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European Conference on Computer Vision : The European Conference on Computer Vision (ECCV) is a biennial research conference with the proceedings published by Springer Science+Business Media. Similar to ICCV in scope and quality, it is held those years which ICCV is not. It is considered to be one of the top conference... |
European Conference on Computer Vision : The conference is usually held in autumn in Europe. |
European Conference on Computer Vision : Computer Vision and Pattern Recognition International Conference on Computer Vision == References == |
International Conference on Acoustics, Speech, and Signal Processing : ICASSP, the International Conference on Acoustics, Speech, and Signal Processing, is an annual flagship conference organized by IEEE Signal Processing Society. Ei Compendex has indexed all papers included in its proceedings. The first ICASSP was hel... |
International Conference on Computer Vision : The International Conference on Computer Vision (ICCV) is a research conference sponsored by the Institute of Electrical and Electronics Engineers (IEEE) held every other year. It is considered to be one of the top conferences in computer vision, alongside CVPR and ECCV, an... |
International Conference on Computer Vision : The conference is usually held in the Spring in various international locations. |
International Conference on Computer Vision : Computer Vision and Pattern Recognition European Conference on Computer Vision == References == |
International Conference on Digital Audio Effects : The annual International Conference on Digital Audio Effects or DAFx Conference is a meeting of enthusiasts working in research areas on audio signal processing, acoustics, and music related disciplines, who come together to present and discuss their findings. The con... |
International Conference on Digital Audio Effects : DAFX, 2020 - Vienna, Austria DAFX, 2019 - Birmingham, UK DAFX, 2018 - Aveiro, Portugal DAFX, 2017 - Edinburgh, UK DAFX, 2016 - Brno, Czech Republic DAFX, 2015 - Trondheim, Norway DAFX, 2014 - Erlangen, Germany DAFX, 2013 - Maynooth, Ireland DAFX, 2012 - York, UK DAFX,... |
International Conference on Digital Audio Effects : International Society for Music Information Retrieval Sound and Music Computing Conference == References == |
Computational learning theory : In computer science, computational learning theory (or just learning theory) is a subfield of artificial intelligence devoted to studying the design and analysis of machine learning algorithms. |
Computational learning theory : Theoretical results in machine learning mainly deal with a type of inductive learning called supervised learning. In supervised learning, an algorithm is given samples that are labeled in some useful way. For example, the samples might be descriptions of mushrooms, and the labels could b... |
Computational learning theory : Error tolerance (PAC learning) Grammar induction Information theory Occam learning Stability (learning theory) |
Computational learning theory : A description of some of these publications is given at important publications in machine learning. |
Algorithmic learning theory : Algorithmic learning theory is a mathematical framework for analyzing machine learning problems and algorithms. Synonyms include formal learning theory and algorithmic inductive inference. Algorithmic learning theory is different from statistical learning theory in that it does not make us... |
Algorithmic learning theory : Unlike statistical learning theory and most statistical theory in general, algorithmic learning theory does not assume that data are random samples, that is, that data points are independent of each other. This makes the theory suitable for domains where observations are (relatively) noise... |
Algorithmic learning theory : The concept was introduced in E. Mark Gold's seminal paper "Language identification in the limit". The objective of language identification is for a machine running one program to be capable of developing another program by which any given sentence can be tested to determine whether it is ... |
Algorithmic learning theory : Learning theorists have investigated other learning criteria, such as the following. Efficiency: minimizing the number of data points required before convergence to a correct hypothesis. Mind Changes: minimizing the number of hypothesis changes that occur before convergence. Mind change bo... |
Algorithmic learning theory : Since 1990, there is an International Conference on Algorithmic Learning Theory (ALT), called Workshop in its first years (1990–1997). Between 1992 and 2016, proceedings were published in the LNCS series. Starting from 2017, they are published by the Proceedings of Machine Learning Researc... |
Algorithmic learning theory : Formal epistemology Sample exclusion dimension |
Bondy's theorem : In mathematics, Bondy's theorem is a bound on the number of elements needed to distinguish the sets in a family of sets from each other. It belongs to the field of combinatorics, and is named after John Adrian Bondy, who published it in 1972. |
Bondy's theorem : The theorem is as follows: Let X be a set with n elements and let A1, A2, ..., An be distinct subsets of X. Then there exists a subset S of X with n − 1 elements such that the sets Ai ∩ S are all distinct. In other words, if we have a 0-1 matrix with n rows and n columns such that each row is distinct... |
Bondy's theorem : Consider the 4 × 4 matrix [ 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 ] 1&1&0&1\\0&1&0&1\\0&0&1&1\\0&1&1&0\end where all rows are pairwise distinct. If we delete, for example, the first column, the resulting matrix [ 1 0 1 1 0 1 0 1 1 1 1 0 ] 1&0&1\\1&0&1\\0&1&1\\1&1&0\end no longer has this property: the first... |
Bondy's theorem : From the perspective of computational learning theory, Bondy's theorem can be rephrased as follows: Let C be a concept class over a finite domain X. Then there exists a subset S of X with the size at most |C| − 1 such that S is a witness set for every concept in C. This implies that every finite conce... |
Concept class : In computational learning theory in mathematics, a concept over a domain X is a total Boolean function over X. A concept class is a class of concepts. Concept classes are a subject of computational learning theory. Concept class terminology frequently appears in model theory associated with probably app... |
Concept class : A sample s is a partial function from X to . Identifying a concept with its characteristic function mapping X to , it is a special case of a sample. Two samples are consistent if they agree on the intersection of their domains. A sample s ′ extends another sample s if the two are consistent and... |
Concept class : Suppose that C = S + ( X ) (X) . Then: the subclass \ is reachable with the sample s = ; the subclass S + ( Y ) (Y) for Y ⊆ X are reachable with a sample that maps the elements of X − Y to zero; the subclass S ( X ) , which consists of the singleton sets, is not reachable. |
Concept class : Let C be some concept class. For any concept c ∈ C , we call this concept 1 / d -good for a positive integer d if, for all x ∈ X , at least 1 / d of the concepts in C agree with c on the classification of x . The fingerprint dimension F D ( C ) of the entire concept class C is the least posit... |
Distribution learning theory : The distributional learning theory or learning of probability distribution is a framework in computational learning theory. It has been proposed from Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert Schapire and Linda Sellie in 1994 and it was inspired from the PAC-frame... |
Distribution learning theory : Let X be the support of the distributions of interest. As in the original work of Kearns et al. if X is finite it can be assumed without loss of generality that X = n ^ where n is the number of bits that have to be used in order to represent any y ∈ X . We focus in probability distri... |
Distribution learning theory : In their seminal work, Kearns et al. deal with the case where A is described in term of a finite polynomial sized circuit and they proved the following for some specific classes of distribution. O R gate distributions for this kind of distributions there is no polynomial-sized evaluator... |
Distribution learning theory : ϵ − Covers One very common technique in order to find a learning algorithm for a class of distributions C is to first find a small ϵ − cover of C . Definition A set C ϵ is called ϵ -cover of C if for every D ∈ C there is a D ′ ∈ C ϵ such that d ( D , D ′ ) ≤ ϵ . An ϵ − cover is... |
Distribution learning theory : Learning of simple well known distributions is a well studied field and there are a lot of estimators that can be used. One more complicated class of distributions is the distribution of a sum of variables that follow simple distributions. These learning procedure have a close relation wi... |
Distribution learning theory : Let the random variables X ∼ N ( μ 1 , Σ 1 ) ,\Sigma _) and Y ∼ N ( μ 2 , Σ 2 ) ,\Sigma _) . Define the random variable Z which takes the same value as X with probability w 1 and the same value as Y with probability w 2 = 1 − w 1 =1-w_ . Then if F 1 is the density of X and F 2 is t... |
Error tolerance (PAC learning) : In PAC learning, error tolerance refers to the ability of an algorithm to learn when the examples received have been corrupted in some way. In fact, this is a very common and important issue since in many applications it is not possible to access noise-free data. Noise can interfere wit... |
Error tolerance (PAC learning) : In the following, let X be our n -dimensional input space. Let H be a class of functions that we wish to use in order to learn a -valued target function f defined over X . Let D be the distribution of the inputs over X . The goal of a learning algorithm A is to choose the best... |
Error tolerance (PAC learning) : In the classification noise model a noise rate 0 ≤ η < 1 2 is introduced. Then, instead of Oracle ( x ) (x) that returns always the correct label of example x , algorithm A can only call a faulty oracle Oracle ( x , η ) (x,\eta ) that will flip the label of x with probability η . A... |
Error tolerance (PAC learning) : Statistical Query Learning is a kind of active learning problem in which the learning algorithm A can decide if to request information about the likelihood P f ( x ) that a function f correctly labels example x , and receives an answer accurate within a tolerance α . Formally, when... |
Error tolerance (PAC learning) : In the malicious classification model an adversary generates errors to foil the learning algorithm. This setting describes situations of error burst, which may occur when for a limited time transmission equipment malfunctions repeatedly. Formally, algorithm A calls an oracle Oracle ( x... |
Error tolerance (PAC learning) : In the nonuniform random attribute noise model the algorithm is learning a Boolean function, a malicious oracle Oracle ( x , ν ) (x,\nu ) may flip each i -th bit of example x = ( x 1 , x 2 , … , x n ) ,x_,\ldots ,x_) independently with probability ν i ≤ ν \leq \nu . This type of error... |
Growth function : The growth function, also called the shatter coefficient or the shattering number, measures the richness of a set family or class of functions. It is especially used in the context of statistical learning theory, where it is used to study properties of statistical learning methods. The term 'growth fu... |
Growth function : 1. The domain is the real line R . The set-family H contains all the half-lines (rays) from a given number to positive infinity, i.e., all sets of the form \mid x\in \mathbb \ for some x 0 ∈ R \in \mathbb . For any set C of m real numbers, the intersection H ∩ C contains m + 1 sets: the emp... |
Growth function : The main property that makes the growth function interesting is that it can be either polynomial or exponential - nothing in-between. The following is a property of the intersection-size:: Lem.1 If, for some set C m of size m , and for some number n ≤ m , | H ∩ C m | ≥ Comp ( n , m ) |\geq \opera... |
Growth function : Let Ω be a set on which a probability measure Pr is defined. Let H be family of subsets of Ω (= a family of events). Suppose we choose a set C m that contains m elements of Ω , where each element is chosen at random according to the probability measure P , independently of the others (i.e., wi... |
Induction of regular languages : In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see lan... |
Induction of regular languages : A regular language is defined as a (finite or infinite) set of strings that can be described by one of the mathematical formalisms called "finite automaton", "regular grammar", or "regular expression", all of which have the same expressive power. Since the latter formalism leads to shor... |
Induction of regular languages : Dupont et al. have shown that the set of all structurally complete finite automata generating a given input set of example strings forms a lattice, with the trivial undergeneralized and the trivial overgeneralized automaton as bottom and top element, respectively. Each member of this la... |
Induction of regular languages : Finding common patterns in DNA and RNA structure descriptions (Bioinformatics) Modelling natural language acquisition by humans Learning of structural descriptions from structured example documents, in particular Document Type Definitions (DTD) from SGML documents Learning the structure... |
Language identification in the limit : Language identification in the limit is a formal model for inductive inference of formal languages, mainly by computers (see machine learning and induction of regular languages). It was introduced by E. Mark Gold in a technical report and a journal article with the same title. In ... |
Language identification in the limit : This model is an early attempt to formally capture the notion of learnability. Gold's journal article introduces for contrast the stronger models Finite identification (where the learner has to announce correctness after a finite number of steps), and Fixed-time identification (wh... |
Language identification in the limit : It is instructive to look at concrete examples (in the tables) of learning sessions the definition of identification in the limit speaks about. A fictitious session to learn a regular language L over the alphabet from text presentation:In each step, the teacher gives a string bel... |
Language identification in the limit : More formally, a language L is a nonempty set, and its elements are called sentences. a language family is a set of languages. a language-learning environment E for a language L is a stream of sentences from L , such that each sentence in L appears at least once. a language l... |
Language identification in the limit : Dana Angluin gave the characterizations of learnability from text (positive information) in a 1980 paper. If a learner is required to be effective, then an indexed class of recursive languages is learnable in the limit if there is an effective procedure that uniformly enumerates t... |
Language identification in the limit : The table shows which language classes are identifiable in the limit in which learning model. On the right-hand side, each language class is a superclass of all lower classes. Each learning model (i.e. type of presentation) can identify in the limit all classes below it. In partic... |
Language identification in the limit : Condition 1 in Angluin's paper is not always easy to verify. Therefore, people come up with various sufficient conditions for the learnability of a language class. See also Induction of regular languages for learnable subclasses of regular languages. |
Language identification in the limit : A bound over the number of hypothesis changes that occur before convergence. |
Language identification in the limit : Finite thickness implies finite elasticity; the converse is not true. Finite elasticity and conservatively learnable implies the existence of a mind change bound. [1] Finite elasticity and M-finite thickness implies the existence of a mind change bound. However, M-finite thickness... |
Language identification in the limit : If a countable class of recursive languages has a mind change bound for noncomputable learners, does the class also have a mind change bound for computable learners, or is the class unlearnable by a computable learner? |
Natarajan dimension : In the theory of Probably Approximately Correct Machine Learning, the Natarajan dimension characterizes the complexity of learning a set of functions, generalizing from the Vapnik-Chervonenkis dimension for boolean functions to multi-class functions. Originally introduced as the Generalized Dimens... |
Natarajan dimension : Let H be a set of functions from a set X to a set Y . H shatters a set C ⊂ X if there exist two functions f 0 , f 1 ∈ H ,f_\in H such that For every x ∈ C , f 0 ( x ) ≠ f 1 ( x ) (x)\neq f_(x) . For every B ⊂ C , there exists a function h ∈ H such that for all x ∈ B , h ( x ) = f 0 ( x ) (x... |
Occam learning : In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (PAC) learning, where the learner is evaluated on its predic... |
Occam learning : Occam Learning is named after Occam's razor, which is a principle stating that, given all other things being equal, a shorter explanation for observed data should be favored over a lengthier explanation. The theory of Occam learning is a formal and mathematical justification for this principle. It was ... |
Occam learning : The succinctness of a concept c in concept class C can be expressed by the length s i z e ( c ) of the shortest bit string that can represent c in C . Occam learning connects the succinctness of a learning algorithm's output to its predictive power on unseen data. Let C and H be concept classes ... |
Occam learning : Occam learnability implies PAC learnability, as the following theorem of Blumer, et al. shows: |
Occam learning : We first prove the Cardinality version. Call a hypothesis h ∈ H bad if e r r o r ( h ) ≥ ϵ , where again e r r o r ( h ) is with respect to the true concept c and the underlying distribution D . The probability that a set of samples S is consistent with h is at most ( 1 − ϵ ) m , by the indepen... |
Occam learning : Though Occam and PAC learnability are equivalent, the Occam framework can be used to produce tighter bounds on the sample complexity of classical problems including conjunctions, conjunctions with few relevant variables, and decision lists. |
Occam learning : Occam algorithms have also been shown to be successful for PAC learning in the presence of errors, probabilistic concepts, function learning and Markovian non-independent examples. |
Occam learning : Structural risk minimization Computational learning theory |
Occam learning : Blumer, A.; Ehrenfeucht, A.; Haussler, D.; Warmuth, M. K. Learnability and the Vapnik-Chervonenkis dimension. Journal of the ACM, 36(4):929–865, 1989. |
Probably approximately correct learning : In computational learning theory, probably approximately correct (PAC) learning is a framework for mathematical analysis of machine learning. It was proposed in 1984 by Leslie Valiant. In this framework, the learner receives samples and must select a generalization function (ca... |
Probably approximately correct learning : In order to give the definition for something that is PAC-learnable, we first have to introduce some terminology. For the following definitions, two examples will be used. The first is the problem of character recognition given an array of n bits encoding a binary-valued image... |
Probably approximately correct learning : Under some regularity conditions these conditions are equivalent: The concept class C is PAC learnable. The VC dimension of C is finite. C is a uniformly Glivenko-Cantelli class. C is compressible in the sense of Littlestone and Warmuth |
Probably approximately correct learning : Occam learning Data mining Error tolerance (PAC learning) Sample complexity |
Probably approximately correct learning : M. Kearns, U. Vazirani. An Introduction to Computational Learning Theory. MIT Press, 1994. A textbook. M. Mohri, A. Rostamizadeh, and A. Talwalkar. Foundations of Machine Learning. MIT Press, 2018. Chapter 2 contains a detailed treatment of PAC-learnability. Readable through op... |
Representer theorem : For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel product... |
Representer theorem : The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : X × X → R \times \to \mathbb on a non-empty set X with a corresponding reproducing kernel Hilbert space H k . Let there be given a training sa... |
Representer theorem : The Theorem stated above is a particular example of a family of results that are collectively referred to as "representer theorems"; here we describe several such. The first statement of a representer theorem was due to Kimeldorf and Wahba for the special case in which E ( ( x 1 , y 1 , f ( x 1 ) ... |
Representer theorem : Representer theorems are useful from a practical standpoint because they dramatically simplify the regularized empirical risk minimization problem ( ‡ ) . In most interesting applications, the search domain H k for the minimization will be an infinite-dimensional subspace of L 2 ( X ) () , and t... |
Representer theorem : Mercer's theorem Kernel methods |
Representer theorem : Argyriou, Andreas; Micchelli, Charles A.; Pontil, Massimiliano (2009). "When Is There a Representer Theorem? Vector Versus Matrix Regularizers". Journal of Machine Learning Research. 10 (Dec): 2507–2529. Cucker, Felipe; Smale, Steve (2002). "On the Mathematical Foundations of Learning". Bulletin o... |
Sample exclusion dimension : In computational learning theory, sample exclusion dimensions arise in the study of exact concept learning with queries. In algorithmic learning theory, a concept over a domain X is a Boolean function over X. Here we only consider finite domains. A partial approximation S of a concept c is ... |
Shattered set : A class of sets is said to shatter another set if it is possible to "pick out" any element of that set using intersection. The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes a... |
Shattered set : Suppose A is a set and C is a class of sets. The class C shatters the set A if for each subset a of A, there is some element c of C such that a = c ∩ A . Equivalently, C shatters A when their intersection is equal to A's power set: P(A) = . We employ the letter C to refer to a "class" or "collection" o... |
Shattered set : We will show that the class of all discs in the plane (two-dimensional space) does not shatter every set of four points on the unit circle, yet the class of all convex sets in the plane does shatter every finite set of points on the unit circle. Let A be a set of four points on the unit circle and let C... |
Shattered set : To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function). For a collection C of sets s ⊂ Ω , Ω being any space, often a sample space, we define the nth shattering coefficient of C as S C ( n ) = max ∀ x 1 , x 2 , … , x n ∈ Ω ... |
Shattered set : If A cannot be shattered by C, there will be a smallest value of n that makes the shatter coefficient(n) less than 2 n because as n gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of n for which the S_C(n) is still 2 n , because as n gets smaller, th... |
Shattered set : Sauer–Shelah lemma, relating the cardinality of a family of sets to the size of its largest shattered set |
Shattered set : Wencour, R. S.; Dudley, R. M. (1981), "Some special Vapnik–Chervonenkis classes", Discrete Mathematics, 33 (3): 313–318, doi:10.1016/0012-365X(81)90274-0. Steele, J. M. (1975), Combinatorial Entropy and Uniform Limit Laws, Ph.D. thesis, Stanford University, Mathematics Department Steele, J. M. (1978), "... |
Shattered set : Origin of "Shattered sets" terminology, by J. Steele |
Vapnik–Chervonenkis dimension : In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest... |
Vapnik–Chervonenkis dimension : f is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d different classifiers, is at most d (this is an upper bound on the VC dimensi... |
Vapnik–Chervonenkis dimension : The VC dimension of the dual set-family of F is strictly less than 2 vc ( F ) + 1 ()+1 , and this is best possible. The VC dimension of a finite set-family H is at most log 2 | H | |H| .: 56 This is because | H ∩ C | ≤ | H | by definition. Given a set-family H , define H s as a... |
Vapnik–Chervonenkis dimension : The VC dimension is defined for spaces of binary functions (functions to ). Several generalizations have been suggested for spaces of non-binary functions. For multi-class functions (e.g., functions to ), the Natarajan dimension, and its generalization the DS dimension can be used. For r... |
Vapnik–Chervonenkis dimension : Growth function Sauer–Shelah lemma, a bound on the number of sets in a set system in terms of the VC dimension. Karpinski–Macintyre theorem, a bound on the VC dimension of general Pfaffian formulas. |
Vapnik–Chervonenkis dimension : Moore, Andrew. "VC dimension tutorial" (PDF). Vapnik, Vladimir (2000). The nature of statistical learning theory. Springer. Blumer, A.; Ehrenfeucht, A.; Haussler, D.; Warmuth, M. K. (1989). "Learnability and the Vapnik–Chervonenkis dimension" (PDF). Journal of the ACM. 36 (4): 929–865. d... |
Vapnik–Chervonenkis theory : Vapnik–Chervonenkis theory (also known as VC theory) was developed during 1960–1990 by Vladimir Vapnik and Alexey Chervonenkis. The theory is a form of computational learning theory, which attempts to explain the learning process from a statistical point of view. |
Vapnik–Chervonenkis theory : VC theory covers at least four parts (as explained in The Nature of Statistical Learning Theory): Theory of consistency of learning processes What are (necessary and sufficient) conditions for consistency of a learning process based on the empirical risk minimization principle? Nonasymptoti... |
Vapnik–Chervonenkis theory : A similar setting is considered, which is more common to machine learning. Let X is a feature space and Y = =\ . A function f : X → Y \to is called a classifier. Let F be a set of classifiers. Similarly to the previous section, define the shattering coefficient (also known as growth fun... |
Vapnik–Chervonenkis theory : See references in articles: Richard M. Dudley, empirical processes, Shattered set. Bousquet, Olivier; Elisseeff, Andr´e (1 March 2002). "Stability and Generalization". The Journal of Machine Learning Research. 2: 499–526. doi:10.1162/153244302760200704. S2CID 1157797. Retrieved 10 December ... |
Win–stay, lose–switch : In psychology, game theory, statistics, and machine learning, win–stay, lose–switch (also win–stay, lose–shift) is a heuristic learning strategy used to model learning in decision situations. It was first invented as an improvement over randomization in bandit problems. It was later applied to t... |
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