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Due to reciprocity, machines enable humans to have their potentialities "strengthened" for tasks such as resolving ambiguous information, exercising the judgment of difficult cases, and contacting dissatisfied clients.” Daugherty and Wilson have observed successful new types of human-computer interaction in occupations... | Wikipedia - Artificial intelligence in hiring - AI and the future of hiring | 197 | 1,112 | null |
Article: Attention (machine learning). Attention is a machine learning method that determines the importance of each component in a sequence relative to the other components in that sequence. In natural language processing, importance is represented by "soft" weights assigned to each word in a sentence. More generally,... | Wikipedia - Attention (machine learning) - Summary | 254 | 1,378 | null |
Section: Overview. The modern era of machine attention was revitalized by grafting an attention mechanism (Fig 1. orange) to an Encoder-Decoder. Figure 2 shows the internal step-by-step operation of the attention block (A) in Fig 1. This attention scheme has been compared to the Query-Key analogy of relational database... | Wikipedia - Attention (machine learning) - Overview | 220 | 1,060 | null |
Section: Overview > Interpreting attention weights. In translating between languages, alignment is the process of matching words from the source sentence to words of the translated sentence. Networks that perform verbatim translation without regard to word order would show the highest scores along the (dominant) diagon... | Wikipedia - Attention (machine learning) - Overview > Interpreting attention weights | 307 | 1,417 | null |
Section: Variants. Many variants of attention implement soft weights, such as fast weight programmers, or fast weight controllers (1992). A "slow" neural network outputs the "fast" weights of another neural network through outer products. The slow network learns by gradient descent. It was later renamed as "linearized ... | Wikipedia - Attention (machine learning) - Variants | 232 | 1,160 | null |
Section: Mathematical representation > Standard Scaled Dot-Product Attention. For matrices: Q ∈ R m × d k , K ∈ R n × d k {\displaystyle \mathbf {Q} \in \mathbb {R} ^{m\times d_{k}},\mathbf {K} \in \mathbb {R} ^{n\times d_{k}}} and V ∈ R n × d v {\displaystyle \mathbf {V} \in \mathbb {R} ^{n\times d_{v}}} , the scaled ... | Wikipedia - Attention (machine learning) - Mathematical representation > Standard Scaled Dot-Product Attention | 305 | 757 | null |
For matrices: Q ∈ R m × d k , K ∈ R n × d k {\displaystyle \mathbf {Q} \in \mathbb {R} ^{m\times d_{k}},\mathbf {K} \in \mathbb {R} ^{n\times d_{k}}} and V ∈ R n × d v {\displaystyle \mathbf {V} \in \mathbb {R} ^{n\times d_{v}}} , the scaled dot-product, or QKV attention is defined as: Attention ( Q , K , V ) = softmax... | Wikipedia - Attention (machine learning) - Mathematical representation > Standard Scaled Dot-Product Attention | 361 | 895 | null |
The matrix Q {\displaystyle \mathbf {Q} } contains m {\displaystyle m} queries, while matrices K , V {\displaystyle \mathbf {K} ,\mathbf {V} } jointly contain an unordered set of n {\displaystyle n} key-value pairs. Value vectors in matrix V {\displaystyle \mathbf {V} } are weighted using the weights resulting from the... | Wikipedia - Attention (machine learning) - Mathematical representation > Standard Scaled Dot-Product Attention | 309 | 938 | null |
To understand the permutation invariance and permutation equivariance properties of QKV attention, let A ∈ R m × m {\displaystyle \mathbf {A} \in \mathbb {R} ^{m\times m}} and B ∈ R n × n {\displaystyle \mathbf {B} \in \mathbb {R} ^{n\times n}} be permutation matrices; and D ∈ R m × n {\displaystyle \mathbf {D} \in \ma... | Wikipedia - Attention (machine learning) - Mathematical representation > Standard Scaled Dot-Product Attention | 436 | 1,163 | null |
The softmax function is permutation equivariant in the sense that: softmax ( A D B ) = A softmax ( D ) B {\displaystyle {\text{softmax}}(\mathbf {A} \mathbf {D} \mathbf {B} )=\mathbf {A} \,{\text{softmax}}(\mathbf {D} )\mathbf {B} } By noting that the transpose of a permutation matrix is also its inverse, it follows th... | Wikipedia - Attention (machine learning) - Mathematical representation > Standard Scaled Dot-Product Attention | 323 | 913 | null |
These properties are inherited when applying linear transforms to the inputs and outputs of QKV attention blocks. For example, a simple self-attention function defined as: X ↦ Attention ( X T q , X T k , X T v ) {\displaystyle \mathbf {X} \mapsto {\text{Attention}}(\mathbf {X} \mathbf {T} _{q},\mathbf {X} \mathbf {T} _... | Wikipedia - Attention (machine learning) - Mathematical representation > Standard Scaled Dot-Product Attention | 187 | 630 | null |
Section: Mathematical representation > Masked Attention. When QKV attention is used as a building block for an autoregressive decoder, and when at training time all input and output matrices have n {\displaystyle n} rows, a masked attention variant is used: Attention ( Q , K , V ) = softmax ( Q K T d k + M ) V {\displa... | Wikipedia - Attention (machine learning) - Mathematical representation > Masked Attention | 289 | 875 | null |
Section: Mathematical representation > Multi-Head Attention. Multi-head attention MultiHead ( Q , K , V ) = Concat ( head 1 , . . . , head h ) W O {\displaystyle {\text{MultiHead}}(\mathbf {Q} ,\mathbf {K} ,\mathbf {V} )={\text{Concat}}({\text{head}}_{1},...,{\text{head}}_{h})\mathbf {W} ^{O}} where each head is comput... | Wikipedia - Attention (machine learning) - Mathematical representation > Multi-Head Attention | 347 | 803 | null |
The permutation properties of (standard, unmasked) QKV attention apply here also. For permutation matrices, A , B {\displaystyle \mathbf {A} ,\mathbf {B} } : MultiHead ( A Q , B K , B V ) = A MultiHead ( Q , K , V ) {\displaystyle {\text{MultiHead}}(\mathbf {A} \mathbf {Q} ,\mathbf {B} \mathbf {K} ,\mathbf {B} \mathbf ... | Wikipedia - Attention (machine learning) - Mathematical representation > Multi-Head Attention | 294 | 730 | null |
Section: Mathematical representation > Bahdanau (Additive) Attention. Attention ( Q , K , V ) = softmax ( tanh ( W Q Q + W K K ) V ) {\displaystyle {\text{Attention}}(\mathbf {Q} ,\mathbf {K} ,\mathbf {V} )={\text{softmax}}(\tanh(\mathbf {W} _{Q}\mathbf {Q} +\mathbf {W} _{K}\mathbf {K} )\mathbf {V} )} where W Q {\dis... | Wikipedia - Attention (machine learning) - Mathematical representation > Bahdanau (Additive) Attention | 170 | 419 | null |
These can then be applied to a dot-product attention mechanism, to obtain h 0 ′ = A t t e n t i o n ( h 0 W Q , H W K , H W V ) h 1 ′ = A t t e n t i o n ( h 1 W Q , H W K , H W V ) ⋯ {\displaystyle {\begin{aligned}h_{0}'&=\mathrm {Attention} (h_{0}W^{Q},HW^{K},HW^{V})\\h_{1}'&=\mathrm {Attention} (h_{1}W^{Q},HW^{K},HW... | Wikipedia - Attention (machine learning) - Mathematical representation > Self Attention | 296 | 692 | null |
Article: Audio inpainting. Audio inpainting (also known as audio interpolation) is an audio restoration task which deals with the reconstruction of missing or corrupted portions of a digital audio signal. Inpainting techniques are employed when parts of the audio have been lost due to various factors such as transmissi... | Wikipedia - Audio inpainting - Summary | 339 | 1,730 | null |
Section: Definition. Consider a digital audio signal x {\displaystyle \mathbf {x} } . A corrupted version of x {\displaystyle \mathbf {x} } , which is the audio signal presenting missing gaps to be reconstructed, can be defined as x ~ = m ∘ x {\displaystyle \mathbf {\tilde {x}} =\mathbf {m} \circ \mathbf {x} } , where ... | Wikipedia - Audio inpainting - Definition | 222 | 765 | null |
This is an ill-posed inverse problem, which is characterized by a non-unique set of solutions. For this reason, similarly to the formulation used for the inpainting problem in other domains, the reconstructed audio signal can be found through an optimization problem that is formally expressed as x ^ ∗ = argmin X ^ L ( ... | Wikipedia - Audio inpainting - Definition | 329 | 1,075 | null |
Since L {\displaystyle L} is computed only on the reliable frames, there are many solutions that can minimize L ( m ∘ x ^ , x ~ ) {\displaystyle L(\mathbf {m} \circ \mathbf {\hat {x}} ,\mathbf {\tilde {x}} )} . It is thus necessary to add a constraint to the minimization, in order to restrict the results only to the va... | Wikipedia - Audio inpainting - Definition | 194 | 727 | null |
Section: Techniques > Model-based techniques. Model-based techniques involve the exploitation of mathematical models or assumptions about the underlying structure of the audio signal. These models can be based on prior knowledge of the audio content or statistical properties observed in the data. By leveraging these mo... | Wikipedia - Audio inpainting - Techniques > Model-based techniques | 325 | 1,783 | null |
Section: Techniques > Data-driven techniques. Data-driven techniques rely on the analysis and exploitation of the available audio data. These techniques often employ deep learning algorithms that learn patterns and relationships directly from the provided data. They involve training models on large datasets of audio ex... | Wikipedia - Audio inpainting - Techniques > Data-driven techniques | 339 | 1,758 | null |
In GAN-based inpaniting methods the generator acts as a context encoder and produces a plausible completion for the gap only given the available information surrounding it. The discriminator is used to train the generator and tests the consistency of the produced inpainted audio. Recently, also diffusion models have es... | Wikipedia - Audio inpainting - Techniques > Data-driven techniques | 256 | 1,306 | null |
Section: Applications. Audio inpainting finds applications in a wide range of fields, including audio restoration and audio forensics among the others. In these fields, audio inpainting can be used to eliminate noise, glitches, or undesired distortions from an audio recording, thus enhancing its quality and intelligibi... | Wikipedia - Audio inpainting - Applications | 176 | 882 | null |
Article: Automated decision-making. Automated decision-making (ADM) is the use of data, machines and algorithms to make decisions in a range of contexts, including public administration, business, health, education, law, employment, transport, media and entertainment, with varying degrees of human oversight or interven... | Wikipedia - Automated decision-making - Summary | 163 | 904 | null |
Section: Overview. There are different definitions of ADM based on the level of automation involved. Some definitions suggests ADM involves decisions made through purely technological means without human input, such as the EU's General Data Protection Regulation (Article 22). However, ADM technologies and applications ... | Wikipedia - Automated decision-making - Overview | 256 | 1,431 | null |
Section: ADM technologies. Automated decision-making technologies (ADMT) are software-coded digital tools that automate the translation of input data to output data, contributing to the function of automated decision-making systems. There are a wide range of technologies in use across ADM applications and systems. ADMT... | Wikipedia - Automated decision-making - ADM technologies | 179 | 961 | null |
Section: ADM technologies > Machine learning. Machine learning (ML) involves training computer programs through exposure to large data sets and examples to learn from experience and solve problems. Machine learning can be used to generate and analyse data as well as make algorithmic calculations and has been applied to... | Wikipedia - Automated decision-making - ADM technologies > Machine learning | 220 | 1,163 | null |
Section: Applications > Media and entertainment. Digital media, entertainment platforms, and information services increasingly provide content to audiences via automated recommender systems based on demographic information, previous selections, collaborative filtering or content-based filtering. This includes music and... | Wikipedia - Automated decision-making - Applications > Media and entertainment | 153 | 939 | null |
Section: Applications > Advertising. Online advertising is closely integrated with many digital media platforms, websites and search engines and often involves automated delivery of display advertisements in diverse formats. 'Programmatic' online advertising involves automating the sale and delivery of digital advertis... | Wikipedia - Automated decision-making - Applications > Advertising | 190 | 1,116 | null |
Section: Applications > Social services. Governments have been implementing digital technologies to provide more efficient administration and social services since the early 2000s, often referred to as e-government. Many governments around the world are now using automated, algorithmic systems for profiling and targeti... | Wikipedia - Automated decision-making - Applications > Social services | 271 | 1,597 | null |
Section: Applications > Transport and mobility. Connected and automated mobility (CAM) involves autonomous vehicles such as self-driving cars and other forms of transport which use automated decision-making systems to replace various aspects of human control of the vehicle. This can range from level 0 (complete human d... | Wikipedia - Automated decision-making - Applications > Transport and mobility | 260 | 1,412 | null |
Section: Ethical and legal issues. There are many social, ethical and legal implications of automated decision-making systems. Concerns raised include lack of transparency and contestability of decisions, incursions on privacy and surveillance, exacerbating systemic bias and inequality due to data and algorithmic bias,... | Wikipedia - Automated decision-making - Ethical and legal issues | 343 | 1,820 | null |
GDPR also includes some rules on the right to explanation however the exact scope and nature of these is currently subject to pending review by the Court of Justice of the European Union. These provisions were not first introduced in the GDPR, but have been present in a similar form across Europe since the Data Protect... | Wikipedia - Automated decision-making - Ethical and legal issues | 150 | 775 | null |
Section: Research fields. Many academic disciplines and fields are increasingly turning their attention to the development, application and implications of ADM including business, computer sciences, human computer interaction (HCI), law, public administration, and media and communications. The automation of media conte... | Wikipedia - Automated decision-making - Research fields | 158 | 877 | null |
Section: Uses of Automation in Construction. Equipment control and management: Automation can be used to control and monitor construction equipment, such as cranes, excavators, and bulldozers. Material handling: Automated systems can be used to handle, transport, and place materials such as concrete, bricks, and stones... | Wikipedia - Automation in construction - Uses of Automation in Construction | 180 | 1,005 | null |
Section: Benefits of Automation in Construction. The use of automation in construction has become increasingly prevalent in recent years due to its numerous benefits. Automation in construction refers to the use of machinery, software, and other technologies to perform tasks that were previously done manually by worker... | Wikipedia - Automation in construction - Benefits of Automation in Construction | 243 | 1,402 | null |
Article: Bag-of-words model. The bag-of-words (BoW) model is a model of text which uses an unordered collection (a "bag") of words. It is used in natural language processing and information retrieval (IR). It disregards word order (and thus most of syntax or grammar) but captures multiplicity. The bag-of-words model is... | Wikipedia - Bag-of-words model - Summary | 152 | 661 | null |
Section: Definition. The following models a text document using bag-of-words. Here are two simple text documents: Based on these two text documents, a list is constructed as follows for each document: Representing each bag-of-words as a JSON object, and attributing to the respective JavaScript variable: Each key is the... | Wikipedia - Bag-of-words model - Definition | 231 | 892 | null |
Section: Definition > Word order. The BoW representation of a text removes all word ordering. For example, the BoW representation of "man bites dog" and "dog bites man" are the same, so any algorithm that operates with a BoW representation of text must treat them in the same way. Despite this lack of syntax or grammar,... | Wikipedia - Bag-of-words model - Definition > Word order | 173 | 871 | null |
Section: Informal description. A ball tree is a binary tree in which every node defines a D-dimensional ball containing a subset of the points to be searched. Each internal node of the tree partitions the data points into two disjoint sets which are associated with different balls. While the balls themselves may inters... | Wikipedia - Ball tree - Informal description | 175 | 838 | null |
This gives rise to the useful property that, for a given test point t outside the ball, the distance to any point in a ball B in the tree is greater than or equal to the distance from t to the surface of the ball. Formally: D B ( t ) = { max ( | t − B.pivot | − B.radius , D B.parent ) , if B ≠ R o o t max ( | t − B.piv... | Wikipedia - Ball tree - Informal description | 345 | 990 | null |
Section: Construction. A number of ball tree construction algorithms are available. The goal of such an algorithm is to produce a tree that will efficiently support queries of the desired type (e.g. nearest-neighbor) in the average case. The specific criteria of an ideal tree will depend on the type of question being a... | Wikipedia - Ball tree - Construction | 180 | 911 | null |
Section: Construction > k-d construction algorithm. The simplest such procedure is termed the "k-d Construction Algorithm", by analogy with the process used to construct k-d trees. This is an offline algorithm, that is, an algorithm that operates on the entire data set at once. The tree is built top-down by recursively... | Wikipedia - Ball tree - Construction > k-d construction algorithm | 167 | 763 | null |
Section: Construction > k-d construction algorithm > Pseudocode. function construct_balltree is input: D, an array of data points. output: B, the root of a constructed ball tree. if a single point remains then create a leaf B containing the single point in D return B else let c be the dimension of greatest spread let p... | Wikipedia - Ball tree - Construction > k-d construction algorithm > Pseudocode | 156 | 650 | null |
Section: Nearest-neighbor search > Description. The ball tree nearest-neighbor algorithm examines nodes in depth-first order, starting at the root. During the search, the algorithm maintains a max-first priority queue (often implemented with a heap), denoted Q here, of the k nearest points encountered so far. At each n... | Wikipedia - Ball tree - Nearest-neighbor search > Description | 221 | 1,057 | null |
Section: Nearest-neighbor search > Pseudocode. function knn_search is input: t, the target point for the query k, the number of nearest neighbors of t to search for Q, max-first priority queue containing at most k points B, a node, or ball, in the tree output: Q, containing the k nearest neighbors from within B if dist... | Wikipedia - Ball tree - Nearest-neighbor search > Pseudocode | 212 | 760 | null |
Article: Base rate. In probability and statistics, the base rate (also known as prior probabilities) is the class of probabilities unconditional on "featural evidence" (likelihoods). It is the proportion of individuals in a population who have a certain characteristic or trait. For example, if 1% of the population were... | Wikipedia - Base rate - Summary | 337 | 1,682 | null |
Section: Base rate fallacy. Many psychological studies have examined a phenomenon called base-rate neglect or base rate fallacy, in which category base rates are not integrated with presented evidence in a normative manner, although not all evidence is consistent regarding how common this fallacy is. Mathematician Keit... | Wikipedia - Base rate - Base rate fallacy | 299 | 1,510 | null |
Article: Bayesian interpretation of kernel regularization. Bayesian interpretation of kernel regularization examines how kernel methods in machine learning can be understood through the lens of Bayesian statistics, a framework that uses probability to model uncertainty. Kernel methods are founded on the concept of simi... | Wikipedia - Bayesian interpretation of kernel regularization - Summary | 308 | 1,772 | null |
Section: The supervised learning problem. The classical supervised learning problem requires estimating the output for some new input point x ′ {\displaystyle \mathbf {x} '} by learning a scalar-valued estimator f ^ ( x ′ ) {\displaystyle {\hat {f}}(\mathbf {x} ')} on the basis of a training set S {\displaystyle S} con... | Wikipedia - Bayesian interpretation of kernel regularization - The supervised learning problem | 191 | 526 | null |
The classical supervised learning problem requires estimating the output for some new input point x ′ {\displaystyle \mathbf {x} '} by learning a scalar-valued estimator f ^ ( x ′ ) {\displaystyle {\hat {f}}(\mathbf {x} ')} on the basis of a training set S {\displaystyle S} consisting of n {\displaystyle n} input-outpu... | Wikipedia - Bayesian interpretation of kernel regularization - The supervised learning problem | 471 | 1,134 | null |
Given a symmetric and positive bivariate function k ( ⋅ , ⋅ ) {\displaystyle k(\cdot ,\cdot )} called a kernel, one of the most popular estimators in machine learning is given by where K ≡ k ( X , X ) {\displaystyle \mathbf {K} \equiv k(\mathbf {X} ,\mathbf {X} )} is the kernel matrix with entries K i j = k ( x i , x j... | Wikipedia - Bayesian interpretation of kernel regularization - The supervised learning problem | 309 | 750 | null |
Section: A regularization perspective > Reproducing kernel Hilbert space. A reproducing kernel Hilbert space (RKHS) H k {\displaystyle {\mathcal {H}}_{k}} is a Hilbert space of functions defined by a symmetric, positive-definite function k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\rightarrow \mat... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > Reproducing kernel Hilbert space | 202 | 610 | null |
There are three main properties that make an RKHS appealing: 1. The reproducing property, after which the RKHS is named, f ( x ) = ⟨ f , k ( x , ⋅ ) ⟩ k , ∀ f ∈ H k , {\displaystyle f(\mathbf {x} )=\langle f,k(\mathbf {x} ,\cdot )\rangle _{k},\quad \forall \ f\in {\mathcal {H}}_{k},} where ⟨ ⋅ , ⋅ ⟩ k {\displaystyle \l... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > Reproducing kernel Hilbert space | 275 | 731 | null |
Section: A regularization perspective > The regularized functional. The estimator is derived as the minimizer of the regularized functional where f ∈ H k {\displaystyle f\in {\mathcal {H}}_{k}} and ‖ ⋅ ‖ k {\displaystyle \|\cdot \|_{k}} is the norm in H k {\displaystyle {\mathcal {H}}_{k}} . The first term in this func... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > The regularized functional | 313 | 1,119 | null |
Section: A regularization perspective > Derivation of the estimator. The explicit form of the estimator in equation (1) is derived in two steps. First, the representer theorem states that the minimizer of the functional (2) can always be written as a linear combination of the kernels centered at the training-set points... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > Derivation of the estimator | 307 | 879 | null |
For a function of the form in equation (3), we have that ‖ f ‖ k 2 = ⟨ f , f ⟩ k , = ⟨ ∑ i = 1 N c i k ( x i , ⋅ ) , ∑ j = 1 N c j k ( x j , ⋅ ) ⟩ k , = ∑ i = 1 N ∑ j = 1 N c i c j ⟨ k ( x i , ⋅ ) , k ( x j , ⋅ ) ⟩ k , = ∑ i = 1 N ∑ j = 1 N c i c j k ( x i , x j ) , = c ⊤ K c . {\displaystyle {\begin{aligned}\|f\|_{k}^... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > Derivation of the estimator | 461 | 810 | null |
{\displaystyle {\begin{aligned}\|f\|_{k}^{2}&=\langle f,f\rangle _{k},\\&=\left\langle \sum _{i=1}^{N}c_{i}k(\mathbf {x} _{i},\cdot ),\sum _{j=1}^{N}c_{j}k(\mathbf {x} _{j},\cdot )\right\rangle _{k},\\&=\sum _{i=1}^{N}\sum _{j=1}^{N}c_{i}c_{j}\langle k(\mathbf {x} _{i},\cdot ),k(\mathbf {x} _{j},\cdot )\rangle _{k},\\&... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > Derivation of the estimator | 492 | 921 | null |
{\displaystyle {\frac {1}{n}}\|\mathbf {y} -\mathbf {K} \mathbf {c} \|^{2}+\lambda \mathbf {c} ^{\top }\mathbf {K} \mathbf {c} .} This functional is convex in c {\displaystyle \mathbf {c} } and therefore we can find its minimum by setting the gradient with respect to c {\displaystyle \mathbf {c} } to zero, − 1 n K ( Y ... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > Derivation of the estimator | 354 | 840 | null |
{\displaystyle {\begin{aligned}-{\frac {1}{n}}\mathbf {K} (\mathbf {Y} -\mathbf {K} \mathbf {c} )+\lambda \mathbf {K} \mathbf {c} &=0,\\(\mathbf {K} +\lambda n\mathbf {I} )\mathbf {c} &=\mathbf {Y} ,\\\mathbf {c} &=(\mathbf {K} +\lambda n\mathbf {I} )^{-1}\mathbf {Y} .\end{aligned}}} Substituting this expression for th... | Wikipedia - Bayesian interpretation of kernel regularization - A regularization perspective > Derivation of the estimator | 260 | 567 | null |
Section: A Bayesian perspective > The Gaussian process. A Gaussian process (GP) is a stochastic process in which any finite number of random variables that are sampled follow a joint Normal distribution. The mean vector and covariance matrix of the Gaussian distribution completely specify the GP. GPs are usually used a... | Wikipedia - Bayesian interpretation of kernel regularization - A Bayesian perspective > The Gaussian process | 154 | 665 | null |
{\displaystyle f\sim {\mathcal {GP}}(m,k).} In terms of the underlying Gaussian distribution, we have that for any finite set X = { x i } i = 1 n {\displaystyle \mathbf {X} =\{\mathbf {x} _{i}\}_{i=1}^{n}} if we let f ( X ) = [ f ( x 1 ) , … , f ( x n ) ] ⊤ {\displaystyle f(\mathbf {X} )=[f(\mathbf {x} _{1}),\ldots ,f(... | Wikipedia - Bayesian interpretation of kernel regularization - A Bayesian perspective > The Gaussian process | 348 | 739 | null |
Section: A Bayesian perspective > Derivation of the estimator. In a regression context, the likelihood function is usually assumed to be a Gaussian distribution and the observations to be independent and identically distributed (iid), p ( y | f , x , σ 2 ) = N ( f ( x ) , σ 2 ) . {\displaystyle p(y|f,\mathbf {x} ,\sigm... | Wikipedia - Bayesian interpretation of kernel regularization - A Bayesian perspective > Derivation of the estimator | 220 | 795 | null |
The iid assumption makes it possible to factorize the likelihood function over the data points given the set of inputs X {\displaystyle \mathbf {X} } and the variance of the noise σ 2 {\displaystyle \sigma ^{2}} , and thus the posterior distribution can be computed analytically. For a test input vector x ′ {\displaysty... | Wikipedia - Bayesian interpretation of kernel regularization - A Bayesian perspective > Derivation of the estimator | 346 | 987 | null |
For a test input vector x ′ {\displaystyle \mathbf {x} '} , given the training data S = { X , Y } {\displaystyle S=\{\mathbf {X} ,\mathbf {Y} \}} , the posterior distribution is given by p ( f ( x ′ ) | S , x ′ , ϕ ) = N ( m ( x ′ ) , σ 2 ( x ′ ) ) , {\displaystyle p(f(\mathbf {x} ')|S,\mathbf {x} ',{\boldsymbol {\phi ... | Wikipedia - Bayesian interpretation of kernel regularization - A Bayesian perspective > Derivation of the estimator | 436 | 985 | null |
{\displaystyle k(\mathbf {x} ,\mathbf {x} ')=\sum _{i=1}^{p}\Phi ^{i}(\mathbf {x} )\Phi ^{i}(\mathbf {x} ').} Functions in the RKHS with kernel K {\displaystyle \mathbf {K} } can then be written as f w ( x ) = ∑ i = 1 p w i Φ i ( x ) = ⟨ w , Φ ( x ) ⟩ , {\displaystyle f_{\mathbf {w} }(\mathbf {x} )=\sum _{i=1}^{p}\math... | Wikipedia - Bayesian interpretation of kernel regularization - The connection between regularization and Bayes | 218 | 447 | null |
{\displaystyle k(\mathbf {x} ,\mathbf {x} ')=\sum _{i=1}^{p}\Phi ^{i}(\mathbf {x} )\Phi ^{i}(\mathbf {x} ').} Functions in the RKHS with kernel K {\displaystyle \mathbf {K} } can then be written as f w ( x ) = ∑ i = 1 p w i Φ i ( x ) = ⟨ w , Φ ( x ) ⟩ , {\displaystyle f_{\mathbf {w} }(\mathbf {x} )=\sum _{i=1}^{p}\math... | Wikipedia - Bayesian interpretation of kernel regularization - The connection between regularization and Bayes | 352 | 792 | null |
{\displaystyle \|f_{\mathbf {w} }\|_{k}=\|\mathbf {w} \|.} We can now build a Gaussian process by assuming w = [ w 1 , … , w p ] ⊤ {\displaystyle \mathbf {w} =[w^{1},\ldots ,w^{p}]^{\top }} to be distributed according to a multivariate Gaussian distribution with zero mean and identity covariance matrix, w ∼ N ( 0 , I )... | Wikipedia - Bayesian interpretation of kernel regularization - The connection between regularization and Bayes | 472 | 1,001 | null |
{\displaystyle \mathbf {w} \sim {\mathcal {N}}(0,\mathbf {I} )\propto \exp(-\|\mathbf {w} \|^{2}).} If we assume a Gaussian likelihood we have P ( Y | X , f ) = N ( f ( X ) , σ 2 I ) ∝ exp ( − 1 σ 2 ‖ f w ( X ) − Y ‖ 2 ) , {\displaystyle P(\mathbf {Y} |\mathbf {X} ,f)={\mathcal {N}}(f(\mathbf {X} ),\sigma ^{2}\mathbf... | Wikipedia - Bayesian interpretation of kernel regularization - The connection between regularization and Bayes | 527 | 1,169 | null |
The resulting posterior distribution is then given by P ( f | X , Y ) ∝ exp ( − 1 σ 2 ‖ f w ( X ) − Y ‖ n 2 + ‖ w ‖ 2 ) {\displaystyle P(f|\mathbf {X} ,\mathbf {Y} )\propto \exp \left(-{\frac {1}{\sigma ^{2}}}\|f_{\mathbf {w} }(\mathbf {X} )-\mathbf {Y} \|_{n}^{2}+\|\mathbf {w} \|^{2}\right)} We can see that a maximu... | Wikipedia - Bayesian interpretation of kernel regularization - The connection between regularization and Bayes | 343 | 1,276 | null |
Section: History > Early Mathematics Foundations > From 1960s to 1980s. The earliest idea of Bayesian optimization sprang in 1964, from a paper by American applied mathematician Harold J. Kushner, “A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise”. Although not directl... | Wikipedia - Bayesian optimization - History > Early Mathematics Foundations > From 1960s to 1980s | 306 | 1,599 | null |
Section: History > Early Mathematics Foundations > From Theory to Practice. In the 1990s, Bayesian optimization began to gradually transition from pure theory to real-world applications. In 1998, Donald R. Jones and his coworkers published a paper titled “Gaussian Optimization”. In this paper, they proposed the Gaussia... | Wikipedia - Bayesian optimization - History > Early Mathematics Foundations > From Theory to Practice | 285 | 1,490 | null |
Section: Strategy. Bayesian optimization is used on problems of the form max x ∈ X f ( x ) {\textstyle \max _{x\in X}f(x)} , with X {\textstyle X} being the set of all possible parameters x {\textstyle x} , typically with less than or equal to 20 dimensions for optimal usage ( X → R d ∣ d ≤ 20 {\textstyle X\rightarrow ... | Wikipedia - Bayesian optimization - Strategy | 346 | 1,485 | null |
The most common two methods use Gaussian processes in a method called kriging. Another less expensive method uses the Parzen-Tree Estimator to construct two distributions for 'high' and 'low' points, and then finds the location that maximizes the expected improvement. Standard Bayesian optimization relies upon each x ∈... | Wikipedia - Bayesian optimization - Strategy | 150 | 763 | null |
Section: Applications. The approach has been applied to solve a wide range of problems, including learning to rank, computer graphics and visual design, robotics, sensor networks, automatic algorithm configuration, automatic machine learning toolboxes, reinforcement learning, planning, visual attention, architecture co... | Wikipedia - Bayesian optimization - Applications | 213 | 1,221 | null |
Article: Bayesian structural time series. Bayesian structural time series (BSTS) model is a statistical technique used for feature selection, time series forecasting, nowcasting, inferring causal impact and other applications. The model is designed to work with time series data. The model has also promising application... | Wikipedia - Bayesian structural time series - Summary | 211 | 1,064 | null |
Section: General model description. The model consists of three main components: Kalman filter. The technique for time series decomposition. In this step, a researcher can add different state variables: trend, seasonality, regression, and others. Spike-and-slab method. In this step, the most important regression predic... | Wikipedia - Bayesian structural time series - General model description | 157 | 833 | null |
Article: Bias–variance tradeoff. In statistics and machine learning, the bias–variance tradeoff describes the relationship between a model's complexity, the accuracy of its predictions, and how well it can make predictions on previously unseen data that were not used to train the model. In general, as the number of tun... | Wikipedia - Bias–variance tradeoff - Summary | 312 | 1,597 | null |
Section: Motivation. The bias–variance tradeoff is a central problem in supervised learning. Ideally, one wants to choose a model that both accurately captures the regularities in its training data, but also generalizes well to unseen data. Unfortunately, it is typically impossible to do both simultaneously. High-varia... | Wikipedia - Bias–variance tradeoff - Motivation | 334 | 1,500 | null |
Consequently, a sample will appear accurate (i.e. have low bias) under the aforementioned selection conditions, but may result in underfitting. In other words, test data may not agree as closely with training data, which would indicate imprecision and therefore inflated variance. A graphical example would be a straight... | Wikipedia - Bias–variance tradeoff - Motivation | 299 | 1,565 | null |
Section: Bias–variance decomposition of mean squared error. Suppose that we have a training set consisting of a set of points x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} and real-valued labels y i {\displaystyle y_{i}} associated with the points x i {\displaystyle x_{i}} . We assume that the data is generated by a... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error | 236 | 674 | null |
That is, y i = f ( x i ) + ε i {\displaystyle y_{i}=f(x_{i})+\varepsilon _{i}} , where ε i {\displaystyle \varepsilon _{i}} is a noise sample. We want to find a function f ^ ( x ; D ) {\displaystyle {\hat {f}}(x;D)} , that approximates the true function f ( x ) {\displaystyle f(x)} as well as possible, by means of some... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error | 325 | 811 | null |
We make "as well as possible" precise by measuring the mean squared error between y {\displaystyle y} and f ^ ( x ; D ) {\displaystyle {\hat {f}}(x;D)} : we want ( y − f ^ ( x ; D ) ) 2 {\displaystyle (y-{\hat {f}}(x;D))^{2}} to be minimal, both for x 1 , … , x n {\displaystyle x_{1},\dots ,x_{n}} and for points outsid... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error | 275 | 897 | null |
conditional to x) as follows:: 34 : 223 E D , ε [ ( y − f ^ ( x ; D ) ) 2 ] = ( Bias D [ f ^ ( x ; D ) ] ) 2 + Var D [ f ^ ( x ; D ) ] + σ 2 {\displaystyle \mathbb {E} _{D,\varepsilon }{\Big [}{\big (}y-{\hat {f}}(x;D){\big )}^{2}{\Big ]}={\Big (}\operatorname {Bias} _{D}{\big [}{\hat {f}}(x;D){\big ]}{\Big )}^{2}+... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error | 338 | 664 | null |
{f}}(x;D)-f(x){\big ]}\\&=\mathbb {E} _{D}{\big [}{\hat {f}}(x;D){\big ]}\,-\,f(x)\\&=\mathbb {E} _{D}{\big [}{\hat {f}}(x;D){\big ]}\,-\,\mathbb {E} _{y|x}{\big [}y(x){\big ]}\end{aligned}}} and Var D [ f ^ ( x ; D ) ] ≜ E D [ ( E D [ f ^ ( x ; D ) ] − f ^ ( x ; D ) ) 2 ] {\displaystyle \operatorname {Var} _{D}{\big... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error | 350 | 592 | null |
{f(x)} _{E_{y|x}[y]}{\big )}^{2}{\Big ]}} The expectation ranges over different choices of the training set D = { ( x 1 , y 1 ) … , ( x n , y n ) } {\displaystyle D=\{(x_{1},y_{1})\dots ,(x_{n},y_{n})\}} , all sampled from the same joint distribution P ( x , y ) {\displaystyle P(x,y)} which can for example be done via ... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error | 303 | 932 | null |
E.g., when approximating a non-linear function f ( x ) {\displaystyle f(x)} using a learning method for linear models, there will be error in the estimates f ^ ( x ) {\displaystyle {\hat {f}}(x)} due to this assumption; the variance of the learning method, or, intuitively, how much the learning method f ^ ( x ) {\displ... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error | 224 | 812 | null |
Let us write the mean-squared error of our model: MSE ≜ E [ ( y − f ^ ( x ) ) 2 ] = E [ ( f ( x ) + ε − f ^ ( x ) ) 2 ] since y ≜ f ( x ) + ε = E [ ( f ( x ) − f ^ ( x ) ) 2 ] + 2 E [ ( f ( x ) − f ^ ( x ) ) ε ] + E [ ε 2 ] {\displaystyle {\begin{aligned}{\text{MSE}}&\triangleq \mathbb {E} {\Big [}{\big (}y-{\hat {f}}(... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 349 | 635 | null |
{E} [\varepsilon ^{2}]\end{aligned}}} We can show that the second term of this equation is null: E [ ( f ( x ) − f ^ ( x ) ) ε ] = E [ f ( x ) − f ^ ( x ) ] E [ ε ] since ε is independent from x = 0 since E [ ε ] = 0 {\displaystyle {\begin{aligned}\mathbb {E} {\Big [}{\big (}f(x)-{\hat {f}}(x){\big )}\varepsilon {\Big ... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 302 | 691 | null |
Let us now expand the remaining term: E [ ( f ( x ) − f ^ ( x ) ) 2 ] = E [ ( f ( x ) − E [ f ^ ( x ) ] + E [ f ^ ( x ) ] − f ^ ( x ) ) 2 ] = E [ ( f ( x ) − E [ f ^ ( x ) ] ) 2 ] + 2 E [ ( f ( x ) − E [ f ^ ( x ) ] ) ( E [ f ^ ( x ) ] − f ^ ( x ) ) ] + E [ ( E [ f ^ ( x ) ] − f ^ ( x ) ) 2 ] {\displaystyle {\begin{ali... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 347 | 632 | null |
]}{\big )}^{2}{\Big ]}}\,+\,2\ {\color {PineGreen}\mathbb {E} {\Big [}{\big (}f(x)-\mathbb {E} {\big [}{\hat {f}}(x){\big ]}{\big )}{\big (}\mathbb {E} {\big [}{\hat {f}}(x){\big ]}-{\hat {f}}(x){\big )}{\Big ]}}\,+\,\mathbb {E} {\Big [}{\big (}\mathbb {E} {\big [}{\hat {f}}(x){\big ]}-{\hat {f}}(x){\big )}^{2}{\Big ]}... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 350 | 622 | null |
[}{\big (}f(x)-\mathbb {E} {\big [}{\hat {f}}(x){\big ]}{\big )}^{2}{\Big ]}}&=\mathbb {E} {\big [}f(x)^{2}{\big ]}\,-\,2\ \mathbb {E} {\Big [}f(x)\ \mathbb {E} {\big [}{\hat {f}}(x){\big ]}{\Big ]}\,+\,\mathbb {E} {\Big [}\mathbb {E} {\big [}{\hat {f}}(x){\big ]}^{2}{\Big ]}\\&=f(x)^{2}\,-\,2\ f(x)\ \mathbb {E} {\big ... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 350 | 566 | null |
not a random variable, but a fixed, deterministic function of x {\displaystyle x} . Therefore, E [ f ( x ) ] = f ( x ) {\displaystyle \mathbb {E} {\big [}f(x){\big ]}=f(x)} . Similarly E [ f ( x ) 2 ] = f ( x ) 2 {\displaystyle \mathbb {E} {\big [}f(x)^{2}{\big ]}=f(x)^{2}} , and E [ f ( x ) E [ f ^ ( x ) ] ] = f ( x )... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 300 | 588 | null |
Using the same reasoning, we can expand the second term and show that it is null: E [ ( f ( x ) − E [ f ^ ( x ) ] ) ( E [ f ^ ( x ) ] − f ^ ( x ) ) ] = E [ f ( x ) E [ f ^ ( x ) ] − f ( x ) f ^ ( x ) − E [ f ^ ( x ) ] 2 + E [ f ^ ( x ) ] f ^ ( x ) ] = f ( x ) E [ f ^ ( x ) ] − f ( x ) E [ f ^ ( x ) ] − E [ f ^ ( x ) ] ... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 350 | 671 | null |
[}{\hat {f}}(x){\big ]}^{2}+\mathbb {E} {\big [}{\hat {f}}(x){\big ]}\ {\hat {f}}(x){\Big ]}\\&=f(x)\ \mathbb {E} {\big [}{\hat {f}}(x){\big ]}\,-\,f(x)\ \mathbb {E} {\big [}{\hat {f}}(x){\big ]}\,-\,\mathbb {E} {\big [}{\hat {f}}(x){\big ]}^{2}\,+\,\mathbb {E} {\big [}{\hat {f}}(x){\big ]}^{2}\\&=0\end{aligned}}} Even... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 344 | 631 | null |
{f}}(x){\big ]}{\Big )}^{2}+\mathbb {E} {\Big [}{\big (}\mathbb {E} {\big [}{\hat {f}}(x){\big ]}-{\hat {f}}(x){\big )}^{2}{\Big ]}+\sigma ^{2}\\&=\operatorname {Bias} {\big (}{\hat {f}}(x){\big )}^{2}\,+\,\operatorname {Var} {\big [}{\hat {f}}(x){\big ]}\,+\,\sigma ^{2}\end{aligned}}} Finally, the MSE loss function (o... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 249 | 506 | null |
{f}}(x){\big ]}{\Big )}^{2}+\mathbb {E} {\Big [}{\big (}\mathbb {E} {\big [}{\hat {f}}(x){\big ]}-{\hat {f}}(x){\big )}^{2}{\Big ]}+\sigma ^{2}\\&=\operatorname {Bias} {\big (}{\hat {f}}(x){\big )}^{2}\,+\,\operatorname {Var} {\big [}{\hat {f}}(x){\big ]}\,+\,\sigma ^{2}\end{aligned}}} Finally, the MSE loss function (o... | Wikipedia - Bias–variance tradeoff - Bias–variance decomposition of mean squared error > Derivation | 356 | 687 | null |
Section: Approaches. Dimensionality reduction and feature selection can decrease variance by simplifying models. Similarly, a larger training set tends to decrease variance. Adding features (predictors) tends to decrease bias, at the expense of introducing additional variance. Learning algorithms typically have some tu... | Wikipedia - Bias–variance tradeoff - Approaches | 291 | 1,486 | null |
Section: Approaches > k-nearest neighbors. In the case of k-nearest neighbors regression, when the expectation is taken over the possible labeling of a fixed training set, a closed-form expression exists that relates the bias–variance decomposition to the parameter k:: 37, 223 E [ ( y − f ^ ( x ) ) 2 ∣ X = x ] = ( f ( ... | Wikipedia - Bias–variance tradeoff - Approaches > k-nearest neighbors | 336 | 958 | null |
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