaystyle \mathbf {w} _{t}} and w y {\displaystyle \mathbf {w} _{y}} are the model parameters. Weston and Watkins provided a similar definition, but with a sum rather than a max: ℓ ( y ) = ∑ y ≠ t max ( 0 , 1 + w y x − w t x ) {\displaystyle \ell (y)=\sum _{y\neq t}\max(0,1+\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} . In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs with margin rescaling use the following variant, where w denotes the SVM's parameters, y the SVM's predictions, φ the joint feature function, and Δ the Hamming loss: ℓ ( y ) = max ( 0 , Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ − ⟨ w , ϕ ( x , t ) ⟩ ) = max ( 0 , max y ∈ Y ( Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ ) − ⟨ w , ϕ ( x , t ) ⟩ ) {\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\max _{y\in {\mathcal {Y}}}\left(\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle \right)-\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\end{aligned}}} . == Optimization == The hinge loss is a convex function, so many of the usual convex optimizers used in machine learning can work with it. It is not differentiable, but has a subgradient with respect to model parameters w of a linear SVM with score function y = w ⋅ x {\displaystyle y=\mathbf {w} \cdot \mathbf {x} } that is given by ∂ ℓ ∂ w i = { − t ⋅ x i if t ⋅ y < 1 , 0 otherwise . {\displaystyle {\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1,\\0&{\text{otherwise}}.\end{cases}}} However, since the derivative of the hinge loss at t y = 1 {\displaystyle ty=1} is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's ℓ ( y ) = { 1 2 − t y if t y ≤ 0 , 1 2 ( 1 − t y ) 2 if 0 < t y < 1 , 0 if 1 ≤ t y {\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0 Read more →

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  Elastic net regularization

  In statistics and, in particular, in the fitting of linear or logistic regression models, the elastic net is a regularized regression method that linearly combines the L1 and L2 penalties of the lasso and ridge methods. Nevertheless, elastic net regularization is typically more accurate than both methods with regard to reconstruction. == Specification == The elastic net method overcomes the limitations of the LASSO (least absolute shrinkage and selection operator) method which uses a penalty function based on ‖ β ‖ 1 = ∑ j = 1 p | β j | . {\displaystyle \|\beta \|_{1}=\textstyle \sum _{j=1}^{p}|\beta _{j}|.} Use of this penalty function has several limitations. For example, in the "large p, small n" case (high-dimensional data with few examples), the LASSO selects at most n variables before it saturates. Also if there is a group of highly correlated variables, then the LASSO tends to select one variable from a group and ignore the others. To overcome these limitations, the elastic net adds a quadratic part ( ‖ β ‖ 2 {\displaystyle \|\beta \|^{2}} ) to the penalty, which when used alone is ridge regression (known also as Tikhonov regularization). The estimates from the elastic net method are defined by β ^ ≡ argmin β ( ‖ y − X β ‖ 2 + λ 2 ‖ β ‖ 2 + λ 1 ‖ β ‖ 1 ) . {\displaystyle {\hat {\beta }}\equiv {\underset {\beta }{\operatorname {argmin} }}(\|y-X\beta \|^{2}+\lambda _{2}\|\beta \|^{2}+\lambda _{1}\|\beta \|_{1}).} The quadratic penalty term makes the loss function strongly convex, and it therefore has a unique minimum. The elastic net method includes the LASSO and ridge regression: in other words, each of them is a special case where λ 1 = λ , λ 2 = 0 {\displaystyle \lambda _{1}=\lambda ,\lambda _{2}=0} or λ 1 = 0 , λ 2 = λ {\displaystyle \lambda _{1}=0,\lambda _{2}=\lambda } . Meanwhile, the naive version of elastic net method finds an estimator in a two-stage procedure : first for each fixed λ 2 {\displaystyle \lambda _{2}} it finds the ridge regression coefficients, and then does a LASSO type shrinkage. This kind of estimation incurs a double amount of shrinkage, which leads to increased bias and poor predictions. To improve the prediction performance, sometimes the coefficients of the naive version of elastic net is rescaled by multiplying the estimated coefficients by ( 1 + λ 2 ) {\displaystyle (1+\lambda _{2})} . Examples of where the elastic net method has been applied are: Support vector machine Metric learning Portfolio optimization Cancer prognosis == Reduction to support vector machine == It was proven in 2014 that the elastic net can be reduced to the linear support vector machine. A similar reduction was previously proven for the LASSO in 2014. The authors showed that for every instance of the elastic net, an artificial binary classification problem can be constructed such that the hyper-plane solution of a linear support vector machine (SVM) is identical to the solution β {\displaystyle \beta } (after re-scaling). The reduction immediately enables the use of highly optimized SVM solvers for elastic net problems. It also enables the use of GPU acceleration, which is often already used for large-scale SVM solvers. The reduction is a simple transformation of the original data and regularization constants X ∈ R n × p , y ∈ R n , λ 1 ≥ 0 , λ 2 ≥ 0 {\displaystyle X\in {\mathbb {R} }^{n\times p},y\in {\mathbb {R} }^{n},\lambda _{1}\geq 0,\lambda _{2}\geq 0} into new artificial data instances and a regularization constant that specify a binary classification problem and the SVM regularization constant X 2 ∈ R 2 p × n , y 2 ∈ { − 1 , 1 } 2 p , C ≥ 0. {\displaystyle X_{2}\in {\mathbb {R} }^{2p\times n},y_{2}\in \{-1,1\}^{2p},C\geq 0.} Here, y 2 {\displaystyle y_{2}} consists of binary labels − 1 , 1 {\displaystyle {-1,1}} . When 2 p > n {\displaystyle 2p>n} it is typically faster to solve the linear SVM in the primal, whereas otherwise the dual formulation is faster. Some authors have referred to the transformation as Support Vector Elastic Net (SVEN), and provided the following MATLAB pseudo-code: == Software == "Glmnet: Lasso and elastic-net regularized generalized linear models" is a software which is implemented as an R source package and as a MATLAB toolbox. This includes fast algorithms for estimation of generalized linear models with ℓ1 (the lasso), ℓ2 (ridge regression) and mixtures of the two penalties (the elastic net) using cyclical coordinate descent, computed along a regularization path. JMP Pro 11 includes elastic net regularization, using the Generalized Regression personality with Fit Model. "pensim: Simulation of high-dimensional data and parallelized repeated penalized regression" implements an alternate, parallelised "2D" tuning method of the ℓ parameters, a method claimed to result in improved prediction accuracy. scikit-learn includes linear regression and logistic regression with elastic net regularization. SVEN, a Matlab implementation of Support Vector Elastic Net. This solver reduces the Elastic Net problem to an instance of SVM binary classification and uses a Matlab SVM solver to find the solution. Because SVM is easily parallelizable, the code can be faster than Glmnet on modern hardware. SpaSM, a Matlab implementation of sparse regression, classification and principal component analysis, including elastic net regularized regression. Apache Spark provides support for Elastic Net Regression in its MLlib machine learning library. The method is available as a parameter of the more general LinearRegression class. SAS (software) The SAS procedure Glmselect and SAS Viya procedure Regselect support the use of elastic net regularization for model selection.

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  Eager learning

  In artificial intelligence, eager learning is a learning method in which the system tries to construct a general, input-independent target function during training of the system, as opposed to lazy learning, where generalization beyond the training data is delayed until a query is made to the system. The main advantage gained in employing an eager learning method, such as an artificial neural network, is that the target function will be approximated globally during training, thus requiring much less space than using a lazy learning system. Eager learning systems also deal much better with noise in the training data. Eager learning is an example of offline learning, in which post-training queries to the system have no effect on the system itself, and thus the same query to the system will always produce the same result. The main disadvantage with eager learning is that it is generally unable to provide good local approximations in the target function.

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  LIBSVM

  LIBSVM and LIBLINEAR are two popular open source machine learning libraries, both developed at the National Taiwan University and both written in C++ though with a C API. LIBSVM implements the sequential minimal optimization (SMO) algorithm for kernelized support vector machines (SVMs), supporting classification and regression. LIBLINEAR implements linear SVMs and logistic regression models trained using a coordinate descent algorithm. The SVM learning code from both libraries is often reused in other open source machine learning toolkits, including GATE, KNIME, Orange and scikit-learn. Bindings and ports exist for programming languages such as Java, MATLAB, R, Julia, and Python. It is available in e1071 library in R and scikit-learn in Python. Both libraries are free software released under the 3-clause BSD license.

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  Latent space

  A latent space, also known as a latent feature space or embedding space, is an embedding of a set of items within a manifold in which items resembling each other are positioned closer to one another. Position within the latent space can be viewed as being defined by a set of latent variables that emerge from the resemblances between the objects. In most cases, the dimensionality of the latent space is chosen to be lower than the dimensionality of the feature space from which the data points are drawn, making the construction of a latent space an example of dimensionality reduction, which can also be viewed as a form of data compression. Latent spaces are usually fit via machine learning, and they can then be used as feature spaces in machine learning models, including classifiers and other supervised predictors. The interpretation of latent spaces in machine learning models is an ongoing area of research, but achieving clear interpretations remains challenging. The black-box nature of these models often makes the latent space unintuitive, while its high-dimensional, complex, and nonlinear characteristics further complicate the task of understanding it. Analysis of the latent space geometry of diffusion models reveals a fractal structure of phase transitions in the latent space, characterized by abrupt changes in the Fisher information metric. Some visualization techniques have been developed to connect the latent space to the visual world, but there is often not a direct connection between the latent space interpretation and the model itself. Such techniques include t-distributed stochastic neighbor embedding (t-SNE), where the latent space is mapped to two dimensions for visualization. Latent space distances lack physical units, so the interpretation of these distances may depend on the application. == Embedding models == Several embedding models have been developed to perform this transformation to create latent space embeddings given a set of data items and a similarity function. These models learn the embeddings by leveraging statistical techniques and machine learning algorithms. Here are some commonly used embedding models: Word2Vec: Word2Vec is a popular embedding model used in natural language processing (NLP). It learns word embeddings by training a neural network on a large corpus of text. Word2Vec captures semantic and syntactic relationships between words, allowing for meaningful computations like word analogies. GloVe: GloVe (Global Vectors for Word Representation) is another widely used embedding model for NLP. It combines global statistical information from a corpus with local context information to learn word embeddings. GloVe embeddings are known for capturing both semantic and relational similarities between words. Siamese Networks: Siamese networks are a type of neural network architecture commonly used for similarity-based embedding. They consist of two identical subnetworks that process two input samples and produce their respective embeddings. Siamese networks are often used for tasks like image similarity, recommendation systems, and face recognition. Variational Autoencoders (VAEs): VAEs are generative models that simultaneously learn to encode and decode data. The latent space in VAEs acts as an embedding space. By training VAEs on high-dimensional data, such as images or audio, the model learns to encode the data into a compact latent representation. VAEs are known for their ability to generate new data samples from the learned latent space. == Multimodality == Multimodality refers to the integration and analysis of multiple modes or types of data within a single model or framework. Embedding multimodal data involves capturing relationships and interactions between different data types, such as images, text, audio, and structured data. Multimodal embedding models aim to learn joint representations that fuse information from multiple modalities, allowing for cross-modal analysis and tasks. These models enable applications like image captioning, visual question answering, and multimodal sentiment analysis. To embed multimodal data, specialized architectures such as deep multimodal networks or multimodal transformers are employed. These architectures combine different types of neural network modules to process and integrate information from various modalities. The resulting embeddings capture the complex relationships between different data types, facilitating multimodal analysis and understanding. == Applications == Embedding latent space and multimodal embedding models have found numerous applications across various domains: Information retrieval: Embedding techniques enable efficient similarity search and recommendation systems by representing data points in a compact space. Natural language processing: Word embeddings have revolutionized NLP tasks like sentiment analysis, machine translation, and document classification. Computer vision: Image and video embeddings enable tasks like object recognition, image retrieval, and video summarization. Recommendation systems: Embeddings help capture user preferences and item characteristics, enabling personalized recommendations. Healthcare: Embedding techniques have been applied to electronic health records, medical imaging, and genomic data for disease prediction, diagnosis, and treatment. Social systems: Embedding techniques can be used to learn latent representations of social systems such as internal migration systems, academic citation networks, and world trade networks.

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