text stringlengths 14 4.79k | source stringlengths 13 304 | tokens float64 75 1.06k β | char_length float64 106 4.79k β | article_title stringlengths 16 300 β |
|---|---|---|---|---|
Section: Unscrolling and reading > Reactions. UniversitΓ degli Studi di Napoli Federico II papyrology professor Federica Nicolardi praised the discovery declaring: "It's extremely exciting to be reading entire words, not just sequences of letters, from within a scroll", and expressing confidence in further future decip... | Wikipedia - PHerc. Paris. 4 - Unscrolling and reading > Reactions | 233 | 1,011 | null |
Article: Phi coefficient. In statistics, the phi coefficient, or mean square contingency coefficient, denoted by Ο or rΟ, is a measure of association for two binary variables. In machine learning, it is known as the Matthews correlation coefficient (MCC) and used as a measure of the quality of binary (two-class) classi... | Wikipedia - Phi coefficient - Summary | 168 | 795 | null |
Section: Definition. A Pearson correlation coefficient estimated for two binary variables will return the phi coefficient. Two binary variables are considered positively associated if most of the data falls along the diagonal cells. In contrast, two binary variables are considered negatively associated if most of the d... | Wikipedia - Phi coefficient - Definition | 344 | 1,174 | null |
The phi coefficient can also be expressed using only n {\displaystyle n} , n 11 {\displaystyle n_{11}} , n 1 β {\displaystyle n_{1\bullet }} , and n β 1 {\displaystyle n_{\bullet 1}} , as Ο = n n 11 β n 1 β n β 1 n 1 β n β 1 ( n β n 1 β ) ( n β n β 1 ) . {\displaystyle \varphi ={\frac {nn_{11}-n_{1\bullet }n_{\bullet 1... | Wikipedia - Phi coefficient - Definition | 178 | 395 | null |
Section: Machine learning. The MCC is defined identically to phi coefficient, introduced by Karl Pearson, also known as the Yule phi coefficient from its introduction by Udny Yule in 1912. Despite these antecedents which predate Matthews's use by several decades, the term MCC is widely used in the field of bioinformati... | Wikipedia - Phi coefficient - Machine learning | 344 | 1,583 | null |
Other measures, such as the proportion of correct predictions (also termed accuracy), are not useful when the two classes are of very different sizes. For example, assigning every object to the larger set achieves a high proportion of correct predictions, but is not generally a useful classification. The MCC can be cal... | Wikipedia - Phi coefficient - Machine learning | 301 | 868 | null |
The MCC can be calculated directly from the confusion matrix using the formula: MCC = T P Γ T N β F P Γ F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) {\displaystyle {\text{MCC}}={\frac {{\mathit {TP}}\times {\mathit {TN}}-{\mathit {FP}}\times {\mathit {FN}}}{\sqrt {({\mathit {TP}}+{\mathit {FP}})({\mathit... | Wikipedia - Phi coefficient - Machine learning | 325 | 917 | null |
both labels and model predictions are all positive or negative), the limit does not exist. The MCC can be calculated with the formula: MCC = P P V Γ T P R Γ T N R Γ N P V β F D R Γ F N R Γ F P R Γ F O R {\displaystyle {\text{MCC}}={\sqrt {{\mathit {PPV}}\times {\mathit {TPR}}\times {\mathit {TNR}}\times {\mathit {NPV}}... | Wikipedia - Phi coefficient - Machine learning | 217 | 636 | null |
The MCC can be calculated with the formula: MCC = P P V Γ T P R Γ T N R Γ N P V β F D R Γ F N R Γ F P R Γ F O R {\displaystyle {\text{MCC}}={\sqrt {{\mathit {PPV}}\times {\mathit {TPR}}\times {\mathit {TNR}}\times {\mathit {NPV}}}}-{\sqrt {{\mathit {FDR}}\times {\mathit {FNR}}\times {\mathit {FPR}}\times {\mathit {FOR}... | Wikipedia - Phi coefficient - Machine learning | 458 | 1,030 | null |
The original formula as given by Matthews was: N = T N + T P + F N + F P S = T P + F N N P = T P + F P N MCC = T P / N β S Γ P P S ( 1 β S ) ( 1 β P ) {\displaystyle {\begin{aligned}N&={\mathit {TN}}+{\mathit {TP}}+{\mathit {FN}}+{\mathit {FP}}\\[6pt]S&={\frac {{\mathit {TP}}+{\mathit {FN}}}{N}}\\[6pt]P&={\frac {{\math... | Wikipedia - Phi coefficient - Machine learning | 316 | 779 | null |
Section: Example. Given a sample of 12 pictures, 8 of cats and 4 of dogs, where cats belong to class 1 and dogs belong to class 0, actual = [1,1,1,1,1,1,1,1,0,0,0,0], assume that a classifier that distinguishes between cats and dogs is trained, and we take the 12 pictures and run them through the classifier, and the cl... | Wikipedia - Phi coefficient - Example | 300 | 1,195 | null |
In abstract terms, the confusion matrix is as follows: where P = positive; N = negative; TP = truepositive; FP = false positive; TN = true negative; FN = false negative. Plugging the numbers from the formula: MCC = 6 Γ 3 β 1 Γ 2 ( 6 + 1 ) Γ ( 6 + 2 ) Γ ( 3 + 1 ) Γ ( 3 + 2 ) = 16 1120 β 0.478 {\displaystyle {\text{MCC}}... | Wikipedia - Phi coefficient - Example | 180 | 442 | null |
MCC = β k β l β m C k k C l m β C k l C m k β k ( β l C k l ) ( β k β² | k β² β k β l β² C k β² l β² ) β k ( β l C l k ) ( β k β² | k β² β k β l β² C l β² k β² ) {\displaystyle {\text{MCC}}={\frac {\sum _{k}\sum _{l}\sum _{m}C_{kk}C_{lm}-C_{kl}C_{mk}}{{\sqrt {\sum _{k}\left(\sum _{l}C_{kl}\right)\left(\sum _{k'|k'\neq k}\sum _{l... | Wikipedia - Phi coefficient - Multiclass case | 316 | 642 | null |
Instead the minimum value will be between β1 and 0 depending on the true distribution. The maximum value is always +1. This formula can be more easily understood by defining intermediate variables: t k = β i C i k {\displaystyle t_{k}=\sum _{i}C_{ik}} the number of times class k truly occurred, p k = β i C k i {\displa... | Wikipedia - Phi coefficient - Multiclass case | 189 | 579 | null |
This allows the formula to be expressed as: MCC = c s β t β β
p β s 2 β p β β
p β s 2 β t β β
t β {\displaystyle {\text{MCC}}={\frac {cs-{\vec {t}}\cdot {\vec {p}}}{{\sqrt {s^{2}-{\vec {p}}\cdot {\vec {p}}}}{\sqrt {s^{2}-{\vec {t}}\cdot {\vec {t}}}}}}} Using above formula to compute MCC measure for the dog and cat exam... | Wikipedia - Phi coefficient - Multiclass case | 347 | 749 | null |
Section: Advantages over accuracy and F1 score. As explained by Davide Chicco in his paper "Ten quick tips for machine learning in computational biology" (BioData Mining, 2017) and "The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation" (BMC Genomics... | Wikipedia - Phi coefficient - Advantages over accuracy and F1 score | 334 | 1,338 | null |
accuracy = T P + T N T P + T N + F P + F N {\displaystyle {\text{accuracy}}={\frac {TP+TN}{TP+TN+FP+FN}}} (Equation 1, accuracy: worst value = 0; best value = 1) F1 score = 2 T P 2 T P + F P + F N {\displaystyle {\text{F1 score}}={\frac {2TP}{2TP+FP+FN}}} (Equation 2, F1 score: worst value = 0; best value = 1) However,... | Wikipedia - Phi coefficient - Advantages over accuracy and F1 score | 324 | 1,185 | null |
These values lead to the following performance scores: accuracy = 95%, and F1 score = 97.44%. By reading these over-optimistic scores, then you will be very happy and will think that your machine learning algorithm is doing an excellent job. Obviously, you would be on the wrong track. On the contrary, to avoid these da... | Wikipedia - Phi coefficient - Advantages over accuracy and F1 score | 340 | 1,280 | null |
Consider this other example. You ran a classification on the same dataset which led to the following values for the confusion matrix categories: TP = 90, FP = 4; TN = 1, FN = 5. In this example, the classifier has performed well in classifying positive instances, but was not able to correctly recognize negative data el... | Wikipedia - Phi coefficient - Advantages over accuracy and F1 score | 333 | 1,623 | null |
Article: Predictive learning. Predictive learning is a machine learning (ML) technique where an artificial intelligence model is fed new data to develop an understanding of its environment, capabilities, and limitations. This technique finds application in many areas, including neuroscience, business, robotics, and com... | Wikipedia - Predictive learning - Summary | 289 | 1,555 | null |
Section: Mathematical procedures > Training process. Similar to ML, predictive learning aims to extrapolate the value of an unknown dependent variable Y {\displaystyle Y} , given independent input data X = ( x 1 , x 2 , β¦ , x n ) {\displaystyle X=(x_{1},x_{2},\dots ,x_{n})} . A set of attributes can be classified into ... | Wikipedia - Predictive learning - Mathematical procedures > Training process | 228 | 1,002 | null |
Section: Mathematical procedures > Maximizing accuracy. In order to ensure maximum accuracy for a predictive learning model, the predicted values y ^ = F ( x ) {\displaystyle {\hat {y}}=F(x)} must not exceed a certain error threshold when compared to actual values y {\displaystyle y} by the risk formula: R ( F ) = E x ... | Wikipedia - Predictive learning - Mathematical procedures > Maximizing accuracy | 292 | 919 | null |
Section: Mathematical procedures > Ensemble learning. In some cases, using a singular machine learning approach is not enough to create an accurate estimate for certain data. Ensemble learning is the combination of several ML algorithms to create a stronger model. Each model is represented by the function F ( x ) = a 0... | Wikipedia - Predictive learning - Mathematical procedures > Ensemble learning | 202 | 697 | null |
Each model is represented by the function F ( x ) = a 0 + β m = 1 M a m f m ( x ) {\displaystyle F(x)=a_{0}+\sum _{m=1}^{M}a_{m}f_{m}(x)} , where M {\displaystyle M} is the number of ensemble models, a 0 {\displaystyle a_{0}} is the bias, a m {\displaystyle a_{m}} is the weight corresponding to each m {\displaystyle m}... | Wikipedia - Predictive learning - Mathematical procedures > Ensemble learning | 398 | 1,067 | null |
Section: Applications > Cognitive development. Sensorimotor signals are neural impulses sent to the brain upon physical touch. Using predictive learning to detect sensorimotor signals plays a key role in early cognitive development, as the human brain represents sensorimotor signals in a predictive manner (it attempts ... | Wikipedia - Predictive learning - Applications > Cognitive development | 165 | 848 | null |
Article: Predictive state representation. In computer science, a predictive state representation (PSR) is a way to model a state of controlled dynamical system from a history of actions taken and resulting observations. PSR captures the state of a system as a vector of predictions for future tests (experiments) that ca... | Wikipedia - Predictive state representation - Summary | 177 | 875 | null |
Section: Definition. Consider a dynamic system based on a discrete set A {\displaystyle {\mathcal {A}}} of actions and a discrete set O {\displaystyle {\mathcal {O}}} of observations. A history h {\displaystyle h} is a sequence a 1 o 1 β¦ a β o β {\displaystyle a_{1}o_{1}\dots a_{\ell }o_{\ell }} where a 1 , β¦ , a β {\d... | Wikipedia - Predictive state representation - Definition | 331 | 926 | null |
To do that, we introduce the notion of test. A test t {\displaystyle t} is of the same type that a history: it is a sequence of action-observation pairs. The idea is now to consider a set of tests { t 1 , β¦ , t n } {\displaystyle \{t_{1},\dots ,t_{n}\}} to full characterize a hidden state. To do that, we first define t... | Wikipedia - Predictive state representation - Definition | 279 | 789 | null |
We say that p ( h ) {\displaystyle p(h)} is a predictive state representation (PSR) if and only if it forms a sufficient statistic for the system. In other words, p ( h ) {\displaystyle p(h)} is a predictive state representation (PSR) if and only if for all possible tests t {\displaystyle t} , there exists a function f... | Wikipedia - Predictive state representation - Definition | 265 | 891 | null |
Section: Tasks > Label ranking. In label ranking, the model has an instance space X = { x i } {\displaystyle X=\{x_{i}\}\,\!} and a finite set of labels Y = { y i | i = 1 , 2 , β― , k } {\displaystyle Y=\{y_{i}|i=1,2,\cdots ,k\}\,\!} . The preference information is given in the form y i β» x y j {\displaystyle y_{i}\succ... | Wikipedia - Preference learning - Tasks > Label ranking | 326 | 955 | null |
It was observed that some conventional classification problems can be generalized in the framework of label ranking problem: if a training instance x {\displaystyle x\,\!} is labeled as class y i {\displaystyle y_{i}\,\!} , it implies that β j β i , y i β» x y j {\displaystyle \forall j\neq i,y_{i}\succ _{x}y_{j}\,\!} .... | Wikipedia - Preference learning - Tasks > Label ranking | 255 | 776 | null |
Section: Tasks > Instance ranking. Instance ranking also has the instance space X {\displaystyle X\,\!} and label set Y {\displaystyle Y\,\!} . In this task, labels are defined to have a fixed order y 1 β» y 2 β» β― β» y k {\displaystyle y_{1}\succ y_{2}\succ \cdots \succ y_{k}\,\!} and each instance x l {\displaystyle x_{... | Wikipedia - Preference learning - Tasks > Instance ranking | 166 | 509 | null |
Section: Techniques. There are two practical representations of the preference information A β» B {\displaystyle A\succ B\,\!} . One is assigning A {\displaystyle A\,\!} and B {\displaystyle B\,\!} with two real numbers a {\displaystyle a\,\!} and b {\displaystyle b\,\!} respectively such that a > b {\displaystyle a>b\,... | Wikipedia - Preference learning - Techniques | 227 | 691 | null |
Section: Techniques > Utility function. If we can find a mapping from data to real numbers, ranking the data can be solved by ranking the real numbers. This mapping is called utility function. For label ranking the mapping is a function f : X Γ Y β R {\displaystyle f:X\times Y\rightarrow \mathbb {R} \,\!} such that y i... | Wikipedia - Preference learning - Techniques > Utility function | 211 | 670 | null |
Section: Techniques > Preference relations. The binary representation of preference information is called preference relation. For each pair of alternatives (instances or labels), a binary predicate can be learned by conventional supervised learning approach. FΓΌrnkranz and HΓΌllermeier proposed this approach in label ra... | Wikipedia - Preference learning - Techniques > Preference relations | 160 | 881 | null |
Article: Prior knowledge for pattern recognition. Pattern recognition is a very active field of research intimately bound to machine learning. Also known as classification or statistical classification, pattern recognition aims at building a classifier that can determine the class of an input pattern. This procedure, k... | Wikipedia - Prior knowledge for pattern recognition - Summary | 166 | 830 | null |
Section: Prior knowledge. Prior knowledge refers to all information about the problem available in addition to the training data. However, in this most general form, determining a model from a finite set of samples without prior knowledge is an ill-posed problem, in the sense that a unique model may not exist. Many cla... | Wikipedia - Prior knowledge for pattern recognition - Prior knowledge | 186 | 1,026 | null |
Section: Class-invariance. A very common type of prior knowledge in pattern recognition is the invariance of the class (or the output of the classifier) to a transformation of the input pattern. This type of knowledge is referred to as transformation-invariance. The mostly used transformations used in image recognition... | Wikipedia - Prior knowledge for pattern recognition - Class-invariance | 216 | 757 | null |
Incorporating the invariance to a transformation T ΞΈ : x β¦ T ΞΈ x {\displaystyle T_{\theta }:{\boldsymbol {x}}\mapsto T_{\theta }{\boldsymbol {x}}} parametrized in ΞΈ {\displaystyle \theta } into a classifier of output f ( x ) {\displaystyle f({\boldsymbol {x}})} for an input pattern x {\displaystyle {\boldsymbol {x}}} c... | Wikipedia - Prior knowledge for pattern recognition - Class-invariance | 282 | 739 | null |
{\displaystyle f({\boldsymbol {x}})=f(T_{\theta }{\boldsymbol {x}}),\quad \forall {\boldsymbol {x}},\theta .} Local invariance can also be considered for a transformation centered at ΞΈ = 0 {\displaystyle \theta =0} , so that T 0 x = x {\displaystyle T_{0}{\boldsymbol {x}}={\boldsymbol {x}}} , by using the constraint β ... | Wikipedia - Prior knowledge for pattern recognition - Class-invariance | 253 | 718 | null |
Another approach is to consider class-invariance with respect to a "domain of the input space" instead of a transformation. In this case, the problem becomes finding f {\displaystyle f} so that f ( x ) = y P , β x β P , {\displaystyle f({\boldsymbol {x}})=y_{\mathcal {P}},\ \forall {\boldsymbol {x}}\in {\mathcal {P}},}... | Wikipedia - Prior knowledge for pattern recognition - Class-invariance | 224 | 754 | null |
Article: Proactive learning. Proactive learning is a generalization of active learning designed to relax unrealistic assumptions and thereby reach practical applications. "In real life, it is possible and more general to have multiple sources of information with differing reliabilities or areas of expertise. Active lea... | Wikipedia - Proactive learning - Summary | 244 | 1,240 | null |
Article: Proaftn. Proaftn is a fuzzy classification method that belongs to the class of supervised learning algorithms. The acronym Proaftn stands for: (PROcΓ©dure d'Affectation Floue pour la problΓ©matique du Tri Nominal), which means in English: Fuzzy Assignment Procedure for Nominal Sorting. The method enables to dete... | Wikipedia - Proaftn - Summary | 331 | 1,729 | null |
Section: Introduction. A numerical method is an algorithm that approximates the solution to a mathematical problem (examples below include the solution to a linear system of equations, the value of an integral, the solution of a differential equation, the minimum of a multivariate function). In a probabilistic numerica... | Wikipedia - Probabilistic numerics - Introduction | 339 | 1,731 | null |
In such cases, the variance of the Gaussian posterior is then associated with a worst-case estimate for the squared error. Probabilistic numerical methods promise several conceptual advantages over classic, point-estimate based approximation techniques: They return structured error estimates (in particular, the ability... | Wikipedia - Probabilistic numerics - Introduction | 332 | 1,822 | null |
Section: Numerical tasks > Integration. Probabilistic numerical methods have been developed for the problem of numerical integration, with the most popular method called Bayesian quadrature. In numerical integration, function evaluations f ( x 1 ) , β¦ , f ( x n ) {\displaystyle f(x_{1}),\ldots ,f(x_{n})} at a number of... | Wikipedia - Probabilistic numerics - Numerical tasks > Integration | 325 | 1,106 | null |
Section: Numerical tasks > Optimization. Probabilistic numerics have also been studied for mathematical optimization, which consist of finding the minimum or maximum of some objective function f {\displaystyle f} given (possibly noisy or indirect) evaluations of that function at a set of points. Perhaps the most notabl... | Wikipedia - Probabilistic numerics - Numerical tasks > Optimization | 251 | 1,472 | null |
Section: Numerical tasks > Local optimization. Probabilistic numerical methods have been developed in the context of stochastic optimization for deep learning, in particular to address main issues such as learning rate tuning and line searches, batch-size selection, early stopping, pruning, and first- and second-order ... | Wikipedia - Probabilistic numerics - Numerical tasks > Local optimization | 329 | 995 | null |
In this setting, the optimization objective is often an empirical risk of the form L ( ΞΈ ) = 1 N β n = 1 N β ( y n , f ΞΈ ( x n ) ) {\displaystyle \textstyle L(\theta )={\frac {1}{N}}\sum _{n=1}^{N}\ell (y_{n},f_{\theta }(x_{n}))} defined by a dataset D = { ( x n , y n ) } n = 1 N {\displaystyle \textstyle {\mathcal {D}... | Wikipedia - Probabilistic numerics - Numerical tasks > Local optimization | 352 | 1,005 | null |
Section: Numerical tasks > Linear algebra. Probabilistic numerical methods for linear algebra have primarily focused on solving systems of linear equations of the form A x = b {\displaystyle Ax=b} and the computation of determinants | A | {\displaystyle |A|} . A large class of methods are iterative in nature and collec... | Wikipedia - Probabilistic numerics - Numerical tasks > Linear algebra | 309 | 1,182 | null |
Section: Numerical tasks > Ordinary differential equations. Probabilistic numerical methods for ordinary differential equations y Λ ( t ) = f ( t , y ( t ) ) {\displaystyle {\dot {y}}(t)=f(t,y(t))} , have been developed for initial and boundary value problems. Many different probabilistic numerical methods designed for... | Wikipedia - Probabilistic numerics - Numerical tasks > Ordinary differential equations | 260 | 1,269 | null |
In its infancy, this class of methods was based on naive Gaussian process regression. This was later improved (in terms of efficient computation) in favor of GaussβMarkov priors modeled by the stochastic differential equation d x ( t ) = A x ( t ) d t + B d v ( t ) {\displaystyle \mathrm {d} x(t)=Ax(t)\,\mathrm {d} t+B... | Wikipedia - Probabilistic numerics - Numerical tasks > Ordinary differential equations | 274 | 1,043 | null |
Section: History and related fields. The interplay between numerical analysis and probability is touched upon by a number of other areas of mathematics, including average-case analysis of numerical methods, information-based complexity, game theory, and statistical decision theory. Precursors to what is now being calle... | Wikipedia - Probabilistic numerics - History and related fields | 300 | 1,185 | null |
A later seminal contribution to the interplay of numerical analysis and probability was provided by Albert Suldin in the context of univariate quadrature. The statistical problem considered by Suldin was the approximation of the definite integral β« u ( t ) d t {\displaystyle \textstyle \int u(t)\,\mathrm {d} t} of a fu... | Wikipedia - Probabilistic numerics - History and related fields | 337 | 1,262 | null |
Note that Suldin's Brownian motion prior on the integrand u {\displaystyle u} is a Gaussian measure and that the operations of integration and of point wise evaluation of u {\displaystyle u} are both linear maps. Thus, the definite integral β« u ( t ) d t {\displaystyle \textstyle \int u(t)\,\mathrm {d} t} is a real-val... | Wikipedia - Probabilistic numerics - History and related fields | 320 | 1,206 | null |
As noted by Houman Owhadi and collaborators, interplays between numerical approximation and statistical inference can also be traced back to Palasti and Renyi, Sard, Kimeldorf and Wahba (on the correspondence between Bayesian estimation and spline smoothing/interpolation) and Larkin (on the correspondence between Gauss... | Wikipedia - Probabilistic numerics - History and related fields | 293 | 1,538 | null |
Article: Probability matching. Probability matching is a decision strategy in which predictions of class membership are proportional to the class base rates. Thus, if in the training set positive examples are observed 60% of the time, and negative examples are observed 40% of the time, then the observer using a probabi... | Wikipedia - Probability matching - Summary | 310 | 1,358 | null |
Article: Product of experts. Product of experts (PoE) is a machine learning technique. It models a probability distribution by combining the output from several simpler distributions. It was proposed by Geoffrey Hinton in 1999, along with an algorithm for training the parameters of such a system. The core idea is to co... | Wikipedia - Product of experts - Summary | 310 | 999 | null |
This allows each expert to make decisions on the basis of a few dimensions without having to cover the full dimensionality of a problem: P ( y | { x k } ) = 1 Z β j = 1 M f j ( y | { x k } ) {\displaystyle P(y|\{x_{k}\})={\frac {1}{Z}}\prod _{j=1}^{M}f_{j}(y|\{x_{k}\})} where f j {\displaystyle f_{j}} are unnormalized ... | Wikipedia - Product of experts - Summary | 384 | 924 | null |
This is related to (but quite different from) a mixture model, where several probability distributions p j ( y | { x j } ) {\displaystyle p_{j}(y|\{x_{j}\})} are combined via an "or" operation, which is a weighted sum of their density functions: P ( y | { x k } ) = β j = 1 M Ξ± j p j ( y | { x k } ) , {\displaystyle P(y... | Wikipedia - Product of experts - Summary | 258 | 816 | null |
Article: Programming by example. In computer science, programming by example (PbE), also termed programming by demonstration or more generally as demonstrational programming, is an end-user development technique for teaching a computer new behavior by demonstrating actions on concrete examples. The system records user ... | Wikipedia - Programming by example - Summary | 293 | 1,462 | null |
Article: Proximal gradient methods for learning. Proximal gradient (forward backward splitting) methods for learning is an area of research in optimization and statistical learning theory which studies algorithms for a general class of convex regularization problems where the regularization penalty may not be different... | Wikipedia - Proximal gradient methods for learning - Summary | 328 | 1,094 | null |
Section: Relevant background. Proximal gradient methods are applicable in a wide variety of scenarios for solving convex optimization problems of the form min x β H F ( x ) + R ( x ) , {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x),} where F {\displaystyle F} is convex and differentiable with Lipschitz continuous ... | Wikipedia - Proximal gradient methods for learning - Relevant background | 289 | 930 | null |
The usual criterion of x {\displaystyle x} minimizes F ( x ) + R ( x ) {\displaystyle F(x)+R(x)} if and only if β ( F + R ) ( x ) = 0 {\displaystyle \nabla (F+R)(x)=0} in the convex, differentiable setting is now replaced by 0 β β ( F + R ) ( x ) , {\displaystyle 0\in \partial (F+R)(x),} where β Ο {\displaystyle \parti... | Wikipedia - Proximal gradient methods for learning - Relevant background | 375 | 982 | null |
Given a convex function Ο : H β R {\displaystyle \varphi :{\mathcal {H}}\to \mathbb {R} } an important operator to consider is its proximal operator prox Ο : H β H {\displaystyle \operatorname {prox} _{\varphi }:{\mathcal {H}}\to {\mathcal {H}}} defined by prox Ο β‘ ( u ) = arg β‘ min x β H Ο ( x ) + 1 2 β u β x β 2 2 , ... | Wikipedia - Proximal gradient methods for learning - Relevant background | 243 | 629 | null |
The proximal operator can be seen as a generalization of a projection. We see that the proximity operator is important because x β {\displaystyle x^{*}} is a minimizer to the problem min x β H F ( x ) + R ( x ) {\displaystyle \min _{x\in {\mathcal {H}}}F(x)+R(x)} if and only if x β = prox Ξ³ R β‘ ( x β β Ξ³ β F ( x β ) ) ... | Wikipedia - Proximal gradient methods for learning - Relevant background | 180 | 485 | null |
Section: Relevant background > Moreau decomposition. One important technique related to proximal gradient methods is the Moreau decomposition, which decomposes the identity operator as the sum of two proximity operators. Namely, let Ο : X β R {\displaystyle \varphi :{\mathcal {X}}\to \mathbb {R} } be a lower semicontin... | Wikipedia - Proximal gradient methods for learning - Relevant background > Moreau decomposition | 172 | 562 | null |
We define its Fenchel conjugate Ο β : X β R {\displaystyle \varphi ^{*}:{\mathcal {X}}\to \mathbb {R} } to be the function Ο β ( u ) := sup x β X β¨ x , u β© β Ο ( x ) . {\displaystyle \varphi ^{*}(u):=\sup _{x\in {\mathcal {X}}}\langle x,u\rangle -\varphi (x).} The general form of Moreau's decomposition states that for ... | Wikipedia - Proximal gradient methods for learning - Relevant background > Moreau decomposition | 323 | 776 | null |
{\displaystyle \varphi ^{*}(u):=\sup _{x\in {\mathcal {X}}}\langle x,u\rangle -\varphi (x).} The general form of Moreau's decomposition states that for any x β X {\displaystyle x\in {\mathcal {X}}} and any Ξ³ > 0 {\displaystyle \gamma >0} that x = prox Ξ³ Ο β‘ ( x ) + Ξ³ prox Ο β / Ξ³ β‘ ( x / Ξ³ ) , {\displaystyle x=\operato... | Wikipedia - Proximal gradient methods for learning - Relevant background > Moreau decomposition | 343 | 1,042 | null |
Section: Lasso regularization. Consider the regularized empirical risk minimization problem with square loss and with the β 1 {\displaystyle \ell _{1}} norm as the regularization penalty: min w β R d 1 n β i = 1 n ( y i β β¨ w , x i β© ) 2 + Ξ» β w β 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1... | Wikipedia - Proximal gradient methods for learning - Lasso regularization | 300 | 850 | null |
Such β 1 {\displaystyle \ell _{1}} regularization problems are interesting because they induce sparse solutions, that is, solutions w {\displaystyle w} to the minimization problem have relatively few nonzero components. Lasso can be seen to be a convex relaxation of the non-convex problem min w β R d 1 n β i = 1 n ( y ... | Wikipedia - Proximal gradient methods for learning - Lasso regularization | 268 | 811 | null |
Section: Lasso regularization > Solving for L1 proximity operator. For simplicity we restrict our attention to the problem where Ξ» = 1 {\displaystyle \lambda =1} . To solve the problem min w β R d 1 n β i = 1 n ( y i β β¨ w , x i β© ) 2 + β w β 1 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}... | Wikipedia - Proximal gradient methods for learning - Lasso regularization > Solving for L1 proximity operator | 318 | 780 | null |
Let us compute the proximity operator for R ( w ) {\displaystyle R(w)} . First we find an alternative characterization of the proximity operator prox R β‘ ( x ) {\displaystyle \operatorname {prox} _{R}(x)} as follows: u = prox R β‘ ( x ) βΊ 0 β β ( R ( u ) + 1 2 β u β x β 2 2 ) βΊ 0 β β R ( u ) + u β x βΊ x β u β β R ( u ) ... | Wikipedia - Proximal gradient methods for learning - Lasso regularization > Solving for L1 proximity operator | 338 | 793 | null |
{\displaystyle \partial |w_{i}|={\begin{cases}1,&w_{i}>0\\-1,&w_{i}<0\\\left[-1,1\right],&w_{i}=0.\end{cases}}} Using the recharacterization of the proximity operator given above, for the choice of R ( w ) = β w β 1 {\displaystyle R(w)=\|w\|_{1}} and Ξ³ > 0 {\displaystyle \gamma >0} we have that prox Ξ³ R β‘ ( x ) {\displ... | Wikipedia - Proximal gradient methods for learning - Lasso regularization > Solving for L1 proximity operator | 349 | 779 | null |
Section: Lasso regularization > Fixed point iterative schemes. To finally solve the lasso problem we consider the fixed point equation shown earlier: x β = prox Ξ³ R β‘ ( x β β Ξ³ β F ( x β ) ) . {\displaystyle x^{*}=\operatorname {prox} _{\gamma R}\left(x^{*}-\gamma \nabla F(x^{*})\right).} Given that we have computed th... | Wikipedia - Proximal gradient methods for learning - Lasso regularization > Fixed point iterative schemes | 293 | 852 | null |
{\displaystyle w^{k+1}=S_{\gamma }\left(w^{k}-\gamma \nabla F\left(w^{k}\right)\right).} Note here the effective trade-off between the empirical error term F ( w ) {\displaystyle F(w)} and the regularization penalty R ( w ) {\displaystyle R(w)} . This fixed point method has decoupled the effect of the two different con... | Wikipedia - Proximal gradient methods for learning - Lasso regularization > Fixed point iterative schemes | 312 | 1,243 | null |
Section: Practical considerations > Adaptive step size. In the fixed point iteration scheme w k + 1 = prox Ξ³ R β‘ ( w k β Ξ³ β F ( w k ) ) , {\displaystyle w^{k+1}=\operatorname {prox} _{\gamma R}\left(w^{k}-\gamma \nabla F\left(w^{k}\right)\right),} one can allow variable step size Ξ³ k {\displaystyle \gamma _{k}} instea... | Wikipedia - Proximal gradient methods for learning - Practical considerations > Adaptive step size | 159 | 590 | null |
Section: Practical considerations > Elastic net (mixed norm regularization). Elastic net regularization offers an alternative to pure β 1 {\displaystyle \ell _{1}} regularization. The problem of lasso ( β 1 {\displaystyle \ell _{1}} ) regularization involves the penalty term R ( w ) = β w β 1 {\displaystyle R(w)=\|w\|_... | Wikipedia - Proximal gradient methods for learning - Practical considerations > Elastic net (mixed norm regularization) | 194 | 665 | null |
This is often avoided by the inclusion of an additional strictly convex term, such as an β 2 {\displaystyle \ell _{2}} norm regularization penalty. For example, one can consider the problem min w β R d 1 n β i = 1 n ( y i β β¨ w , x i β© ) 2 + Ξ» ( ( 1 β ΞΌ ) β w β 1 + ΞΌ β w β 2 2 ) , {\displaystyle \min _{w\in \mathbb {R}... | Wikipedia - Proximal gradient methods for learning - Practical considerations > Elastic net (mixed norm regularization) | 213 | 479 | null |
For example, one can consider the problem min w β R d 1 n β i = 1 n ( y i β β¨ w , x i β© ) 2 + Ξ» ( ( 1 β ΞΌ ) β w β 1 + ΞΌ β w β 2 2 ) , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}(y_{i}-\langle w,x_{i}\rangle )^{2}+\lambda \left((1-\mu )\|w\|_{1}+\mu \|w\|_{2}^{2}\right),} where x i β R d an... | Wikipedia - Proximal gradient methods for learning - Practical considerations > Elastic net (mixed norm regularization) | 323 | 669 | null |
{\displaystyle x_{i}\in \mathbb {R} ^{d}{\text{ and }}y_{i}\in \mathbb {R} .} For 0 < ΞΌ β€ 1 {\displaystyle 0<\mu \leq 1} the penalty term Ξ» ( ( 1 β ΞΌ ) β w β 1 + ΞΌ β w β 2 2 ) {\displaystyle \lambda \left((1-\mu )\|w\|_{1}+\mu \|w\|_{2}^{2}\right)} is now strictly convex, and hence the minimization problem now admits a... | Wikipedia - Proximal gradient methods for learning - Practical considerations > Elastic net (mixed norm regularization) | 221 | 615 | null |
Section: Exploiting group structure > Group lasso. Group lasso is a generalization of the lasso method when features are grouped into disjoint blocks. Suppose the features are grouped into blocks { w 1 , β¦ , w G } {\displaystyle \{w_{1},\ldots ,w_{G}\}} . Here we take as a regularization penalty R ( w ) = β g = 1 G β w... | Wikipedia - Proximal gradient methods for learning - Exploiting group structure > Group lasso | 219 | 793 | null |
Where the lasso penalty has a proximity operator which is soft thresholding on each individual component, the proximity operator for the group lasso is soft thresholding on each group. For the group w g {\displaystyle w_{g}} we have that proximity operator of Ξ» Ξ³ ( β g = 1 G β w g β 2 ) {\displaystyle \lambda \gamma \l... | Wikipedia - Proximal gradient methods for learning - Exploiting group structure > Group lasso | 325 | 843 | null |
Article: Pythia (machine learning). Pythia is an ancient text restoration model that recovers missing characters from a damaged text input using deep neural networks. It was created by Yannis Assael, Thea Sommerschield, and Jonathan Prag, researchers from Google DeepMind and the University of Oxford. To study the socie... | Wikipedia - Pythia (machine learning) - Summary | 274 | 1,355 | null |
Article: Quantification (machine learning). In machine learning and data mining, quantification (variously called learning to quantify, or supervised prevalence estimation, or class prior estimation) is the task of using supervised learning in order to train models (quantifiers) that estimate the relative frequencies (... | Wikipedia - Quantification (machine learning) - Summary | 346 | 1,716 | null |
It has been shown in multiple research works that performing quantification by classifying all unlabelled instances and then counting the instances that have been attributed to each class (the 'classify and count' method) usually leads to suboptimal quantification accuracy. This suboptimality may be seen as a direct co... | Wikipedia - Quantification (machine learning) - Summary | 207 | 1,034 | null |
Section: Quantification tasks. The main variants of quantification, according to the characteristics of the set of classes used, are: Binary quantification, corresponding to the case in which there are only n = 2 {\displaystyle n=2} classes and each data item belongs to exactly one of them; Single-label multiclass quan... | Wikipedia - Quantification (machine learning) - Quantification tasks | 328 | 1,433 | null |
Binary-only methods include the Mixture Model (MM) method, the HDy method, SVM(KLD), and SVM(Q). Methods that can deal with both the binary case and the single-label multiclass case include probabilistic classify and count (PCC), adjusted classify and count (ACC), probabilistic adjusted classify and count (PACC), and t... | Wikipedia - Quantification (machine learning) - Quantification tasks | 196 | 738 | null |
Article: Quantum machine learning. Quantum machine learning is the integration of quantum algorithms within machine learning programs. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer, i.e. quantum-enhanced machine learning. While ma... | Wikipedia - Quantum machine learning - Summary | 287 | 1,734 | null |
Section: Machine learning with quantum computers > Quantum associative memories and quantum pattern recognition. Associative (or content-addressable) memories are able to recognize stored content on the basis of a similarity measure, while random access memories are accessed by the address of stored information and not... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum associative memories and quantum pattern recognition | 341 | 1,678 | null |
Section: Machine learning with quantum computers > Linear algebra simulation with quantum amplitudes. A number of quantum algorithms for machine learning are based on the idea of amplitude encoding, that is, to associate the amplitudes of a quantum state with the inputs and outputs of computations. Since a state of n {... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Linear algebra simulation with quantum amplitudes | 312 | 1,612 | null |
For reference, any known classical algorithm for matrix inversion requires a number of operations that grows more than quadratically in the dimension of the matrix (e.g. O ( n 2.373 ) {\displaystyle O{\mathord {\left(n^{2.373}\right)}}} ), but they are not restricted to sparse matrices. Quantum matrix inversion can be ... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Linear algebra simulation with quantum amplitudes | 197 | 944 | null |
Section: Machine learning with quantum computers > Variational quantum circuits (VQCs). Variational Quantum Circuits also known as Parametrized Quantum Circuits (PQCs) are based on Variational Quantum Algorithms (VQAs). VQCs consist of three parts: preparation of initial states, quantum circuit, and measurement. Resear... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Variational quantum circuits (VQCs) | 218 | 1,133 | null |
Section: Machine learning with quantum computers > Quantum machine learning algorithms based on Grover search. Another approach to improving classical machine learning with quantum information processing uses amplitude amplification methods based on Grover's search algorithm, which has been shown to solve unstructured ... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum machine learning algorithms based on Grover search | 250 | 1,285 | null |
The minimization is initialized by some random element in our data set, and iteratively does this subroutine to find the minimum element in the data set. This minimization is notably used in quantum k-medians, and it has a speed up of at least O ( n k ) {\displaystyle {\mathcal {O}}\left({\sqrt {\frac {n}{k}}}\right)} ... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum machine learning algorithms based on Grover search | 183 | 742 | null |
Section: Machine learning with quantum computers > Quantum-enhanced reinforcement learning. Reinforcement learning is a branch of machine learning distinct from supervised and unsupervised learning, which also admits quantum enhancements. In quantum-enhanced reinforcement learning, a quantum agent interacts with a clas... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum-enhanced reinforcement learning | 206 | 1,113 | null |
Section: Machine learning with quantum computers > Quantum annealing. Quantum annealing is an optimization technique used to determine the local minima and maxima of a function over a given set of candidate functions. This is a method of discretizing a function with many local minima or maxima in order to determine the... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum annealing | 173 | 888 | null |
Section: Machine learning with quantum computers > NISQ Circuit as Quantum Model. As the depth of the quantum circuit advances on NISQ devices, the noise level rises, posing a significant challenge to accurately computing costs and gradients on training models. The noise tolerance will be improved by using the quantum ... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > NISQ Circuit as Quantum Model | 166 | 933 | null |
Section: Machine learning with quantum computers > Quantum sampling techniques. Sampling from high-dimensional probability distributions is at the core of a wide spectrum of computational techniques with important applications across science, engineering, and society. Examples include deep learning, probabilistic progr... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum sampling techniques | 350 | 2,022 | null |
Complementary work that appeared roughly simultaneously showed that quantum annealing can be used for supervised learning in classification tasks. The same device was later used to train a fully connected Boltzmann machine to generate, reconstruct, and classify down-scaled, low-resolution handwritten digits, among othe... | Wikipedia - Quantum machine learning - Machine learning with quantum computers > Quantum sampling techniques | 323 | 1,673 | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.