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Let w g {\displaystyle w_{g}} be the vector of coefficients in group g {\displaystyle g} , we can define a regularization term and its group norm as Ξ» R ( w ) = Ξ» β g = 1 G β w g β g {\displaystyle \lambda R(w)=\lambda \sum _{g=1}^{G}\|w_{g}\|_{g}} , where β w g β g {\displaystyle \|w_{g}\|_{g}} is the group β 2 {\disp... | Wikipedia - Structured sparsity regularization - Structures and norms > Non-overlapping groups: group Lasso | 348 | 958 | null |
Section: Structures and norms > Overlapping groups > Intersection of complements: group Lasso. The intersection of complements approach is used in cases when we want to select only those input variables that have positive coefficients in all groups they belong to. Consider again the group Lasso for a regularized empiri... | Wikipedia - Structured sparsity regularization - Structures and norms > Overlapping groups > Intersection of complements: group Lasso | 302 | 1,038 | null |
The formulation of the union of groups approach is also referred to as latent group Lasso, and requires to modify the group β 2 {\displaystyle \ell _{2}} norm considered above and introduce the following regularizer R ( w ) = i n f { β g β w g β g : w = β g = 1 G w Β― g } {\displaystyle R(w)=inf\left\{\sum _{g}\|w_{g}\|... | Wikipedia - Structured sparsity regularization - Structures and norms > Overlapping groups > Union of groups: latent group Lasso | 350 | 866 | null |
Section: Structures and norms > Issues with Group Lasso regularization and alternative approaches. The objective function using group lasso consists of an error function, which is generally required to be convex but not necessarily strongly convex, and a group β 1 {\displaystyle \ell _{1}} regularization term. An issue... | Wikipedia - Structured sparsity regularization - Structures and norms > Issues with Group Lasso regularization and alternative approaches | 328 | 1,495 | null |
Section: Structures and norms > Norms based on the structure over Input variables > Hierarchical Norms. See: Unsupervised learning Unsupervised learning methods are often used to learn the parameters of latent variable models. Latent variable models are statistical models where in addition to the observed variables, a ... | Wikipedia - Structured sparsity regularization - Structures and norms > Norms based on the structure over Input variables > Hierarchical Norms | 205 | 1,032 | null |
Section: Algorithms for computation > Best subset selection problem. The problem of choosing the best subset of input variables can be naturally formulated under a penalization framework as: min w β R d 1 n β i = 1 n V ( y i , w , x i ) + Ξ» β w β 0 , {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}... | Wikipedia - Structured sparsity regularization - Algorithms for computation > Best subset selection problem | 277 | 977 | null |
Section: Algorithms for computation > Convex relaxation. A natural approximation for the best subset selection problem is the β 1 {\displaystyle \ell _{1}} norm regularization: min w β R d 1 n β i = 1 n V ( y i , w , x i ) + Ξ» β w β 1 {\displaystyle \min _{w\in \mathbb {R} ^{d}}{\frac {1}{n}}\sum _{i=1}^{n}V(y_{i},w,x_... | Wikipedia - Structured sparsity regularization - Algorithms for computation > Convex relaxation | 196 | 525 | null |
Section: Algorithms for computation > Proximal gradient methods. Proximal gradient methods, also called forward-backward splitting, are optimization methods useful for minimizing functions with a convex and differentiable component, and a convex potentially non-differentiable component. As such, proximal gradient metho... | Wikipedia - Structured sparsity regularization - Algorithms for computation > Proximal gradient methods | 263 | 842 | null |
Section: Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning. Structured Sparsity regularization can be applied in the context of multiple kernel learning. Multiple kernel learning refers to a set of machine learning methods that use a predefined set of kernels and learn an optimal l... | Wikipedia - Structured sparsity regularization - Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning | 304 | 1,314 | null |
. . , q } {\displaystyle B=\{b_{t}:X\rightarrow \mathbb {R} ,t=1,...,q\}} where q {\displaystyle q} and p {\displaystyle p} are integers, will be considered. The atoms in A {\displaystyle A} as well as the atoms in B {\displaystyle B} are assumed to be linearly independent. Let D = { d k : X β R , k = 1 , . . . , p + q... | Wikipedia - Structured sparsity regularization - Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning | 179 | 442 | null |
, p + q } = A βͺ B {\displaystyle D=\{d_{k}:X\rightarrow \mathbb {R} ,k=1,...,p+q\}=A\cup B} be the union of the two dictionaries. Consider the linear space of functions H {\displaystyle H} given by linear combinations of the form f ( x ) = β i = 1 p + q w j d j ( x ) = β j = 1 p w A j a j ( x ) + β t = 1 q w B t b t ( ... | Wikipedia - Structured sparsity regularization - Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning | 330 | 641 | null |
Consider the linear space of functions H {\displaystyle H} given by linear combinations of the form f ( x ) = β i = 1 p + q w j d j ( x ) = β j = 1 p w A j a j ( x ) + β t = 1 q w B t b t ( x ) , x β X {\displaystyle f(x)=\sum _{i=1}^{p+q}{w^{j}d_{j}(x)}=\sum _{j=1}^{p}{w_{A}^{j}a_{j}(x)}+\sum _{t=1}^{q}{w_{B}^{t}b_{t}... | Wikipedia - Structured sparsity regularization - Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning | 331 | 693 | null |
Assume the atoms in D {\displaystyle D} to still be linearly independent, or equivalently, that the map w = ( w A , w B ) β¦ f {\displaystyle w=(w_{A},w_{B})\mapsto f} is one to one. The functions in the space H {\displaystyle H} can be seen as the sums of two components, one in the space H A {\displaystyle H_{A}} , the... | Wikipedia - Structured sparsity regularization - Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning | 255 | 735 | null |
Note that we can now view H {\displaystyle H} as a function space in which H A {\displaystyle H_{A}} , H B {\displaystyle H_{B}} are subspaces. In view of the linear independence assumption, H {\displaystyle H} can be identified with R p + q {\displaystyle \mathbb {R} ^{p+q}} and H A , H B {\displaystyle H_{A},H_{B}} w... | Wikipedia - Structured sparsity regularization - Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning | 312 | 906 | null |
. . , a p ( x ) ) {\displaystyle \Phi _{A}(x)=(a_{1}(x),...,a_{p}(x))} , Ξ¦ B : X β R q {\displaystyle \Phi _{B}:X\rightarrow \mathbb {R} ^{q}} , given by Ξ¦ B ( x ) = ( b 1 ( x ) , . . . , b q ( x ) ) {\displaystyle \Phi _{B}(x)=(b_{1}(x),...,b_{q}(x))} , and Ξ¦ : X β R p + q {\displaystyle \Phi :X\rightarrow \mathbb {R}... | Wikipedia - Structured sparsity regularization - Connections to Other Areas of Machine Learning > Connection to Multiple Kernel Learning | 331 | 1,007 | null |
Article: Surrogate model. A surrogate model is an engineering method used when an outcome of interest cannot be easily measured or computed, so an approximate mathematical model of the outcome is used instead. Most engineering design problems require experiments and/or simulations to evaluate design objective and const... | Wikipedia - Surrogate model - Summary | 330 | 1,705 | null |
Section: Goals. The scientific challenge of surrogate modeling is the generation of a surrogate that is as accurate as possible, using as few simulation evaluations as possible. The process comprises three major steps which may be interleaved iteratively: Sample selection (also known as sequential design, optimal exper... | Wikipedia - Surrogate model - Goals | 166 | 821 | null |
Section: Types of surrogate models. Popular surrogate modeling approaches are: polynomial response surfaces; kriging; more generalized Bayesian approaches; gradient-enhanced kriging (GEK); radial basis function; support vector machines; space mapping; artificial neural networks and Bayesian networks. Other methods rece... | Wikipedia - Surrogate model - Types of surrogate models | 159 | 809 | null |
Section: Applications. An important distinction can be made between two different applications of surrogate models: design optimization and design space approximation (also known as emulation). In surrogate model-based optimization, an initial surrogate is constructed using some of the available budgets of expensive ex... | Wikipedia - Surrogate model - Applications | 310 | 1,575 | null |
Section: Surrogate modeling software. Surrogate Modeling Toolbox (SMT: https://github.com/SMTorg/smt) is a Python package that contains a collection of surrogate modeling methods, sampling techniques, and benchmarking functions. This package provides a library of surrogate models that is simple to use and facilitates t... | Wikipedia - Surrogate model - Surrogate modeling software | 202 | 1,021 | null |
Section: Surrogate-Assisted Evolutionary Algorithms (SAEAs). SAEAs are an advanced class of optimization techniques that integrate evolutionary algorithms (EAs) with surrogate models. In traditional EAs, evaluating the fitness of candidate solutions often requires computationally expensive simulations or experiments. S... | Wikipedia - Surrogate model - Surrogate-Assisted Evolutionary Algorithms (SAEAs) | 300 | 1,721 | null |
Article: Tensor (machine learning). In machine learning, the term tensor informally refers to two different concepts (i) a way of organizing data and (ii) a multilinear (tensor) transformation. Data may be organized in a multidimensional array (M-way array), informally referred to as a "data tensor"; however, in the st... | Wikipedia - Tensor (machine learning) - Summary | 268 | 1,279 | null |
Section: History. A tensor is by definition a multilinear map. In mathematics, this may express a multilinear relationship between sets of algebraic objects. In physics, tensor fields, considered as tensors at each point in space, are useful in expressing mechanics such as stress or elasticity. In machine learning, the... | Wikipedia - Tensor (machine learning) - History | 331 | 1,704 | null |
Tensor (multilinear) factor analysis disentangles and reduces the influence of different causal factors with multilinear subspace learning. When treating an image or a video as a 2- or 3-way array, i.e., "data matrix/tensor", tensor methods reduce spatial or time redundancies as demonstrated by Wang and Ahuja. Yoshua B... | Wikipedia - Tensor (machine learning) - History | 288 | 1,363 | null |
Section: Definition. Let F {\displaystyle \mathbb {F} } be a field such as the real numbers R {\displaystyle \mathbb {R} } or the complex numbers C {\displaystyle \mathbb {C} } . A tensor T β F I 0 Γ I 2 Γ β¦ Γ I C {\displaystyle {\mathcal {T}}\in {\mathbb {F} }^{I_{0}\times I_{2}\times \ldots \times I_{C}}} is a multil... | Wikipedia - Tensor (machine learning) - Definition | 340 | 836 | null |
A tensor T β F I 0 Γ I 2 Γ β¦ Γ I C {\displaystyle {\mathcal {T}}\in {\mathbb {F} }^{I_{0}\times I_{2}\times \ldots \times I_{C}}} is a multilinear transformation from a set of domain vector spaces to a range vector space: T : { F I 1 Γ F I 2 Γ β¦ F I C } β¦ F I 0 {\displaystyle {\mathcal {T}}:\{{\mathbb {F} }^{I_{1}}\tim... | Wikipedia - Tensor (machine learning) - Definition | 333 | 781 | null |
In statistics and machine learning, an image is vectorized when viewed as a single observation, and a collection of vectorized images is organized as a "data tensor". For example, a set of facial images { d i p , i e , i l , i v β R I X } {\displaystyle \{{\mathbb {d} }_{i_{p},i_{e},i_{l},i_{v}}\in {\mathbb {R} }^{I_{X... | Wikipedia - Tensor (machine learning) - Definition | 331 | 822 | null |
For example, a set of facial images { d i p , i e , i l , i v β R I X } {\displaystyle \{{\mathbb {d} }_{i_{p},i_{e},i_{l},i_{v}}\in {\mathbb {R} }^{I_{X}}\}} with I X {\displaystyle I_{X}} pixels that are the consequences of multiple causal factors, such as a facial geometry i p ( 1 β€ i p β€ I P ) {\displaystyle i_{p}(... | Wikipedia - Tensor (machine learning) - Definition | 455 | 1,091 | null |
multiway array) D β R I X Γ I P Γ I E Γ I L Γ V {\displaystyle {\mathcal {D}}\in {\mathbb {R} }^{I_{X}\times I_{P}\times I_{E}\times I_{L}\times V}} where I P {\displaystyle I_{P}} are the total number of facial geometries, I E {\displaystyle I_{E}} are the total number of expressions, I L {\displaystyle I_{L}} are the... | Wikipedia - Tensor (machine learning) - Definition | 249 | 892 | null |
Another approach to using tensors in machine learning is to embed various data types directly. For example, a grayscale image, commonly represented as a discrete 2-way array D β R I R X Γ I C X {\displaystyle {\mathbf {D} }\in {\mathbb {R} }^{I_{RX}\times I_{CX}}} with dimensionality I R X Γ I C X {\displaystyle I_{RX}... | Wikipedia - Tensor (machine learning) - Definition | 335 | 1,081 | null |
Thus v {\displaystyle v} becomes a mode-1 tensor v β¦ A β R N . {\displaystyle v\mapsto {\mathcal {A}}\in \mathbb {R} ^{N}.} The embedding of subject-object-verb semantics requires embedding relationships among three words. Because a word is itself a vector, subject-object-verb semantics could be expressed using mode-3 ... | Wikipedia - Tensor (machine learning) - Definition | 219 | 769 | null |
Section: Definition > As unit values. Tensors may be used as the unit values of neural networks which extend the concept of scalar, vector and matrix values to multiple dimensions. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units and the connection weights filtered t... | Wikipedia - Tensor (machine learning) - Definition > As unit values | 327 | 1,144 | null |
Section: Definition > In fully connected layers. Tensors may also be used to compute the layers of a fully connected neural network, where the tensor is applied to the entire layer instead of individual unit values. The output value of single layer unit y m {\displaystyle y_{m}} is the sum-product of its input units an... | Wikipedia - Tensor (machine learning) - Definition > In fully connected layers | 308 | 983 | null |
In this example the unit values are scalars while the tensor takes on the dimensions of the network layers: x n β¦ X β R 1 Γ N , {\displaystyle x_{n}\mapsto {\mathcal {X}}\in \mathbb {R} ^{1\times N},} y n β¦ Y β R M Γ 1 , {\displaystyle y_{n}\mapsto {\mathcal {Y}}\in \mathbb {R} ^{M\times 1},} u n β¦ U β R N Γ M . {\disp... | Wikipedia - Tensor (machine learning) - Definition > In fully connected layers | 272 | 819 | null |
Section: Definition > In convolutional layers. A different reformulation of neural networks allows tensors to express the convolution layers of a neural network. A convolutional layer has multiple inputs, each of which is a spatial structure such as an image or volume. The inputs are convolved by filtering before being... | Wikipedia - Tensor (machine learning) - Definition > In convolutional layers | 275 | 958 | null |
This definition can be rephrased as a matrix-vector product in terms of tensors that express the kernel, data and inverse transform of the kernel. Y = A [ ( C g ) β ( B d ) ] , {\displaystyle {\mathcal {Y}}={\mathcal {A}}[(Cg)\odot (Bd)],} where A , B {\displaystyle {\mathcal {A}},{\mathcal {B}}} and C {\displaystyle {... | Wikipedia - Tensor (machine learning) - Definition > In convolutional layers | 205 | 725 | null |
Section: Definition > Tensor factorization > Tucker decomposition. Tucker decomposition, for example, takes a 3-way array X β R I Γ J Γ K {\displaystyle {\mathcal {X}}\in \mathbb {R} ^{I\times J\times K}} and decomposes the tensor into three matrices A , B , C {\displaystyle {\mathcal {A,B,C}}} and a smaller tensor G {... | Wikipedia - Tensor (machine learning) - Definition > Tensor factorization > Tucker decomposition | 313 | 890 | null |
{\displaystyle {\mathcal {G}}\in \mathbb {R} ^{P\times Q\times R}.} Then the original tensor can be expressed as the tensor product of these four tensors: X = G Γ A Γ B Γ C . {\displaystyle {\mathcal {X}}={\mathcal {G}}\times {\mathcal {A}}\times {\mathcal {B}}\times {\mathcal {C}}.} In the example shown in the figure,... | Wikipedia - Tensor (machine learning) - Definition > Tensor factorization > Tucker decomposition | 250 | 589 | null |
{\displaystyle {\mathcal {X}}={\mathcal {G}}\times {\mathcal {A}}\times {\mathcal {B}}\times {\mathcal {C}}.} In the example shown in the figure, the dimensions of the tensors are X {\displaystyle {\mathcal {X}}} : I=8, J=6, K=3, A {\displaystyle {\mathcal {A}}} : I=8, P=5, B {\displaystyle {\mathcal {B}}} : J=6, Q=4, ... | Wikipedia - Tensor (machine learning) - Definition > Tensor factorization > Tucker decomposition | 313 | 684 | null |
The total number of elements in the Tucker factorization is | A | + | B | + | C | + | G | = {\displaystyle |{\mathcal {A}}|+|{\mathcal {B}}|+|{\mathcal {C}}|+|{\mathcal {G}}|=} ( I Γ P ) + ( J Γ Q ) + ( K Γ R ) + ( P Γ Q Γ R ) = 8 Γ 5 + 6 Γ 4 + 3 Γ 2 + 5 Γ 4 Γ 2 = 110. {\displaystyle (I\times P)+(J\times Q)+(K\times R)... | Wikipedia - Tensor (machine learning) - Definition > Tensor factorization > Tucker decomposition | 295 | 881 | null |
Section: Definition > Tensor factorization > Tensor trains. Another technique for decomposing tensors rewrites the initial tensor as a sequence (train) of smaller sized tensors. A tensor-train (TT) is a sequence of tensors of reduced rank, called canonical factors. The original tensor can be expressed as the sum-produc... | Wikipedia - Tensor (machine learning) - Definition > Tensor factorization > Tensor trains | 206 | 742 | null |
Section: Hardware. Tensors provide a unified way to train neural networks for more complex data sets. However, training is expensive to compute on classical CPU hardware. In 2014, Nvidia developed cuDNN, CUDA Deep Neural Network, a library for a set of optimized primitives written in the parallel CUDA language. CUDA an... | Wikipedia - Tensor (machine learning) - Hardware | 321 | 1,526 | null |
The development of GPU hardware, combined with the unified architecture of tensor cores, has enabled the training of much larger neural networks. In 2022, the largest neural network was Google's PaLM with 540 billion learned parameters (network weights) (the older GPT-3 language model has over 175 billion learned param... | Wikipedia - Tensor (machine learning) - Hardware | 154 | 690 | null |
Article: Time series. In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, coun... | Wikipedia - Time series - Summary | 334 | 1,772 | null |
explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as... | Wikipedia - Time series - Summary | 192 | 990 | null |
Section: Methods for analysis. Methods for time series analysis may be divided into two classes: frequency-domain methods and time-domain methods. The former include spectral analysis and wavelet analysis; the latter include auto-correlation and cross-correlation analysis. In the time domain, correlation and analysis c... | Wikipedia - Time series - Methods for analysis | 225 | 1,172 | null |
Section: Panel data. A time series is one type of panel data. Panel data is the general class, a multidimensional data set, whereas a time series data set is a one-dimensional panel (as is a cross-sectional dataset). A data set may exhibit characteristics of both panel data and time series data. One way to tell is to a... | Wikipedia - Time series - Panel data | 174 | 774 | null |
Section: Analysis > Estimation, filtering, and smoothing. This approach may be based on harmonic analysis and filtering of signals in the frequency domain using the Fourier transform, and spectral density estimation. Its development was significantly accelerated during World War II by mathematician Norbert Wiener, elec... | Wikipedia - Time series - Analysis > Estimation, filtering, and smoothing | 196 | 1,123 | null |
Section: Analysis > Curve fitting. Curve fitting is the process of constructing a curve, or mathematical function, that has the best fit to a series of data points, possibly subject to constraints. Curve fitting can involve either interpolation, where an exact fit to the data is required, or smoothing, in which a "smoo... | Wikipedia - Time series - Analysis > Curve fitting | 341 | 1,757 | null |
This is often done by using a related series known for all relevant dates. Alternatively polynomial interpolation or spline interpolation is used where piecewise polynomial functions are fitted in time intervals such that they fit smoothly together. A different problem which is closely related to interpolation is the a... | Wikipedia - Time series - Analysis > Curve fitting | 185 | 1,024 | null |
Section: Analysis > Function approximation. In general, a function approximation problem asks us to select a function among a well-defined class that closely matches ("approximates") a target function in a task-specific way. One can distinguish two major classes of function approximation problems: First, for known targ... | Wikipedia - Time series - Analysis > Function approximation | 327 | 1,653 | null |
Section: Analysis > Prediction and forecasting. In statistics, prediction is a part of statistical inference. One particular approach to such inference is known as predictive inference, but the prediction can be undertaken within any of the several approaches to statistical inference. Indeed, one description of statist... | Wikipedia - Time series - Analysis > Prediction and forecasting | 240 | 1,303 | null |
Section: Models. Models for time series data can have many forms and represent different stochastic processes. When modeling variations in the level of a process, three broad classes of practical importance are the autoregressive (AR) models, the integrated (I) models, and the moving-average (MA) models. These three cl... | Wikipedia - Time series - Models | 309 | 1,423 | null |
Non-linear dependence of the level of a series on previous data points is of interest, partly because of the possibility of producing a chaotic time series. However, more importantly, empirical investigations can indicate the advantage of using predictions derived from non-linear models, over those from linear models, ... | Wikipedia - Time series - Models | 323 | 1,527 | null |
Section: Models > Conditions. There are two sets of conditions under which much of the theory is built: Stationary process Ergodic process Ergodicity implies stationarity, but the converse is not necessarily the case. Stationarity is usually classified into strict stationarity and wide-sense or second-order stationarit... | Wikipedia - Time series - Models > Conditions | 155 | 793 | null |
Section: Models > Tools. Tools for investigating time-series data include: Consideration of the autocorrelation function and the spectral density function (also cross-correlation functions and cross-spectral density functions) Scaled cross- and auto-correlation functions to remove contributions of slow components Perfo... | Wikipedia - Time series - Models > Tools | 247 | 1,477 | null |
Section: Models > Measures. Time-series metrics or features that can be used for time series classification or regression analysis: Univariate linear measures Moment (mathematics) Spectral band power Spectral edge frequency Accumulated energy (signal processing) Characteristics of the autocorrelation function Hjorth pa... | Wikipedia - Time series - Models > Measures | 349 | 1,889 | null |
Article: Toronto Declaration. The Toronto Declaration: Protecting the Rights to Equality and Non-Discrimination in Machine Learning Systems is a declaration that advocates responsible practices for machine learning practitioners and governing bodies. It is a joint statement issued by groups including Amnesty Internatio... | Wikipedia - Toronto Declaration - Summary | 193 | 1,105 | null |
Section: Contents > Using the framework of international human rights law. The framework of international human rights law enumerates various rights, provides mechanisms to hold violators to account, and ensures remedy for the violated. The document cites the United Nations Human Rights Committee's definition of discri... | Wikipedia - Toronto Declaration - Contents > Using the framework of international human rights law | 195 | 1,093 | null |
Section: Contents > Duties of states: human rights obligations. Governments today are deploying machine learning systems, often in collaboration with private entities. Even when development is contracted to such third parties, governments retain their obligation to protect human rights. Before implementation, and on an... | Wikipedia - Toronto Declaration - Contents > Duties of states: human rights obligations | 181 | 1,106 | null |
Article: Transduction (machine learning). In logic, statistical inference, and supervised learning, transduction or transductive inference is reasoning from observed, specific (training) cases to specific (test) cases. In contrast, induction is reasoning from observed training cases to general rules, which are then app... | Wikipedia - Transduction (machine learning) - Summary | 344 | 1,771 | null |
The most well-known example of a case-bases learning algorithm is the k-nearest neighbor algorithm, which is related to transductive learning algorithms. Another example of an algorithm in this category is the Transductive Support Vector Machine (TSVM). A third possible motivation of transduction arises through the nee... | Wikipedia - Transduction (machine learning) - Summary | 151 | 765 | null |
Section: Historical context. The mode of inference from particulars to particulars, which Vapnik came to call transduction, was already distinguished from the mode of inference from particulars to generalizations in part III of the Cambridge philosopher and logician W.E. Johnson's 1924 textbook, Logic. In Johnson's wor... | Wikipedia - Transduction (machine learning) - Historical context | 196 | 957 | null |
Section: Example problem. The following example problem contrasts some of the unique properties of transduction against induction. A collection of points is given, such that some of the points are labeled (A, B, or C), but most of the points are unlabeled (?). The goal is to predict appropriate labels for all of the un... | Wikipedia - Transduction (machine learning) - Example problem | 341 | 1,627 | null |
Section: History. In 1976, Bozinovski and Fulgosi published a paper addressing transfer learning in neural network training. The paper gives a mathematical and geometrical model of the topic. In 1981, a report considered the application of transfer learning to a dataset of images representing letters of computer termin... | Wikipedia - Transfer learning - History | 203 | 1,010 | null |
Section: Definition. The definition of transfer learning is given in terms of domains and tasks. A domain D {\displaystyle {\mathcal {D}}} consists of: a feature space X {\displaystyle {\mathcal {X}}} and a marginal probability distribution P ( X ) {\displaystyle P(X)} , where X = { x 1 , . . . , x n } β X {\displaysty... | Wikipedia - Transfer learning - Definition | 277 | 802 | null |
The function f {\displaystyle f} is used to predict the corresponding label f ( x ) {\displaystyle f(x)} of a new instance x {\displaystyle x} . This task, denoted by T = { Y , f ( x ) } {\displaystyle {\mathcal {T}}=\{{\mathcal {Y}},f(x)\}} , is learned from the training data consisting of pairs { x i , y i } {\displa... | Wikipedia - Transfer learning - Definition | 176 | 453 | null |
This task, denoted by T = { Y , f ( x ) } {\displaystyle {\mathcal {T}}=\{{\mathcal {Y}},f(x)\}} , is learned from the training data consisting of pairs { x i , y i } {\displaystyle \{x_{i},y_{i}\}} , where x i β X {\displaystyle x_{i}\in {\mathcal {X}}} and y i β Y {\displaystyle y_{i}\in {\mathcal {Y}}} . Given a sou... | Wikipedia - Transfer learning - Definition | 402 | 966 | null |
Section: Applications. Algorithms are available for transfer learning in Markov logic networks and Bayesian networks. Transfer learning has been applied to cancer subtype discovery, building utilization, general game playing, text classification, digit recognition, medical imaging and spam filtering. In 2020, it was di... | Wikipedia - Transfer learning - Applications | 228 | 1,193 | null |
Section: Mathematical formula. Suppose there are n things in the universe, and one wants to put them into classes or categories. One has no preconceived ideas or biases about what sorts of categories are "natural" or "normal" and what are not. So one has to consider all the possible classes that could be, all the possi... | Wikipedia - Ugly duckling theorem - Mathematical formula | 305 | 1,270 | null |
One may pick two elements and reorder the bits so they are the first two, and imagine the numbers sorted lexicographically. The first 2 n / 2 {\displaystyle 2^{n}/2} numbers will have bit #1 set to zero, and the second 2 n / 2 {\displaystyle 2^{n}/2} will have it set to one. Within each of those blocks, the top 2 n / 4... | Wikipedia - Ugly duckling theorem - Mathematical formula | 252 | 920 | null |
Section: Mathematical formula > Boolean functions. Let x 1 , x 2 , β¦ , x n {\displaystyle x_{1},x_{2},\dots ,x_{n}} be a set of vectors of k {\displaystyle k} booleans each. The ugly duckling is the vector which is least like the others. Given the booleans, this can be computed using Hamming distance. However, the choi... | Wikipedia - Ugly duckling theorem - Mathematical formula > Boolean functions | 333 | 1,388 | null |
If x and y are different, then there exists a coordinate i {\displaystyle i} where the i {\displaystyle i} -th coordinate of x {\displaystyle x} differs from the i {\displaystyle i} -th coordinate of y {\displaystyle y} . Now the completed features contain every Boolean function on k {\displaystyle k} Boolean variables... | Wikipedia - Ugly duckling theorem - Mathematical formula > Boolean functions | 255 | 941 | null |
Section: Discussion. A possible way around the ugly duckling theorem would be to introduce a constraint on how similarity is measured by limiting the properties involved in classification, for instance, between A and B. However Medin et al. (1993) point out that this does not actually resolve the arbitrariness or bias ... | Wikipedia - Ugly duckling theorem - Discussion | 300 | 1,436 | null |
If you are in the jungle and you see a tiger but you decide not to stereotype (perhaps because you believe that similarity is a false friend), then you will probably be eaten. In other words, in the biological world stereotyping based on veridical judgments of overall similarity statistically results in greater surviva... | Wikipedia - Ugly duckling theorem - Discussion | 342 | 1,581 | null |
Article: Uncertain data. In computer science, uncertain data is data that contains noise that makes it deviate from the correct, intended or original values. In the age of big data, uncertainty or data veracity is one of the defining characteristics of data. Data is constantly growing in volume, variety, velocity and u... | Wikipedia - Uncertain data - Summary | 319 | 1,716 | null |
Section: Example data model for uncertain data. One way to represent uncertain data is through probability distributions. Let us take the example of a relational database. There are three main ways to do represent uncertainty as probability distributions in such a database model. In attribute uncertainty, each uncertai... | Wikipedia - Uncertain data - Example data model for uncertain data | 285 | 1,526 | null |
Article: Uniform convergence in probability. Uniform convergence in probability is a form of convergence in probability in statistical asymptotic theory and probability theory. It means that, under certain conditions, the empirical frequencies of all events in a certain event-family converge to their theoretical probab... | Wikipedia - Uniform convergence in probability - Summary | 220 | 1,207 | null |
Section: Definitions. For a class of predicates H {\displaystyle H} defined on a set X {\displaystyle X} and a set of samples x = ( x 1 , x 2 , β¦ , x m ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{m})} , where x i β X {\displaystyle x_{i}\in X} , the empirical frequency of h β H {\displaystyle h\in H} on x {\displaystyle ... | Wikipedia - Uniform convergence in probability - Definitions | 251 | 583 | null |
{\displaystyle {\widehat {Q}}_{x}(h)={\frac {1}{m}}|\{i:1\leq i\leq m,h(x_{i})=1\}|.} The theoretical probability of h β H {\displaystyle h\in H} is defined as Q P ( h ) = P { y β X : h ( y ) = 1 } . {\displaystyle Q_{P}(h)=P\{y\in X:h(y)=1\}.} The Uniform Convergence Theorem states, roughly, that if H {\displaystyle H... | Wikipedia - Uniform convergence in probability - Definitions | 292 | 996 | null |
Section: Uniform convergence theorem. The statement of the uniform convergence theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for Ξ΅ > 0 {\... | Wikipedia - Uniform convergence in probability - Uniform convergence theorem | 153 | 488 | null |
The statement of the uniform convergence theorem is as follows: If H {\displaystyle H} is a set of { 0 , 1 } {\displaystyle \{0,1\}} -valued functions defined on a set X {\displaystyle X} and P {\displaystyle P} is a probability distribution on X {\displaystyle X} then for Ξ΅ > 0 {\displaystyle \varepsilon >0} and m {\d... | Wikipedia - Uniform convergence in probability - Uniform convergence theorem | 421 | 915 | null |
{\displaystyle P^{m}\{|Q_{P}(h)-{\widehat {Q_{x}}}(h)|\geq \varepsilon {\text{ for some }}h\in H\}\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}.} where, for any x β X m , {\displaystyle x\in X^{m},} , Q P ( h ) = P { ( y β X : h ( y ) = 1 } , {\displaystyle Q_{P}(h)=P\{(y\in X:h(y)=1\},} Q ^ x ( h ) = 1 m | { i : 1 β€ i β€ ... | Wikipedia - Uniform convergence in probability - Uniform convergence theorem | 327 | 646 | null |
draws from the distribution P {\displaystyle P} . Ξ H {\displaystyle \Pi _{H}} is defined as: For any { 0 , 1 } {\displaystyle \{0,1\}} -valued functions H {\displaystyle H} over X {\displaystyle X} and D β X {\displaystyle D\subseteq X} , Ξ H ( D ) = { h β© D : h β H } . {\displaystyle \Pi _{H}(D)=\{h\cap D:h\in H\}.} ... | Wikipedia - Uniform convergence in probability - Uniform convergence theorem | 294 | 835 | null |
Section: SauerβShelah lemma. The SauerβShelah lemma relates the shattering number Ξ h ( m ) {\displaystyle \Pi _{h}(m)} to the VC Dimension. Lemma: Ξ H ( m ) β€ ( e m d ) d {\displaystyle \Pi _{H}(m)\leq \left({\frac {em}{d}}\right)^{d}} , where d {\displaystyle d} is the VC Dimension of the concept class H {\displaysty... | Wikipedia - Uniform convergence in probability - SauerβShelah lemma | 159 | 394 | null |
Section: Proof of uniform convergence theorem. and are the sources of the proof below. Before we get into the details of the proof of the Uniform Convergence Theorem we will present a high level overview of the proof. Symmetrization: We transform the problem of analyzing | Q P ( h ) β Q ^ x ( h ) | β₯ Ξ΅ {\displaystyle |... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem | 347 | 1,136 | null |
Permutation: Since r {\displaystyle r} and s {\displaystyle s} are picked identically and independently, so swapping elements between them will not change the probability distribution on r {\displaystyle r} and s {\displaystyle s} . So, we will try to bound the probability of | Q ^ r ( h ) β Q ^ s ( h ) | β₯ Ξ΅ / 2 {\dis... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem | 300 | 883 | null |
Section: Proof of uniform convergence theorem > Symmetrization. Lemma: Let V = { x β X m : | Q P ( h ) β Q ^ x ( h ) | β₯ Ξ΅ for some h β H } {\displaystyle V=\{x\in X^{m}:|Q_{P}(h)-{\widehat {Q}}_{x}(h)|\geq \varepsilon {\text{ for some }}h\in H\}} and R = { ( r , s ) β X m Γ X m : | Q r ^ ( h ) β Q ^ s ( h ) | β₯ Ξ΅ / 2 ... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Symmetrization | 318 | 619 | null |
{\displaystyle R=\{(r,s)\in X^{m}\times X^{m}:|{\widehat {Q_{r}}}(h)-{\widehat {Q}}_{s}(h)|\geq \varepsilon /2{\text{ for some }}h\in H\}.} Then for m β₯ 2 Ξ΅ 2 {\displaystyle m\geq {\frac {2}{\varepsilon ^{2}}}} , P m ( V ) β€ 2 P 2 m ( R ) {\displaystyle P^{m}(V)\leq 2P^{2m}(R)} . Proof: By the triangle inequality, if |... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Symmetrization | 355 | 650 | null |
Therefore, P 2 m ( R ) β₯ P 2 m { β h β H , | Q P ( h ) β Q ^ r ( h ) | β₯ Ξ΅ and | Q P ( h ) β Q ^ s ( h ) | β€ Ξ΅ / 2 } = β« V P m { s : β h β H , | Q P ( h ) β Q ^ r ( h ) | β₯ Ξ΅ and | Q P ( h ) β Q ^ s ( h ) | β€ Ξ΅ / 2 } d P m ( r ) = A {\displaystyle {\begin{aligned}&P^{2m}(R)\\[5pt]\geq {}&P^{2m}\{\exists h\in H,|Q_{P}(h... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Symmetrization | 344 | 583 | null |
/2\}\,dP^{m}(r)\\[5pt]={}&A\end{aligned}}} since r {\displaystyle r} and s {\displaystyle s} are independent. Now for r β V {\displaystyle r\in V} fix an h β H {\displaystyle h\in H} such that | Q P ( h ) β Q ^ r ( h ) | β₯ Ξ΅ {\displaystyle |Q_{P}(h)-{\widehat {Q}}_{r}(h)|\geq \varepsilon } . For this h {\displaystyle h... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Symmetrization | 177 | 392 | null |
For this h {\displaystyle h} , we shall show that P m { | Q P ( h ) β Q ^ s ( h ) | β€ Ξ΅ 2 } β₯ 1 2 . {\displaystyle P^{m}\left\{|Q_{P}(h)-{\widehat {Q}}_{s}(h)|\leq {\frac {\varepsilon }{2}}\right\}\geq {\frac {1}{2}}.} Thus for any r β V {\displaystyle r\in V} , A β₯ P m ( V ) 2 {\displaystyle A\geq {\frac {P^{m}(V)}{2}... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Symmetrization | 346 | 725 | null |
Notice, m β
Q ^ s ( h ) {\displaystyle m\cdot {\widehat {Q}}_{s}(h)} is a binomial random variable with expectation m β
Q P ( h ) {\displaystyle m\cdot Q_{P}(h)} and variance m β
Q P ( h ) ( 1 β Q P ( h ) ) {\displaystyle m\cdot Q_{P}(h)(1-Q_{P}(h))} . By Chebyshev's inequality we get P m { | Q P ( h ) β Q s ( h ) ^ | ... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Symmetrization | 338 | 654 | null |
By Chebyshev's inequality we get P m { | Q P ( h ) β Q s ( h ) ^ | > Ξ΅ 2 } β€ m β
Q P ( h ) ( 1 β Q P ( h ) ) ( Ξ΅ m / 2 ) 2 β€ 1 Ξ΅ 2 m β€ 1 2 {\displaystyle P^{m}\left\{|Q_{P}(h)-{\widehat {Q_{s}(h)}}|>{\frac {\varepsilon }{2}}\right\}\leq {\frac {m\cdot Q_{P}(h)(1-Q_{P}(h))}{(\varepsilon m/2)^{2}}}\leq {\frac {1}{\vareps... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Symmetrization | 265 | 504 | null |
Section: Proof of uniform convergence theorem > Permutations. Let Ξ m {\displaystyle \Gamma _{m}} be the set of all permutations of { 1 , 2 , 3 , β¦ , 2 m } {\displaystyle \{1,2,3,\dots ,2m\}} that swaps i {\displaystyle i} and m + i {\displaystyle m+i} β i {\displaystyle \forall i} in some subset of { 1 , 2 , 3 , β¦ , 2... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Permutations | 349 | 912 | null |
Then, P 2 m ( R ) = E [ Pr [ Ο ( x ) β R ] ] β€ max x β X 2 m ( Pr [ Ο ( x ) β R ] ) , {\displaystyle P^{2m}(R)=E[\Pr[\sigma (x)\in R]]\leq \max _{x\in X^{2m}}(\Pr[\sigma (x)\in R]),} where the expectation is over x {\displaystyle x} chosen according to P 2 m {\displaystyle P^{2m}} , and the probability is over Ο {\disp... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Permutations | 405 | 947 | null |
{\displaystyle {\begin{aligned}\therefore P^{2m}(R)={}&\int _{X^{2m}}1_{R}(x)\,dP^{2m}(x)\\[5pt]={}&{\frac {1}{|\Gamma _{m}|}}\sum _{\sigma \in \Gamma _{m}}\int _{X^{2m}}1_{R}(\sigma (x))\,dP^{2m}(x)\\[5pt]={}&\int _{X^{2m}}{\frac {1}{|\Gamma _{m}|}}\sum _{\sigma \in \Gamma _{m}}1_{R}(\sigma (x))\,dP^{2m}(x)\\[5pt]&{\t... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Permutations | 350 | 533 | null |
Section: Proof of uniform convergence theorem > Reduction to a finite class. Lemma: Basing on the previous lemma, max x β X 2 m ( Pr [ Ο ( x ) β R ] ) β€ 4 Ξ H ( 2 m ) e β Ξ΅ 2 m / 8 {\displaystyle \max _{x\in X^{2m}}(\Pr[\sigma (x)\in R])\leq 4\Pi _{H}(2m)e^{-\varepsilon ^{2}m/8}} . Proof: Let us define x = ( x 1 , x 2 ... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Reduction to a finite class | 222 | 485 | null |
Proof: Let us define x = ( x 1 , x 2 , β¦ , x 2 m ) {\displaystyle x=(x_{1},x_{2},\ldots ,x_{2m})} and t = | H | x | {\displaystyle t=|H|_{x}|} which is at most Ξ H ( 2 m ) {\displaystyle \Pi _{H}(2m)} . This means there are functions h 1 , h 2 , β¦ , h t β H {\displaystyle h_{1},h_{2},\ldots ,h_{t}\in H} such that for a... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Reduction to a finite class | 232 | 504 | null |
This means there are functions h 1 , h 2 , β¦ , h t β H {\displaystyle h_{1},h_{2},\ldots ,h_{t}\in H} such that for any h β H , β i {\displaystyle h\in H,\exists i} between 1 {\displaystyle 1} and t {\displaystyle t} with h i ( x k ) = h ( x k ) {\displaystyle h_{i}(x_{k})=h(x_{k})} for 1 β€ k β€ 2 m . {\displaystyle 1\l... | Wikipedia - Uniform convergence in probability - Proof of uniform convergence theorem > Reduction to a finite class | 356 | 717 | null |
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