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Examples for K {\displaystyle K} include the linear kernel [ K ( μ P , μ Q ) = ⟨ μ P , μ Q ⟩ H ( k ) ] {\displaystyle \left[K(\mu _{P},\mu _{Q})=\langle \mu _{P},\mu _{Q}\rangle _{{\mathcal {H}}(k)}\right]} , the Gaussian kernel [ K ( μ P , μ Q ) = e − ‖ μ P − μ Q ‖ H ( k ) 2 / ( 2 σ 2 ) ] {\displaystyle \left[K(\mu _{... | Wikipedia - Kernel embedding of distributions - Applications > Distribution regression | 350 | 681 | null |
] {\displaystyle \left[K(\mu _{P},\mu _{Q})=\left(1+\left\|\mu _{P}-\mu _{Q}\right\|_{H(k)}^{2}/\sigma ^{2}\right)^{-1}\right]} , the generalized t-student kernel [ K ( μ P , μ Q ) = ( 1 + ‖ μ P − μ Q ‖ H ( k ) σ ) − 1 , ( σ ≤ 2 ) ] {\displaystyle \left[K(\mu _{P},\mu _{Q})=\left(1+\left\|\mu _{P}-\mu _{Q}\right\|_{H(k... | Wikipedia - Kernel embedding of distributions - Applications > Distribution regression | 335 | 607 | null |
The prediction on a new distribution ( X ^ ) {\displaystyle ({\hat {X}})} takes the simple, analytical form y ^ ( X ^ ) = k [ G + λ ℓ ] − 1 y , {\displaystyle {\hat {y}}{\big (}{\hat {X}}{\big )}=\mathbf {k} [\mathbf {G} +\lambda \ell ]^{-1}\mathbf {y} ,} where k = [ K ( μ X ^ i , μ X ^ ) ] ∈ R 1 × ℓ {\displaystyle \ma... | Wikipedia - Kernel embedding of distributions - Applications > Distribution regression | 350 | 685 | null |
… ; y ℓ ] ∈ R ℓ {\displaystyle \mathbf {y} =[y_{1};\ldots ;y_{\ell }]\in \mathbb {R} ^{\ell }} . Under mild regularity conditions this estimator can be shown to be consistent and it can achieve the one-stage sampled (as if one had access to the true X i {\displaystyle X_{i}} -s) minimax optimal rate. In the J {\display... | Wikipedia - Kernel embedding of distributions - Applications > Distribution regression | 196 | 626 | null |
Section: Example. In this simple example, which is taken from Song et al., X , Y {\displaystyle X,Y} are assumed to be discrete random variables which take values in the set { 1 , … , K } {\displaystyle \{1,\ldots ,K\}} and the kernel is chosen to be the Kronecker delta function, so k ( x , x ′ ) = δ ( x , x ′ ) {\disp... | Wikipedia - Kernel embedding of distributions - Example | 159 | 488 | null |
The kernel embeddings of such a distributions are thus vectors of marginal probabilities while the embeddings of joint distributions in this setting are K × K {\displaystyle K\times K} matrices specifying joint probability tables, and the explicit form of these embeddings is μ X = E [ e X ] = ( P ( X = 1 ) ⋮ P ( X = K ... | Wikipedia - Kernel embedding of distributions - Example | 349 | 826 | null |
{\displaystyle {\mathcal {C}}_{Y\mid X}={\mathcal {C}}_{YX}{\mathcal {C}}_{XX}^{-1},} is in this setting a conditional probability table C Y ∣ X = ( P ( Y = s ∣ X = t ) ) s , t ∈ { 1 , … , K } {\displaystyle {\mathcal {C}}_{Y\mid X}=(P(Y=s\mid X=t))_{s,t\in \{1,\dots ,K\}}} and C X X = ( P ( X = 1 ) … 0 ⋮ ⋱ ⋮ 0 … P ( X... | Wikipedia - Kernel embedding of distributions - Example | 337 | 705 | null |
(x)={\begin{pmatrix}P(Y=1\mid X=x)\\\vdots \\P(Y=K\mid X=x)\\\end{pmatrix}}} In this discrete-valued setting with the Kronecker delta kernel, the kernel sum rule becomes ( P ( X = 1 ) ⋮ P ( X = N ) ) ⏟ μ X π = ( P ( X = s ∣ Y = t ) ) ⏟ C X ∣ Y ( π ( Y = 1 ) ⋮ π ( Y = N ) ) ⏟ μ Y π {\displaystyle \underbrace {\begin{pma... | Wikipedia - Kernel embedding of distributions - Example | 350 | 688 | null |
( Y = 1 ) … 0 ⋮ ⋱ ⋮ 0 … π ( Y = K ) ) ⏟ C Y Y π {\displaystyle \underbrace {\begin{pmatrix}\\P(X=s,Y=t)\\\\\end{pmatrix}} _{{\mathcal {C}}_{XY}^{\pi }}=\underbrace {\begin{pmatrix}\\P(X=s\mid Y=t)\\\\\end{pmatrix}} _{{\mathcal {C}}_{X\mid Y}}\underbrace {\begin{pmatrix}\pi (Y=1)&\dots &0\\\vdots &\ddots &\vdots \\0&\do... | Wikipedia - Kernel embedding of distributions - Example | 226 | 380 | null |
Section: Definition. A knowledge graph G = { E , R , F } {\displaystyle {\mathcal {G}}=\{E,R,F\}} is a collection of entities E {\displaystyle E} , relations R {\displaystyle R} , and facts F {\displaystyle F} . A fact is a triple ( h , r , t ) ∈ F {\displaystyle (h,r,t)\in F} that denotes a link r ∈ R {\displaystyle r... | Wikipedia - Knowledge graph embedding - Definition | 350 | 1,321 | null |
Section: Embedding procedure. All algorithms for creating a knowledge graph embedding follow the same approach. First, the embedding vectors are initialized to random values. Then, they are iteratively optimized using a training set of triples. In each iteration, a batch of size b {\displaystyle b} triples is sampled f... | Wikipedia - Knowledge graph embedding - Embedding procedure | 224 | 1,046 | null |
algorithm Compute entity and relation embeddings input: The training set S = { ( h , r , t ) } {\displaystyle S=\{(h,r,t)\}} , entity set E {\displaystyle E} , relation set R {\displaystyle R} , embedding dimension k {\displaystyle k} output: Entity and relation embeddings initialization: the entities e {\displaystyle ... | Wikipedia - Knowledge graph embedding - Embedding procedure > Pseudocode | 350 | 967 | null |
Section: Performance indicators > Hits@K. Hits@K or in short, H@K, is a performance index that measures the probability to find the correct prediction in the first top K model predictions. Usually, it is used k = 10 {\displaystyle k=10} . Hits@K reflects the accuracy of an embedding model to predict the relation betwee... | Wikipedia - Knowledge graph embedding - Performance indicators > Hits@K | 152 | 508 | null |
Section: Performance indicators > Mean reciprocal rank (MRR). Mean reciprocal rank measures the number of triples predicted correctly. If the first predicted triple is correct, then 1 is added, if the second is correct 1 2 {\displaystyle {\frac {1}{2}}} is summed, and so on. Mean reciprocal rank is generally used to qu... | Wikipedia - Knowledge graph embedding - Performance indicators > Mean reciprocal rank (MRR) | 162 | 517 | null |
Section: Applications > Machine learning tasks. Knowledge graph completion (KGC) is a collection of techniques to infer knowledge from an embedded knowledge graph representation. In particular, this technique completes a triple inferring the missing entity or relation. The corresponding sub-tasks are named link or enti... | Wikipedia - Knowledge graph embedding - Applications > Machine learning tasks | 190 | 990 | null |
Section: Applications > Real world applications. The use of knowledge graph embedding is increasingly pervasive in many applications. In the case of recommender systems, the use of knowledge graph embedding can overcome the limitations of the usual reinforcement learning, as well as limitations of the conventional coll... | Wikipedia - Knowledge graph embedding - Applications > Real world applications | 204 | 1,113 | null |
Section: Models. Given a collection of triples (or facts) F = { < h e a d , r e l a t i o n , t a i l > } {\displaystyle {\mathcal {F}}=\{<head,relation,tail>\}} , the knowledge graph embedding model produces, for each entity and relation present in the knowledge graph a continuous vector representation. ( h , r , t ) ... | Wikipedia - Knowledge graph embedding - Models | 328 | 1,106 | null |
Section: Models > Tensor decomposition model. The tensor decomposition is a family of knowledge graph embedding models that use a multi-dimensional matrix to represent a knowledge graph, that is partially knowable due to gaps of the graph describing a particular domain thoroughly. In particular, these models use a thir... | Wikipedia - Knowledge graph embedding - Models > Tensor decomposition model | 159 | 777 | null |
Section: Models > Tensor decomposition model > Bilinear models. This family of models uses a linear equation to embed the connection between the entities through a relation. In particular, the embedded representation of the relations is a bidimensional matrix. These models, during the embedding procedure, only use the ... | Wikipedia - Knowledge graph embedding - Models > Tensor decomposition model > Bilinear models | 302 | 1,489 | null |
SimplE: This model is the improvement of canonical polyadic decomposition (CP), in which an embedding vector for the relation and two independent embedding vectors for each entity are learned, depending on whether it is a head or a tail in the knowledge graph fact. SimplE resolves the problem of independent learning of... | Wikipedia - Knowledge graph embedding - Models > Tensor decomposition model > Bilinear models | 165 | 662 | null |
Section: Models > Tensor decomposition model > Non-bilinear models. HolE: HolE uses circular correlation to create an embedded representation of the knowledge graph, which can be seen as a compression of the matrix product, but is more computationally efficient and scalable while keeping the capabilities to express asy... | Wikipedia - Knowledge graph embedding - Models > Tensor decomposition model > Non-bilinear models | 343 | 1,706 | null |
Section: Models > Geometric models. The geometric space defined by this family of models encodes the relation as a geometric transformation between the head and tail of a fact. For this reason, to compute the embedding of the tail, it is necessary to apply a transformation τ {\displaystyle \tau } to the head embedding,... | Wikipedia - Knowledge graph embedding - Models > Geometric models | 204 | 803 | null |
Section: Models > Geometric models > Pure translational models. This class of models is inspired by the idea of translation invariance introduced in word2vec. A pure translational model relies on the fact that the embedding vector of the entities are close to each other after applying a proper relational translation in... | Wikipedia - Knowledge graph embedding - Models > Geometric models > Pure translational models | 336 | 1,550 | null |
TransR: A modification of TransH that uses different spaces embedding entities versus relations, thus separating the semantic spaces of entities and relations. TransR also uses a relational projection matrix to translate the embedding of the entities to the relation space. TransD: In TransR, the head and the tail of a ... | Wikipedia - Knowledge graph embedding - Models > Geometric models > Pure translational models | 316 | 1,501 | null |
Section: Models > Geometric models > Translational models with additional embeddings. It is possible to associate additional information to each element in the knowledge graph and their common representation facts. Each entity and relation can be enriched with text descriptions, weights, constraints, and others in orde... | Wikipedia - Knowledge graph embedding - Models > Geometric models > Translational models with additional embeddings | 340 | 1,421 | null |
In other words, given a fact ( h , r , t ) {\displaystyle (h,r,t)} , before applying the vector translation, the head h {\displaystyle h} is multiplied by W r h {\displaystyle W_{r}^{h}} and the tail is multiplied by W r t {\displaystyle W_{r}^{t}} . CrossE: Crossover interactions can be used for related information se... | Wikipedia - Knowledge graph embedding - Models > Geometric models > Translational models with additional embeddings | 275 | 1,179 | null |
Section: Models > Geometric models > Roto-translational models. This family of models, in addition or in substitution of a translation they employ a rotation-like transformation. TorusE: The regularization term of TransE makes the entity embedding to build a spheric space, and consequently loses the translation propert... | Wikipedia - Knowledge graph embedding - Models > Geometric models > Roto-translational models | 264 | 1,192 | null |
Section: Models > Deep learning models > Convolutional neural networks. This family of models, instead of using fully connected layers, employs one or more convolutional layers that convolve the input data applying a low-dimensional filter capable of embedding complex structures with few parameters by learning nonlinea... | Wikipedia - Knowledge graph embedding - Models > Deep learning models > Convolutional neural networks | 338 | 1,453 | null |
ConvE is also particularly efficient in the evaluation procedure: using a 1-N scoring, the model matches, given a head and a relation, all the tails at the same time, saving a lot of evaluation time when compared to the 1-1 evaluation program of the other models. ConvR: ConvR is an adaptive convolutional network aimed ... | Wikipedia - Knowledge graph embedding - Models > Deep learning models > Convolutional neural networks | 317 | 1,260 | null |
Section: Models > Deep learning models > Capsule neural networks. This family of models uses capsule neural networks to create a more stable representation that is able to recognize a feature in the input without losing spatial information. The network is composed of convolutional layers, but they are organized in caps... | Wikipedia - Knowledge graph embedding - Models > Deep learning models > Capsule neural networks | 216 | 864 | null |
Article: Knowledge integration. Knowledge integration is the process of synthesizing multiple knowledge models (or representations) into a common model (representation). Compared to information integration, which involves merging information having different schemas and representation models, knowledge integration focu... | Wikipedia - Knowledge integration - Summary | 350 | 2,089 | null |
Article: Labeled data. Labeled data is a group of samples that have been tagged with one or more labels. Labeling typically takes a set of unlabeled data and augments each piece of it with informative tags. For example, a data label might indicate whether a photo contains a horse or a cow, which words were uttered in a... | Wikipedia - Labeled data - Summary | 179 | 851 | null |
Section: Challenges > Data-driven bias. Algorithmic decision-making is subject to programmer-driven bias as well as data-driven bias. Training data that relies on bias labeled data will result in prejudices and omissions in a predictive model, despite the machine learning algorithm being legitimate. The labeled data us... | Wikipedia - Labeled data - Challenges > Data-driven bias | 182 | 882 | null |
Article: Lazy learning. (Not to be confused with the lazy learning regime, see Neural tangent kernel). In machine learning, lazy learning is a learning method in which generalization of the training data is, in theory, delayed until a query is made to the system, as opposed to eager learning, where the system tries to ... | Wikipedia - Lazy learning - Summary | 315 | 1,487 | null |
Section: Disadvantages. Theoretical disadvantages with lazy learning include: The large space requirement to store the entire training dataset. In practice, this is not an issue because of advances in hardware and the relatively small number of attributes (e.g., as co-occurrence frequency) that need to be stored. Parti... | Wikipedia - Lazy learning - Disadvantages | 346 | 1,726 | null |
There are standard techniques to improve re-computation efficiency so that a particular answer is not recomputed unless the data that impact this answer has changed (e.g., new items, new purchases, new views). In other words, the stored answers are updated incrementally. This approach, used by large e-commerce or media... | Wikipedia - Lazy learning - Disadvantages | 188 | 899 | null |
Section: Leakage modes > Training example leakage. Row-wise leakage is caused by improper sharing of information between rows of data. Types of row-wise leakage include: Premature featurization; leaking from premature featurization before Cross-validation/Train/Test split (must fit MinMax/ngrams/etc on only the train s... | Wikipedia - Leakage (machine learning) - Leakage modes > Training example leakage | 316 | 1,438 | null |
Section: Detection. Data leakage in machine learning can be detected through various methods, focusing on performance analysis, feature examination, data auditing, and model behavior analysis. Performance-wise, unusually high accuracy or significant discrepancies between training and test results often indicate leakage... | Wikipedia - Leakage (machine learning) - Detection | 291 | 1,628 | null |
Section: Definition > Background. Let Ω = X × Y = { ( x , y ) } {\displaystyle \Omega ={\mathcal {X}}\times {\mathcal {Y}}=\{(x,y)\}} be the sample space, where y {\displaystyle y} are the labels and x {\displaystyle x} are the covariates (predictors). F = { f : X ↦ Y } {\displaystyle {\mathcal {F}}=\{f:{\mathcal {X}}\... | Wikipedia - Learnable function class - Definition > Background | 223 | 598 | null |
L : Y × Y ↦ R {\displaystyle L:{\mathcal {Y}}\times {\mathcal {Y}}\mapsto \mathbb {R} } is a pre-given loss function (usually non-negative). Given a probability distribution P ( x , y ) {\displaystyle P(x,y)} on Ω {\displaystyle \Omega } , define the expected risk I P ( f ) {\displaystyle I_{P}(f)} to be: I P ( f ) = ∫... | Wikipedia - Learnable function class - Definition > Background | 291 | 811 | null |
That is, to find solutions to the following problem: f ^ = arg min f ∈ F I P ( f ) {\displaystyle {\hat {f}}=\arg \min _{f\in {\mathcal {F}}}I_{P}(f)} But in practice the distribution P {\displaystyle P} is unknown, and any learning task can only be based on finite samples. Thus we seek instead to find an algorithm t... | Wikipedia - Learnable function class - Definition > Background | 290 | 762 | null |
Section: Definition > Learnable function class. We can make the condition given in the above equation stronger by requiring that the convergence is uniform for all probability distributions. That is: The intuition behind the more strict requirement is as such: the rate at which sequence { f ^ n } {\displaystyle \{{\hat... | Wikipedia - Learnable function class - Definition > Learnable function class | 344 | 1,403 | null |
Section: Interpretations. If the true relationship between y {\displaystyle y} and x {\displaystyle x} is y ∼ f ∗ ( x ) {\displaystyle y\sim f^{*}(x)} , then by selecting the appropriate loss function, f ∗ {\displaystyle f^{*}} can always be expressed as the minimizer of the expected loss across all possible functions.... | Wikipedia - Learnable function class - Interpretations | 324 | 1,014 | null |
Thus we often consider a subset of F ∗ {\displaystyle {\mathcal {F}}^{*}} , F {\displaystyle {\mathcal {F}}} , to carry out searches on. By doing so, we risk that f ∗ {\displaystyle f^{*}} might not be an element of F {\displaystyle {\mathcal {F}}} . This tradeoff can be mathematically expressed as In the above decompo... | Wikipedia - Learnable function class - Interpretations | 295 | 988 | null |
Section: Example: Tikhonov regularization. A good example where learnable classes are used is the so-called Tikhonov regularization in reproducing kernel Hilbert space (RKHS). Specifically, let F ∗ {\displaystyle {\mathcal {F^{*}}}} be an RKHS, and | | ⋅ | | 2 {\displaystyle ||\cdot ||_{2}} be the norm on F ∗ {\display... | Wikipedia - Learnable function class - Example: Tikhonov regularization | 337 | 882 | null |
The empirical minimization algorithm to the dual form of this problem is arg min f ∈ F ∗ { ∑ i = 1 n L ( f ( x i ) , y i ) + λ | | f | | 2 } {\displaystyle \arg \min _{f\in {\mathcal {F}}^{*}}\left\{\sum _{i=1}^{n}L(f(x_{i}),y_{i})+\lambda ||f||_{2}\right\}} This was first introduced by Tikhonov to solve ill-posed pr... | Wikipedia - Learnable function class - Example: Tikhonov regularization | 330 | 951 | null |
Section: Relationship to empirical process theory. Part ( a ) {\displaystyle (a)} in (2) is closely linked to empirical process theory in statistics, where the empirical risk { ∑ i = 1 n L ( y i , f ( x i ) ) , f ∈ F } {\displaystyle \{\sum _{i=1}^{n}L(y_{i},f(x_{i})),f\in {\mathcal {F}}\}} are known as empirical proce... | Wikipedia - Learnable function class - Relationship to empirical process theory | 264 | 957 | null |
Section: Definition. A learning automaton is an adaptive decision-making unit situated in a random environment that learns the optimal action through repeated interactions with its environment. The actions are chosen according to a specific probability distribution which is updated based on the environment response the... | Wikipedia - Learning automaton - Definition | 322 | 1,388 | null |
The states of such an automaton correspond to the states of a "discrete-state discrete-parameter Markov process". At each time step t=0,1,2,3,..., the automaton reads an input from its environment, updates p(t) to p(t+1) by A, randomly chooses a successor state according to the probabilities p(t+1) and outputs the corr... | Wikipedia - Learning automaton - Definition | 259 | 1,018 | null |
Section: Literature. Philip Aranzulla and John Mellor (Home page): Mellor J and Aranzulla P (2000): "Using an S-Model Response Environment with Learnng [sic] Automata Based Routing Schemes for IP Networks ", Proc. Eighth IFIP Workshop on Performance Modelling and Evaluation of ATM and IP Networks, pp 56/1-56/12, Ilkley... | Wikipedia - Learning automaton - Literature | 259 | 900 | null |
Article: Learning curve (machine learning). In machine learning (ML), a learning curve (or training curve) is a graphical representation that shows how a model's performance on a training set (and usually a validation set) changes with the number of training iterations (epochs) or the amount of training data. Typically... | Wikipedia - Learning curve (machine learning) - Summary | 273 | 1,398 | null |
Section: Formal definition. When creating a function to approximate the distribution of some data, it is necessary to define a loss function L ( f θ ( X ) , Y ) {\displaystyle L(f_{\theta }(X),Y)} to measure how good the model output is (e.g., accuracy for classification tasks or mean squared error for regression). We ... | Wikipedia - Learning curve (machine learning) - Formal definition | 150 | 539 | null |
If the training data is { x 1 , x 2 , … , x n } , { y 1 , y 2 , … y n } {\displaystyle \{x_{1},x_{2},\dots ,x_{n}\},\{y_{1},y_{2},\dots y_{n}\}} and the validation data is { x 1 ′ , x 2 ′ , … x m ′ } , { y 1 ′ , y 2 ′ , … y m ′ } {\displaystyle \{x_{1}',x_{2}',\dots x_{m}'\},\{y_{1}',y_{2}',\dots y_{m}'\}} , a learning... | Wikipedia - Learning curve (machine learning) - Formal definition > Training curve for amount of data | 348 | 639 | null |
Section: Formal definition > Training curve for number of iterations. Many optimization algorithms are iterative, repeating the same step (such as backpropagation) until the process converges to an optimal value. Gradient descent is one such algorithm. If θ i ∗ {\displaystyle \theta _{i}^{*}} is the approximation of th... | Wikipedia - Learning curve (machine learning) - Formal definition > Training curve for number of iterations | 213 | 618 | null |
Article: Learning rate. In machine learning and statistics, the learning rate is a tuning parameter in an optimization algorithm that determines the step size at each iteration while moving toward a minimum of a loss function. Since it influences to what extent newly acquired information overrides old information, it m... | Wikipedia - Learning rate - Summary | 290 | 1,519 | null |
Section: Learning rate schedule. Initial rate can be left as system default or can be selected using a range of techniques. A learning rate schedule changes the learning rate during learning and is most often changed between epochs/iterations. This is mainly done with two parameters: decay and momentum. There are many ... | Wikipedia - Learning rate - Learning rate schedule | 300 | 1,480 | null |
Time-based learning schedules alter the learning rate depending on the learning rate of the previous time iteration. Factoring in the decay the mathematical formula for the learning rate is: η n + 1 = η n 1 + d n {\displaystyle \eta _{n+1}={\frac {\eta _{n}}{1+dn}}} where η {\displaystyle \eta } is the learning rate, d... | Wikipedia - Learning rate - Learning rate schedule | 350 | 1,195 | null |
Section: Applications > In information retrieval. Ranking is a central part of many information retrieval problems, such as document retrieval, collaborative filtering, sentiment analysis, and online advertising. A possible architecture of a machine-learned search engine is shown in the accompanying figure. Training da... | Wikipedia - Learning to rank - Applications > In information retrieval | 311 | 1,576 | null |
Typically, users expect a search query to complete in a short time (such as a few hundred milliseconds for web search), which makes it impossible to evaluate a complex ranking model on each document in the corpus, and so a two-phase scheme is used. First, a small number of potentially relevant documents are identified ... | Wikipedia - Learning to rank - Applications > In information retrieval | 166 | 784 | null |
Section: Feature vectors. For the convenience of MLR algorithms, query-document pairs are usually represented by numerical vectors, which are called feature vectors. Such an approach is sometimes called bag of features and is analogous to the bag of words model and vector space model used in information retrieval for r... | Wikipedia - Learning to rank - Feature vectors | 333 | 1,588 | null |
Section: Evaluation measures. There are several measures (metrics) which are commonly used to judge how well an algorithm is doing on training data and to compare the performance of different MLR algorithms. Often a learning-to-rank problem is reformulated as an optimization problem with respect to one of these metrics... | Wikipedia - Learning to rank - Evaluation measures | 252 | 1,156 | null |
Section: Approaches. Learning to Rank approaches are often categorized using one of three approaches: pointwise (where individual documents are ranked), pairwise (where pairs of documents are ranked into a relative order), and listwise (where an entire list of documents are ordered). Tie-Yan Liu of Microsoft Research A... | Wikipedia - Learning to rank - Approaches | 271 | 1,244 | null |
Section: Approaches > Pointwise approach. In this case, it is assumed that each query-document pair in the training data has a numerical or ordinal score. Then the learning-to-rank problem can be approximated by a regression problem — given a single query-document pair, predict its score. Formally speaking, the pointwi... | Wikipedia - Learning to rank - Approaches > Pointwise approach | 202 | 850 | null |
Section: Approaches > Pairwise approach. In this case, the learning-to-rank problem is approximated by a classification problem — learning a binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} that can tell which document is better in a given pair of documents. The classifier shall take two documents as it... | Wikipedia - Learning to rank - Approaches > Pairwise approach | 209 | 698 | null |
In many cases, the binary classifier h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} is implemented with a scoring function f ( x ) {\displaystyle f(x)} . As an example, RankNet adapts a probability model and defines h ( x u , x v ) {\displaystyle h(x_{u},x_{v})} as the estimated probability of the document x u {\displa... | Wikipedia - Learning to rank - Approaches > Pairwise approach | 300 | 733 | null |
Section: History > Practical usage by search engines. Commercial web search engines began using machine-learned ranking systems since the 2000s (decade). One of the first search engines to start using it was AltaVista (later its technology was acquired by Overture, and then Yahoo), which launched a gradient boosting-tr... | Wikipedia - Learning to rank - History > Practical usage by search engines | 307 | 1,581 | null |
Article: Learning with errors. In cryptography, learning with errors (LWE) is a mathematical problem that is widely used to create secure encryption algorithms. It is based on the idea of representing secret information as a set of equations with errors. In other words, LWE is a way to hide the value of a secret by int... | Wikipedia - Learning with errors - Summary | 276 | 993 | null |
Let Z q {\displaystyle \mathbb {Z} _{q}} denote the ring of integers modulo q {\displaystyle q} and let Z q n {\displaystyle \mathbb {Z} _{q}^{n}} denote the set of n {\displaystyle n} -vectors over Z q {\displaystyle \mathbb {Z} _{q}} . There exists a certain unknown linear function f : Z q n → Z q {\displaystyle f:\m... | Wikipedia - Learning with errors - Summary | 336 | 1,031 | null |
Section: Definition. Denote by T = R / Z {\displaystyle \mathbb {T} =\mathbb {R} /\mathbb {Z} } the additive group on reals modulo one. Let s ∈ Z q n {\displaystyle \mathbf {s} \in \mathbb {Z} _{q}^{n}} be a fixed vector. Let ϕ {\displaystyle \phi } be a fixed probability distribution over T {\displaystyle \mathbb {T} ... | Wikipedia - Learning with errors - Definition | 175 | 487 | null |
Pick a vector a ∈ Z q n {\displaystyle \mathbf {a} \in \mathbb {Z} _{q}^{n}} from the uniform distribution over Z q n {\displaystyle \mathbb {Z} _{q}^{n}} , Pick a number e ∈ T {\displaystyle e\in \mathbb {T} } from the distribution ϕ {\displaystyle \phi } , Evaluate t = ⟨ a , s ⟩ / q + e {\displaystyle t=\langle \math... | Wikipedia - Learning with errors - Definition | 350 | 892 | null |
The learning with errors problem L W E q , ϕ {\displaystyle \mathrm {LWE} _{q,\phi }} is to find s ∈ Z q n {\displaystyle \mathbf {s} \in \mathbb {Z} _{q}^{n}} , given access to polynomially many samples of choice from A s , ϕ {\displaystyle A_{\mathbf {s} ,\phi }} . For every α > 0 {\displaystyle \alpha >0} , denote b... | Wikipedia - Learning with errors - Definition | 322 | 849 | null |
For every α > 0 {\displaystyle \alpha >0} , denote by D α {\displaystyle D_{\alpha }} the one-dimensional Gaussian with zero mean and variance α 2 / ( 2 π ) {\displaystyle \alpha ^{2}/(2\pi )} , that is, the density function is D α ( x ) = ρ α ( x ) / α {\displaystyle D_{\alpha }(x)=\rho _{\alpha }(x)/\alpha } where ρ ... | Wikipedia - Learning with errors - Definition | 267 | 708 | null |
Section: Decision version > Solving decision assuming search. Intuitively, if we have a procedure for the search problem, the decision version can be solved easily: just feed the input samples for the decision problem to the solver for the search problem. Denote the given samples by { ( a i , b i ) } ⊂ Z q n × T {\disp... | Wikipedia - Learning with errors - Decision version > Solving decision assuming search | 258 | 867 | null |
Section: Decision version > Solving search assuming decision. For the other direction, given a solver for the decision problem, the search version can be solved as follows: Recover s {\displaystyle \mathbf {s} } one coordinate at a time. To obtain the first coordinate, s 1 {\displaystyle \mathbf {s} _{1}} , make a gues... | Wikipedia - Learning with errors - Decision version > Solving search assuming decision | 303 | 843 | null |
Send the transformed samples to the decision solver. If the guess k {\displaystyle k} was correct, the transformation takes the distribution A s , χ {\displaystyle A_{\mathbf {s} ,\chi }} to itself, and otherwise, since q {\displaystyle q} is prime, it takes it to the uniform distribution. So, given a polynomial-time s... | Wikipedia - Learning with errors - Decision version > Solving search assuming decision | 321 | 1,091 | null |
Namely, we transform our b i {\displaystyle \mathbf {b} _{i}} samples the same way, and transform our a i {\displaystyle \mathbf {a} _{i}} samples by calculating a i + ( 0 , … , r , … , 0 ) {\displaystyle \mathbf {a} _{i}+(0,\ldots ,r,\ldots ,0)} , where the r {\displaystyle r} is in the j th {\displaystyle j^{\text{th... | Wikipedia - Learning with errors - Decision version > Solving search assuming decision | 303 | 852 | null |
Section: Decision version > Average case hardness. Regev showed the random self-reducibility of the LWE and DLWE problems for arbitrary q {\displaystyle q} and χ {\displaystyle \chi } . Given samples { ( a i , b i ) } {\displaystyle \{(\mathbf {a} _{i},\mathbf {b} _{i})\}} from A s , χ {\displaystyle A_{\mathbf {s} ,\c... | Wikipedia - Learning with errors - Decision version > Average case hardness | 230 | 573 | null |
Given samples { ( a i , b i ) } {\displaystyle \{(\mathbf {a} _{i},\mathbf {b} _{i})\}} from A s , χ {\displaystyle A_{\mathbf {s} ,\chi }} , it is easy to see that { ( a i , b i + ⟨ a i , t ⟩ ) / q } {\displaystyle \{(\mathbf {a} _{i},\mathbf {b} _{i}+\langle \mathbf {a} _{i},\mathbf {t} \rangle )/q\}} are samples fro... | Wikipedia - Learning with errors - Decision version > Average case hardness | 351 | 778 | null |
So, suppose there was some set S ⊂ Z q n {\displaystyle {\mathcal {S}}\subset \mathbb {Z} _{q}^{n}} such that | S | / | Z q n | = 1 / poly ( n ) {\displaystyle |{\mathcal {S}}|/|\mathbb {Z} _{q}^{n}|=1/\operatorname {poly} (n)} , and for distributions A s ′ , χ {\displaystyle A_{\mathbf {s} ',\chi }} , with s ′ ← S {... | Wikipedia - Learning with errors - Decision version > Average case hardness | 269 | 665 | null |
Then there would be some distinguisher A {\displaystyle {\mathcal {A}}} , who, given samples { ( a i , b i ) } {\displaystyle \{(\mathbf {a} _{i},\mathbf {b} _{i})\}} , could tell whether they were uniformly random or from A s ′ , χ {\displaystyle A_{\mathbf {s} ',\chi }} . If we need to distinguish uniformly random sa... | Wikipedia - Learning with errors - Decision version > Average case hardness | 319 | 853 | null |
If we need to distinguish uniformly random samples from A s , χ {\displaystyle A_{\mathbf {s} ,\chi }} , where s {\displaystyle \mathbf {s} } is chosen uniformly at random from Z q n {\displaystyle \mathbb {Z} _{q}^{n}} , we could simply try different values t {\displaystyle \mathbf {t} } sampled uniformly at random fr... | Wikipedia - Learning with errors - Decision version > Average case hardness | 347 | 974 | null |
Since S {\displaystyle {\mathcal {S}}} comprises a large fraction of Z q n {\displaystyle \mathbb {Z} _{q}^{n}} , with high probability, if we choose a polynomial number of values for t {\displaystyle \mathbf {t} } , we will find one such that s + t ∈ S {\displaystyle \mathbf {s} +\mathbf {t} \in {\mathcal {S}}} , and ... | Wikipedia - Learning with errors - Decision version > Average case hardness | 179 | 565 | null |
Section: Hardness results > Regev's result. For a n-dimensional lattice L {\displaystyle L} , let smoothing parameter η ε ( L ) {\displaystyle \eta _{\varepsilon }(L)} denote the smallest s {\displaystyle s} such that ρ 1 / s ( L ∗ ∖ { 0 } ) ≤ ε {\displaystyle \rho _{1/s}(L^{*}\setminus \{\mathbf {0} \})\leq \varepsilo... | Wikipedia - Learning with errors - Hardness results > Regev's result | 320 | 884 | null |
The probability of each x ∈ L {\displaystyle x\in L} is proportional to ρ r ( x ) {\displaystyle \rho _{r}(x)} . The discrete Gaussian sampling problem(DGS) is defined as follows: An instance of D G S ϕ {\displaystyle DGS_{\phi }} is given by an n {\displaystyle n} -dimensional lattice L {\displaystyle L} and a number ... | Wikipedia - Learning with errors - Hardness results > Regev's result | 265 | 720 | null |
Regev shows that there is a reduction from GapSVP 100 n γ ( n ) {\displaystyle \operatorname {GapSVP} _{100{\sqrt {n}}\gamma (n)}} to D G S n γ ( n ) / λ ( L ∗ ) {\displaystyle DGS_{{\sqrt {n}}\gamma (n)/\lambda (L^{*})}} for any function γ ( n ) ≥ 1 {\displaystyle \gamma (n)\geq 1} . Regev then shows that there exists... | Wikipedia - Learning with errors - Hardness results > Regev's result | 329 | 898 | null |
Section: Hardness results > Peikert's result. Peikert proves that there is a probabilistic polynomial time reduction from the GapSVP ζ , γ {\displaystyle \operatorname {GapSVP} _{\zeta ,\gamma }} problem in the worst case to solving L W E q , Ψ α {\displaystyle \mathrm {LWE} _{q,\Psi _{\alpha }}} using poly ( n ) {\d... | Wikipedia - Learning with errors - Hardness results > Peikert's result | 255 | 676 | null |
Section: Use in cryptography. The LWE problem serves as a versatile problem used in construction of several cryptosystems. In 2005, Regev showed that the decision version of LWE is hard assuming quantum hardness of the lattice problems G a p S V P γ {\displaystyle \mathrm {GapSVP} _{\gamma }} (for γ {\displaystyle \gam... | Wikipedia - Learning with errors - Use in cryptography | 221 | 751 | null |
Section: Use in cryptography > Public-key cryptosystem. Regev proposed a public-key cryptosystem based on the hardness of the LWE problem. The cryptosystem as well as the proof of security and correctness are completely classical. The system is characterized by m , q {\displaystyle m,q} and a probability distribution χ... | Wikipedia - Learning with errors - Use in cryptography > Public-key cryptosystem | 159 | 578 | null |
The setting of the parameters used in proofs of correctness and security is q ≥ 2 {\displaystyle q\geq 2} , usually a prime number between n 2 {\displaystyle n^{2}} and 2 n 2 {\displaystyle 2n^{2}} . m = ( 1 + ε ) ( n + 1 ) log q {\displaystyle m=(1+\varepsilon )(n+1)\log q} for an arbitrary constant ε {\displaystyle... | Wikipedia - Learning with errors - Use in cryptography > Public-key cryptosystem | 315 | 907 | null |
The cryptosystem is then defined by: Private key: Private key is an s ∈ Z q n {\displaystyle \mathbf {s} \in \mathbb {Z} _{q}^{n}} chosen uniformly at random. Public key: Choose m {\displaystyle m} vectors a 1 , … , a m ∈ Z q n {\displaystyle \mathbf {a} _{1},\ldots ,\mathbf {a} _{m}\in \mathbb {Z} _{q}^{n}} uniformly ... | Wikipedia - Learning with errors - Use in cryptography > Public-key cryptosystem | 180 | 482 | null |
The public key consists of ( a i , b i = ⟨ a i , s ⟩ / q + e i ) i = 1 m {\displaystyle (\mathbf {a} _{i},b_{i}=\langle \mathbf {a} _{i},\mathbf {s} \rangle /q+e_{i})_{i=1}^{m}} Encryption: The encryption of a bit x ∈ { 0 , 1 } {\displaystyle x\in \{0,1\}} is done by choosing a random subset S {\displaystyle S} of [ m ... | Wikipedia - Learning with errors - Use in cryptography > Public-key cryptosystem | 344 | 791 | null |
Section: Use in cryptography > Key exchange. The idea of using LWE and Ring LWE for key exchange was proposed and filed at the University of Cincinnati in 2011 by Jintai Ding. The idea comes from the associativity of matrix multiplications, and the errors are used to provide the security. The paper appeared in 2012 aft... | Wikipedia - Learning with errors - Use in cryptography > Key exchange | 172 | 802 | null |
Section: Definition. The correlation coefficient ρ, expressed as an autocorrelation function or cross-correlation function, depends on the lag-time between the times being considered. Typically such functions, ρ(t), decay to zero with increasing lag-time, but they can assume values across all levels of correlations: st... | Wikipedia - Life-time of correlation - Definition | 171 | 835 | null |
Section: Definition. The basic form of a linear predictor function f ( i ) {\displaystyle f(i)} for data point i (consisting of p explanatory variables), for i = 1, ..., n, is f ( i ) = β 0 + β 1 x i 1 + ⋯ + β p x i p , {\displaystyle f(i)=\beta _{0}+\beta _{1}x_{i1}+\cdots +\beta _{p}x_{ip},} where x i k {\displaystyl... | Wikipedia - Linear predictor function - Definition | 223 | 624 | null |
Section: Definition > Notations > Matrix Notation. An equivalent form using matrix notation is as follows: f ( i ) = β T x i = x i T β {\displaystyle f(i)={\boldsymbol {\beta }}^{\mathrm {T} }\mathbf {x} _{i}=\mathbf {x} _{i}^{\mathrm {T} }{\boldsymbol {\beta }}} where β {\displaystyle {\boldsymbol {\beta }}} and x i {... | Wikipedia - Linear predictor function - Definition > Notations > Matrix Notation | 339 | 853 | null |
Section: Linear regression. An example of the usage of a linear predictor function is in linear regression, where each data point is associated with a continuous outcome yi, and the relationship written y i = f ( i ) + ε i = β T x i + ε i , {\displaystyle y_{i}=f(i)+\varepsilon _{i}={\boldsymbol {\beta }}^{\mathrm {T} ... | Wikipedia - Linear predictor function - Linear regression | 171 | 571 | null |
Section: Stacking. In some models (standard linear regression, in particular), the equations for each of the data points i = 1, ..., n are stacked together and written in vector form as y = X β + ε , {\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},\,} where y = ( y 1 y 2 ⋮ y ... | Wikipedia - Linear predictor function - Stacking | 172 | 451 | null |
In some models (standard linear regression, in particular), the equations for each of the data points i = 1, ..., n are stacked together and written in vector form as y = X β + ε , {\displaystyle \mathbf {y} =\mathbf {X} {\boldsymbol {\beta }}+{\boldsymbol {\varepsilon }},\,} where y = ( y 1 y 2 ⋮ y n ) , X = ( x 1 ′ x... | Wikipedia - Linear predictor function - Stacking | 516 | 1,086 | null |
{\displaystyle \mathbf {y} ={\begin{pmatrix}y_{1}\\y_{2}\\\vdots \\y_{n}\end{pmatrix}},\quad \mathbf {X} ={\begin{pmatrix}\mathbf {x} '_{1}\\\mathbf {x} '_{2}\\\vdots \\\mathbf {x} '_{n}\end{pmatrix}}={\begin{pmatrix}x_{11}&\cdots &x_{1p}\\x_{21}&\cdots &x_{2p}\\\vdots &\ddots &\vdots \\x_{n1}&\cdots &x_{np}\end{pmatri... | Wikipedia - Linear predictor function - Stacking | 397 | 889 | null |
The variables ε i {\displaystyle \varepsilon _{i}} are random variables, which in standard linear regression are distributed according to a standard normal distribution; they express the influence of any unknown factors on the outcome. This makes it possible to find optimal coefficients through the method of least squa... | Wikipedia - Linear predictor function - Stacking | 309 | 1,092 | null |
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