AI Headshot Examples

AI Headshot Examples — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Concurrent MetateM

  Concurrent MetateM is a multi-agent language in which each agent is programmed using a set of (augmented) temporal logic specifications of the behaviour it should exhibit. These specifications are executed directly to generate the behaviour of the agent. As a result, there is no risk of invalidating the logic as with systems where logical specification must first be translated to a lower-level implementation. The root of the MetateM concept is Gabbay's separation theorem; any arbitrary temporal logic formula can be rewritten in a logically equivalent past → future form. Execution proceeds by a process of continually matching rules against a history, and firing those rules when antecedents are satisfied. Any instantiated future-time consequents become commitments which must subsequently be satisfied, iteratively generating a model for the formula made up of the program rules. == Temporal Connectives == The Temporal Connectives of Concurrent MetateM can divided into two categories, as follows: Strict past time connectives: '●' (weak last), '◎' (strong last), '◆' (was), '■' (heretofore), 'S' (since), and 'Z' (zince, or weak since). Present and future time connectives: '◯' (next), '◇' (sometime), '□' (always), 'U' (until), and 'W' (unless). The connectives {◎,●,◆,■,◯,◇,□} are unary; the remainder are binary. === Strict past time connectives === ==== Weak last ==== ●ρ is satisfied now if ρ was true in the previous time. If ●ρ is interpreted at the beginning of time, it is satisfied despite there being no actual previous time. Hence "weak" last. ==== Strong last ==== ◎ρ is satisfied now if ρ was true in the previous time. If ◎ρ is interpreted at the beginning of time, it is not satisfied because there is no actual previous time. Hence "strong" last. ==== Was ==== ◆ρ is satisfied now if ρ was true in any previous moment in time. ==== Heretofore ==== ■ρ is satisfied now if ρ was true in every previous moment in time. ==== Since ==== ρSψ is satisfied now if ψ is true at any previous moment and ρ is true at every moment after that moment. ==== Zince, or weak since ==== ρZψ is satisfied now if (ψ is true at any previous moment and ρ is true at every moment after that moment) OR ψ has not happened in the past. === Present and future time connectives === ==== Next ==== ◯ρ is satisfied now if ρ is true in the next moment in time. ==== Sometime ==== ◇ρ is satisfied now if ρ is true now or in any future moment in time. ==== Always ==== □ρ is satisfied now if ρ is true now and in every future moment in time. ==== Until ==== ρUψ is satisfied now if ψ is true at any future moment and ρ is true at every moment prior. ==== Unless ==== ρWψ is satisfied now if (ψ is true at any future moment and ρ is true at every moment prior) OR ψ does not happen in the future.

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  Multilinear subspace learning

  Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality reduction can be performed on a data tensor that contains a collection of observations that have been vectorized, or observations that are treated as matrices and concatenated into a data tensor. Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D). The mapping from a high-dimensional vector space to a set of lower dimensional vector spaces is a multilinear projection. When observations are retained in the same organizational structure as matrices or higher order tensors, their representations are computed by performing linear projections into the column space, row space and fiber space. Multilinear subspace learning algorithms are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA), independent component analysis (ICA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA). == Background == Multilinear methods may be causal in nature and perform causal inference, or they may be simple regression methods from which no causal conclusion are drawn. Linear subspace learning algorithms are traditional dimensionality reduction techniques that are well suited for datasets that are the result of varying a single causal factor. Unfortunately, they often become inadequate when dealing with datasets that are the result of multiple causal factors. . Multilinear subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor for causally aware dimensionality reduction. These methods may also be employed in reducing horizontal and vertical redundancies irrespective of the causal factors when the observations are treated as a "matrix" (ie. a collection of independent column/row observations) and concatenated into a tensor. == Algorithms == === Multilinear principal component analysis === Historically, multilinear principal component analysis has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg. In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between multilinear tensor decompositions that computed 2nd order statistics associated with each data tensor mode, and subsequent work on Multilinear Independent Component Analysis that computed higher order statistics for each tensor mode. MPCA is an extension of PCA. === Multilinear independent component analysis === Multilinear independent component analysis is an extension of ICA. === Multilinear linear discriminant analysis === Multilinear extension of LDA TTP-based: Discriminant Analysis with Tensor Representation (DATER) TTP-based: General tensor discriminant analysis (GTDA) TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA) === Multilinear canonical correlation analysis === Multilinear extension of CCA TTP-based: Tensor Canonical Correlation Analysis (TCCA) TVP-based: Multilinear Canonical Correlation Analysis (MCCA) TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF) A TTP is a direct projection of a high-dimensional tensor to a low-dimensional tensor of the same order, using N projection matrices for an Nth-order tensor. It can be performed in N steps with each step performing a tensor-matrix multiplication (product). The N steps are exchangeable. This projection is an extension of the higher-order singular value decomposition (HOSVD) to subspace learning. Hence, its origin is traced back to the Tucker decomposition in 1960s. A TVP is a direct projection of a high-dimensional tensor to a low-dimensional vector, which is also referred to as the rank-one projections. As TVP projects a tensor to a vector, it can be viewed as multiple projections from a tensor to a scalar. Thus, the TVP of a tensor to a P-dimensional vector consists of P projections from the tensor to a scalar. The projection from a tensor to a scalar is an elementary multilinear projection (EMP). In EMP, a tensor is projected to a point through N unit projection vectors. It is the projection of a tensor on a single line (resulting a scalar), with one projection vector in each mode. Thus, the TVP of a tensor object to a vector in a P-dimensional vector space consists of P EMPs. This projection is an extension of the canonical decomposition, also known as the parallel factors (PARAFAC) decomposition. === Typical approach in MSL === There are N sets of parameters to be solved, one in each mode. The solution to one set often depends on the other sets (except when N=1, the linear case). Therefore, the suboptimal iterative procedure in is followed. Initialization of the projections in each mode For each mode, fixing the projection in all the other mode, and solve for the projection in the current mode. Do the mode-wise optimization for a few iterations or until convergence. This is originated from the alternating least square method for multi-way data analysis. == Code == MATLAB Tensor Toolbox by Sandia National Laboratories. The MPCA algorithm written in Matlab (MPCA+LDA included). The UMPCA algorithm written in Matlab (data included). The UMLDA algorithm written in Matlab (data included). == Tensor data sets == 3D gait data (third-order tensors): 128x88x20(21.2M); 64x44x20(9.9M); 32x22x10(3.2M);

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  Sufficient dimension reduction

  In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle {\textbf {x}}} , regression analysis aims to study the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , the conditional distribution of y {\displaystyle y} given x {\displaystyle {\textbf {x}}} . A dimension reduction is a function R ( x ) {\displaystyle R({\textbf {x}})} that maps x {\displaystyle {\textbf {x}}} to a subset of R k {\displaystyle \mathbb {R} ^{k}} , k < p, thereby reducing the dimension of x {\displaystyle {\textbf {x}}} . For example, R ( x ) {\displaystyle R({\textbf {x}})} may be one or more linear combinations of x {\displaystyle {\textbf {x}}} . A dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} is said to be sufficient if the distribution of y ∣ R ( x ) {\displaystyle y\mid R({\textbf {x}})} is the same as that of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, no information about the regression is lost in reducing the dimension of x {\displaystyle {\textbf {x}}} if the reduction is sufficient. == Graphical motivation == In a regression setting, it is often useful to summarize the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} graphically. For instance, one may consider a scatterplot of y {\displaystyle y} versus one or more of the predictors or a linear combination of the predictors. A scatterplot that contains all available regression information is called a sufficient summary plot. When x {\displaystyle {\textbf {x}}} is high-dimensional, particularly when p ≥ 3 {\displaystyle p\geq 3} , it becomes increasingly challenging to construct and visually interpret sufficiency summary plots without reducing the data. Even three-dimensional scatter plots must be viewed via a computer program, and the third dimension can only be visualized by rotating the coordinate axes. However, if there exists a sufficient dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} with small enough dimension, a sufficient summary plot of y {\displaystyle y} versus R ( x ) {\displaystyle R({\textbf {x}})} may be constructed and visually interpreted with relative ease. Hence sufficient dimension reduction allows for graphical intuition about the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , which might not have otherwise been available for high-dimensional data. Most graphical methodology focuses primarily on dimension reduction involving linear combinations of x {\displaystyle {\textbf {x}}} . The rest of this article deals only with such reductions. == Dimension reduction subspace == Suppose R ( x ) = A T x {\displaystyle R({\textbf {x}})=A^{T}{\textbf {x}}} is a sufficient dimension reduction, where A {\displaystyle A} is a p × k {\displaystyle p\times k} matrix with rank k ≤ p {\displaystyle k\leq p} . Then the regression information for y ∣ x {\displaystyle y\mid {\textbf {x}}} can be inferred by studying the distribution of y ∣ A T x {\displaystyle y\mid A^{T}{\textbf {x}}} , and the plot of y {\displaystyle y} versus A T x {\displaystyle A^{T}{\textbf {x}}} is a sufficient summary plot. Without loss of generality, only the space spanned by the columns of A {\displaystyle A} need be considered. Let η {\displaystyle \eta } be a basis for the column space of A {\displaystyle A} , and let the space spanned by η {\displaystyle \eta } be denoted by S ( η ) {\displaystyle {\mathcal {S}}(\eta )} . It follows from the definition of a sufficient dimension reduction that F y ∣ x = F y ∣ η T x , {\displaystyle F_{y\mid x}=F_{y\mid \eta ^{T}x},} where F {\displaystyle F} denotes the appropriate distribution function. Another way to express this property is y ⊥ ⊥ x ∣ η T x , {\displaystyle y\perp \!\!\!\perp {\textbf {x}}\mid \eta ^{T}{\textbf {x}},} or y {\displaystyle y} is conditionally independent of x {\displaystyle {\textbf {x}}} , given η T x {\displaystyle \eta ^{T}{\textbf {x}}} . Then the subspace S ( η ) {\displaystyle {\mathcal {S}}(\eta )} is defined to be a dimension reduction subspace (DRS). === Structural dimensionality === For a regression y ∣ x {\displaystyle y\mid {\textbf {x}}} , the structural dimension, d {\displaystyle d} , is the smallest number of distinct linear combinations of x {\displaystyle {\textbf {x}}} necessary to preserve the conditional distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, the smallest dimension reduction that is still sufficient maps x {\displaystyle {\textbf {x}}} to a subset of R d {\displaystyle \mathbb {R} ^{d}} . The corresponding DRS will be d-dimensional. === Minimum dimension reduction subspace === A subspace S {\displaystyle {\mathcal {S}}} is said to be a minimum DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} if it is a DRS and its dimension is less than or equal to that of all other DRSs for y ∣ x {\displaystyle y\mid {\textbf {x}}} . A minimum DRS S {\displaystyle {\mathcal {S}}} is not necessarily unique, but its dimension is equal to the structural dimension d {\displaystyle d} of y ∣ x {\displaystyle y\mid {\textbf {x}}} , by definition. If S {\displaystyle {\mathcal {S}}} has basis η {\displaystyle \eta } and is a minimum DRS, then a plot of y versus η T x {\displaystyle \eta ^{T}{\textbf {x}}} is a minimal sufficient summary plot, and it is (d + 1)-dimensional. == Central subspace == If a subspace S {\displaystyle {\mathcal {S}}} is a DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} , and if S ⊂ S drs {\displaystyle {\mathcal {S}}\subset {\mathcal {S}}_{\text{drs}}} for all other DRSs S drs {\displaystyle {\mathcal {S}}_{\text{drs}}} , then it is a central dimension reduction subspace, or simply a central subspace, and it is denoted by S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . In other words, a central subspace for y ∣ x {\displaystyle y\mid {\textbf {x}}} exists if and only if the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} of all dimension reduction subspaces is also a dimension reduction subspace, and that intersection is the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . The central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does not necessarily exist because the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} is not necessarily a DRS. However, if S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does exist, then it is also the unique minimum dimension reduction subspace. === Existence of the central subspace === While the existence of the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} is not guaranteed in every regression situation, there are some rather broad conditions under which its existence follows directly. For example, consider the following proposition from Cook (1998): Let S 1 {\displaystyle {\mathcal {S}}_{1}} and S 2 {\displaystyle {\mathcal {S}}_{2}} be dimension reduction subspaces for y ∣ x {\displaystyle y\mid {\textbf {x}}} . If x {\displaystyle {\textbf {x}}} has density f ( a ) > 0 {\displaystyle f(a)>0} for all a ∈ Ω x {\displaystyle a\in \Omega _{x}} and f ( a ) = 0 {\displaystyle f(a)=0} everywhere else, where Ω x {\displaystyle \Omega _{x}} is convex, then the intersection S 1 ∩ S 2 {\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}} is also a dimension reduction subspace. It follows from this proposition that the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} exists for such x {\displaystyle {\textbf {x}}} . == Methods for dimension reduction == There are many existing methods for dimension reduction, both graphical and numeric. For example, sliced inverse regression (SIR) and sliced average variance estimation (SAVE) were introduced in the 1990s and continue to be widely used. Although SIR was originally designed to estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different. More recent methods for dimension reduction include likelihood-based sufficient dimension reduction, estimating the central subspace based on the inverse third moment (or kth moment), estimating the central solution space, graphical regression, envelope model, and the principal support vector machine. For more details on these and other methods, consult the statistical literature. Principal components analysis (PCA) and similar methods for dimension reduction are not based on the sufficiency principle. === Example: linear regression === Consider the regression model y = α + β T x + ε , where ε ⊥ ⊥ x . {\displaystyle y=\alpha +\beta ^{T}{\textbf {x}}+\varepsilon ,{\text{ where }}\varepsilon \perp \!\!\!\perp {\textbf {x}}.} Note that the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} is the same as the distribution of y ∣ β T x {\displ

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  Genotypic and phenotypic repair

  Genotypic and phenotypic repair are optional components of an evolutionary algorithm (EA). An EA reproduces essential elements of biological evolution as a computer algorithm in order to solve demanding optimization or planning tasks, at least approximately. A candidate solution is represented by a - usually linear - data structure that plays the role of an individual's chromosome. New solution candidates are generated by mutation and crossover operators following the example of biology. These offspring may be defective, which is corrected or compensated for by genotypic or phenotypic repair. == Description == Genotypic repair, also known as genetic repair, is the removal or correction of impermissible entries in the chromosome that violate restrictions. In phenotypic repair, the corrections are only made in the genotype-phenotype mapping and the chromosome remains unchanged. Michalewicz wrote about the importance of restrictions in real-world applications: "In general, constraints are an integral part of the formulation of any problem". Restriction violations are application-specific and therefore it depends on the current problem whether and which type of repair is useful. They can usually also be treated by a correspondingly extended evaluation and it depends on the problem which measures are possible and which is the most suitable. If a phenotypic repair is feasible, then it is usually the most efficient compared to the other measures. A survey on repair methods used as constraint handling techniques can be found in. Violations of the range limits of genes should be avoided as far as possible by the formulation of the genome. If this is not possible or if restrictions within the search space defined by the genome are involved, their violations are usually handled by the evaluation. This can be done, for example, by penalty functions that lower the fitness. Repair is often also required for combinatorial tasks. The application of a 1- or n-point crossover operator can, for example, lead to genes being missing in one of the child genomes that are present in duplicate in the other. In this case, a suitable genotypic repair measure is to move the surplus genes to the other genome in a positional manner. The use of the aforementioned operators in combinatorial tasks has also proven to be useful in combination with crossover types specially developed for permutations, at least for certain problems. Particularly in combinatorial problems, it has been observed that genotypic repair can promote premature convergence to a suboptimum, but can also significantly accelerate a successful search. Studies on various tasks have shown that this is application-dependent. An effective measure to avoid premature convergence is generally the use of structured populations instead of the usual panmictic ones. Sequence restrictions play a role in many scheduling tasks, for example when it comes to planning workflows. If, for example, it is specified that step A must be carried out before step B and the gene of step B is located before the gene of A in the chromosome, then there is an impermissible gene sequence. This is because the scheduling operation of step B requires the planned end of step A for correct scheduling, but this is not yet scheduled at the time gene B is processed. The problem can be solved in two ways: The scheduling operation of step B is postponed until the gene from step A has been processed. The genome remains unchanged and the repair only influences the genotype-phenotype mapping. Since only the phenotype is changed, this is referred to as phenotypic repair. If, on the other hand, the gene of step B is moved behind the gene of step A, this is a genotypic repair. The same applies to the alternative shift of gene A in front of gene B. In this case, genotypic repair has the disadvantage that it prevents a meaningful restructuring of the gene sequence in the chromosome if this requires several intermediate steps (mutations) that at least partially violate restrictions.

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  Visual hull

  A visual hull is a geometric entity created by shape-from-silhouette 3D reconstruction technique introduced by A. Laurentini. This technique assumes the foreground object in an image can be separated from the background. Under this assumption, the original image can be thresholded into a foreground/background binary image, which we call a silhouette image. The foreground mask, known as a silhouette, is the 2D projection of the corresponding 3D foreground object. Along with the camera viewing parameters, the silhouette defines a back-projected generalized cone that contains the actual object; this cone is called a silhouette cone. The intersection of the two silhouette cones defines a visual hull. which is a bounding geometry of the actual 3D object. When the reconstructed geometry is only used for rendering from a different viewpoint, the implicit reconstruction together with rendering can be done using graphics hardware. == In two dimensions == A technique used in some modern touchscreen devices employs cameras placed in the corners situated opposite infrared LEDs. The one-dimensional projection (shadow) of objects on the surface may be used to reconstruct the convex hull of the object. Visual hull generation method has also been used within experimental tele-meeting systems that aim to allow a user in a remote location to interact with virtual objects. The method uses multiple cameras to capture the real-world movements and interactions of the "sender", employing hardware-accelerated volumetric visual hull representation to create 3D volume from 2D multi-view images. Its ultimate aim is to allow 3D collaboration between the two users in the virtual realm, with the visual hull technique reducing the computational power required to allow this type of interaction and enabling the use of consumer goods such as the Wii Remote as a tool for interaction.

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  Nearest centroid classifier

  In machine learning, a nearest centroid classifier or nearest prototype classifier is a classification model that assigns to observations the label of the class of training samples whose mean (centroid) is closest to the observation. When applied to text classification using word vectors containing tfidf weights to represent documents, the nearest centroid classifier is known as the Rocchio classifier because of its similarity to the Rocchio algorithm for relevance feedback. An extended version of the nearest centroid classifier has found applications in the medical domain, specifically classification of tumors. == Algorithm == === Training === Given labeled training samples { ( x → 1 , y 1 ) , … , ( x → n , y n ) } {\displaystyle \textstyle \{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}} with class labels y i ∈ Y {\displaystyle y_{i}\in \mathbf {Y} } , compute the per-class centroids μ → ℓ = 1 | C ℓ | ∑ i ∈ C ℓ x → i {\displaystyle \textstyle {\vec {\mu }}_{\ell }={\frac {1}{|C_{\ell }|}}{\underset {i\in C_{\ell }}{\sum }}{\vec {x}}_{i}} where C ℓ {\displaystyle C_{\ell }} is the set of indices of samples belonging to class ℓ ∈ Y {\displaystyle \ell \in \mathbf {Y} } . === Prediction === The class assigned to an observation x → {\displaystyle {\vec {x}}} is y ^ = arg ⁡ min ℓ ∈ Y ‖ μ → ℓ − x → ‖ {\displaystyle {\hat {y}}={\arg \min }_{\ell \in \mathbf {Y} }\|{\vec {\mu }}_{\ell }-{\vec {x}}\|} .

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  Markov model

  In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property. == Introduction == Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain. There are four common Markov models used in different situations, depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made: == Markov chain == The simplest Markov model is the Markov chain. It models the state of a system with a random variable that changes through time. In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution. == Hidden Markov model == A hidden Markov model is a Markov chain for which the state is only partially observable or noisily observable. In other words, observations are related to the state of the system, but they are typically insufficient to precisely determine the state. Several well-known algorithms for hidden Markov models exist. For example, given a sequence of observations, the Viterbi algorithm will compute the most-likely corresponding sequence of states, the forward algorithm will compute the probability of the sequence of observations, and the Baum–Welch algorithm will estimate the starting probabilities, the transition function, and the observation function of a hidden Markov model. One common use is for speech recognition, where the observed data is the speech audio waveform and the hidden state is the spoken text. In this example, the Viterbi algorithm finds the most likely sequence of spoken words given the speech audio. == Markov decision process == A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards. == Partially observable Markov decision process == A partially observable Markov decision process (POMDP) is a Markov decision process in which the state of the system is only partially observed. POMDPs are known to be NP complete, but recent approximation techniques have made them useful for a variety of applications, such as controlling simple agents or robots. == Markov random field == A Markov random field, or Markov network, may be considered to be a generalization of a Markov chain in multiple dimensions. In a Markov chain, state depends only on the previous state in time, whereas in a Markov random field, each state depends on its neighbors in any of multiple directions. A Markov random field may be visualized as a field or graph of random variables, where the distribution of each random variable depends on the neighboring variables with which it is connected. More specifically, the joint distribution for any random variable in the graph can be computed as the product of the "clique potentials" of all the cliques in the graph that contain that random variable. Modeling a problem as a Markov random field is useful because it implies that the joint distributions at each vertex in the graph may be computed in this manner. == Hierarchical Markov models == Hierarchical Markov models can be applied to categorize human behavior at various levels of abstraction. For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model and the Abstract Hidden Markov Model. Both have been used for behavior recognition and certain conditional independence properties between different levels of abstraction in the model allow for faster learning and inference. == Tolerant Markov model == A Tolerant Markov model (TMM) is a probabilistic-algorithmic Markov chain model. It assigns the probabilities according to a conditioning context that considers the last symbol, from the sequence to occur, as the most probable instead of the true occurring symbol. A TMM can model three different natures: substitutions, additions or deletions. Successful applications have been efficiently implemented in DNA sequences compression. == Markov-chain forecasting models == Markov-chains have been used as a forecasting methods for several topics, for example price trends, wind power and solar irradiance. The Markov-chain forecasting models utilize a variety of different settings, from discretizing the time-series to hidden Markov-models combined with wavelets and the Markov-chain mixture distribution model (MCM).

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  Tucker decomposition

  In mathematics, Tucker decomposition decomposes a tensor into a set of matrices and one small core tensor. It is named after Ledyard R. Tucker although it goes back to Hitchcock in 1927. Initially described as a three-mode extension of factor analysis and principal component analysis it may actually be generalized to higher mode analysis, which is also called higher-order singular value decomposition (HOSVD) or the M-mode SVD. The algorithm to which the literature typically refers when discussing the Tucker decomposition or the HOSVD is the M-mode SVD algorithm introduced by Vasilescu and Terzopoulos, but misattributed to Tucker or De Lathauwer etal. It may be regarded as a more flexible PARAFAC (parallel factor analysis) model. In PARAFAC the core tensor is restricted to be "diagonal". In practice, Tucker decomposition is used as a modelling tool. For instance, it is used to model three-way (or higher way) data by means of relatively small numbers of components for each of the three or more modes, and the components are linked to each other by a three- (or higher-) way core array. The model parameters are estimated in such a way that, given fixed numbers of components, the modelled data optimally resemble the actual data in the least squares sense. The model gives a summary of the information in the data, in the same way as principal components analysis does for two-way data. For a 3rd-order tensor T ∈ F n 1 × n 2 × n 3 {\displaystyle T\in F^{n_{1}\times n_{2}\times n_{3}}} , where F {\displaystyle F} is either R {\displaystyle \mathbb {R} } or C {\displaystyle \mathbb {C} } , Tucker Decomposition can be denoted as follows, T = T × 1 U ( 1 ) × 2 U ( 2 ) × 3 U ( 3 ) {\displaystyle T={\mathcal {T}}\times _{1}U^{(1)}\times _{2}U^{(2)}\times _{3}U^{(3)}} where T ∈ F d 1 × d 2 × d 3 {\displaystyle {\mathcal {T}}\in F^{d_{1}\times d_{2}\times d_{3}}} is the core tensor, a 3rd-order tensor that contains the 1-mode, 2-mode and 3-mode singular values of T {\displaystyle T} , which are defined as the Frobenius norm of the 1-mode, 2-mode and 3-mode slices of tensor T {\displaystyle {\mathcal {T}}} respectively. U ( 1 ) , U ( 2 ) , U ( 3 ) {\displaystyle U^{(1)},U^{(2)},U^{(3)}} are unitary matrices in F d 1 × n 1 , F d 2 × n 2 , F d 3 × n 3 {\displaystyle F^{d_{1}\times n_{1}},F^{d_{2}\times n_{2}},F^{d_{3}\times n_{3}}} respectively. The k-mode product (k = 1, 2, 3) of T {\displaystyle {\mathcal {T}}} by U ( k ) {\displaystyle U^{(k)}} is denoted as T × U ( k ) {\displaystyle {\mathcal {T}}\times U^{(k)}} with entries as ( T × 1 U ( 1 ) ) ( i 1 , j 2 , j 3 ) = ∑ j 1 = 1 d 1 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) ( T × 2 U ( 2 ) ) ( j 1 , i 2 , j 3 ) = ∑ j 2 = 1 d 2 T ( j 1 , j 2 , j 3 ) U ( 2 ) ( j 2 , i 2 ) ( T × 3 U ( 3 ) ) ( j 1 , j 2 , i 3 ) = ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle {\begin{aligned}({\mathcal {T}}\times _{1}U^{(1)})(i_{1},j_{2},j_{3})&=\sum _{j_{1}=1}^{d_{1}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})\\({\mathcal {T}}\times _{2}U^{(2)})(j_{1},i_{2},j_{3})&=\sum _{j_{2}=1}^{d_{2}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(2)}(j_{2},i_{2})\\({\mathcal {T}}\times _{3}U^{(3)})(j_{1},j_{2},i_{3})&=\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(3)}(j_{3},i_{3})\end{aligned}}} Altogether, the decomposition may also be written more directly as T ( i 1 , i 2 , i 3 ) = ∑ j 1 = 1 d 1 ∑ j 2 = 1 d 2 ∑ j 3 = 1 d 3 T ( j 1 , j 2 , j 3 ) U ( 1 ) ( j 1 , i 1 ) U ( 2 ) ( j 2 , i 2 ) U ( 3 ) ( j 3 , i 3 ) {\displaystyle T(i_{1},i_{2},i_{3})=\sum _{j_{1}=1}^{d_{1}}\sum _{j_{2}=1}^{d_{2}}\sum _{j_{3}=1}^{d_{3}}{\mathcal {T}}(j_{1},j_{2},j_{3})U^{(1)}(j_{1},i_{1})U^{(2)}(j_{2},i_{2})U^{(3)}(j_{3},i_{3})} Taking d i = n i {\displaystyle d_{i}=n_{i}} for all i {\displaystyle i} is always sufficient to represent T {\displaystyle T} exactly, but often T {\displaystyle T} can be compressed or efficiently approximately by choosing d i < n i {\displaystyle d_{i} Read more →

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  Facebook Messenger

  Messenger (formerly known as Facebook Messenger) is an American proprietary instant messaging service developed by Meta Platforms, the company that operates Facebook. Originally developed as Facebook Chat in 2008, the client application of Messenger is currently available on iOS and Android mobile platforms, Windows and macOS desktop platforms, through the Messenger.com web application, and on the standalone Meta Portal hardware. Messenger is used to send messages and exchange photos, videos, stickers, audio, and files, and also react to other users' messages and interact with bots. The service also supports voice and video calling. The standalone apps support using multiple accounts, conversations with end-to-end encryption, and playing games. There are also group chats where you can connect with multiple people at once in a private space such as Panama Chat. With a monthly userbase of over 1 billion people, it is among the largest social media platforms. == History == Following tests of a new instant messaging platform on Facebook in March 2008, the feature, then-titled "Facebook Chat", was gradually released to users in April 2008. Facebook revamped its messaging platform in November 2010, and subsequently acquired group messaging service Beluga in March 2011, which the company used to launch its standalone iOS and Android mobile apps on August 9, 2011. Facebook later launched a BlackBerry version in October 2011. An app for Windows Phone, though lacking features including voice messaging and chat heads, was released in March 2014. In April 2014, Facebook announced that the messaging feature would be removed from the main Facebook app and users will be required to download the separate Messenger app. An iPad-optimized version of the iOS app was released in July 2014. On April 8, 2015, Facebook launched a website interface for Messenger. A Tizen app was released on July 13, 2015. Facebook launched Messenger for Windows 10 in April 2016. In October 2016, Facebook released Messenger Lite, a stripped-down version of Messenger with a reduced feature set. The app is aimed primarily at old Android phones and regions where high-speed Internet is not widely available. In April 2017, Messenger Lite was expanded to 132 more countries. In May 2017, Facebook revamped the design for Messenger on Android and iOS, bringing a new home screen with tabs and categorization of content and interactive media, red dots indicating new activity, and relocated sections. Facebook announced a Messenger program for Windows 7 in a limited beta test in November 2011. The following month, Israeli blog TechIT leaked a download link for the program, with Facebook subsequently confirming and officially releasing the program. The program was eventually discontinued in March 2014. A Firefox web browser add-on was released in December 2012, but was also discontinued in March 2014. In December 2017, Facebook announced Messenger Kids, a new app aimed for persons under 13 years of age. The app comes with some differences compared to the standard version. In 2019, Messenger announced to be the 2nd most downloaded mobile app of the decade, from 2011 to 2019. In December 2019, Messenger dropped support for users to sign in using only a mobile number, meaning that users must sign in to a Facebook account in order to use the service. In March 2020, Facebook started to ship its dedicated Messenger for macOS app through the Mac App Store. The app is currently live in regions including France, Australia, Mexico, Poland, and many others. In April 2020, Facebook began rolling out a new feature called Messenger Rooms, a video chat feature that allows users to chat with up to 50 people at a time. The feature rivals Zoom, an application that gained a lot of popularity during the COVID-19 pandemic. Privacy concerns arose since the feature uses the same data collection policies as mainstream Facebook. In July 2020, Facebook added a new feature in Messenger that lets iOS users to use Apple's Face ID or Touch ID to lock their chats. The feature is called App Lock and is a part of several changes in Messenger regarding privacy and security. The option to view only "Unread Threads" was removed from the inbox, requiring the account holder to scroll through the entire inbox to be certain every unread message has been seen. On October 13, 2020, the Messenger application introduced cross-app messaging with Instagram, which was launched in September 2021. In addition to the integrated messaging, the application announced the introduction of a new logo, which should be an amalgamation of the Messenger and Instagram logo. The desktop app of Messenger was shut down on December 15, 2025. Messaging services were moved to the Facebook website or Messenger's site for those without an account on the former. The Messenger site was discontinued on April 16, 2026. Messaging services were moved to the Facebook website on the morning of April 17, 2026 without an Messenger account on the former to use Facebook account. == Features == The following is a table of features available in Messenger, as well as their geographical coverage and what devices they are available on. In addition there is a vanishing message feature. In addition there is an audio recording feature which allows audio recordings of up to one minute which may or may not be vanishing: === Messenger Rooms === It is a video conferencing feature of Messenger. It allows users to add up to 50 people at a time. Messenger Rooms does not require a Facebook account. Messenger Rooms competes with other services such as Zoom. Back in 2014, Facebook introduced an unrelated, stand-alone application named Rooms, letting users create places for users with similar interests, with users being anonymous to others. This was shut down in December 2015. In April 2020, during the COVID-19 pandemic, Facebook revealed video conferencing features for Messenger called Messenger Rooms. This was seen as a response to the popularity of other video conferencing platforms such as Zoom and Skype in the midst of the COVID-19 pandemic. Messenger Rooms allows users to add up to 50 people per room, without restrictions on time. It does not require a Facebook account or a separate app from Messenger. When used, it only prompts the user for basic information. Users can add 360° virtual backgrounds, mood lighting, and other AR effects as well as share screens. To prevent unwanted participants from joining, users can lock rooms and remove participants. Some have voiced concerns in regards to Messenger Room's privacy and how its parent, Facebook, handles data. Messenger Rooms, unlike some of its competitors, does not use end-to-end encryption. In addition, there have been concerns over how Messenger Rooms collects user data. == Monetization == In January 2017, Facebook announced that it was testing showing advertisements in Messenger's home feed. At the time, the testing was limited to a "small number of users in Australia and Thailand", with the ad format being swipe-based carousel ads. In July, the company announced that they were expanding the testing to a global audience. Stan Chudnovsky, head of Messenger, told VentureBeat that "We'll start slow ... When the average user can be sure to see them we truly don't know because we're just going to be very data-driven and user feedback-driven on making that decision". Facebook told TechCrunch that the advertisements' placement in the inbox depends on factors such as thread count, phone screen size, and pixel density. In a TechCrunch editorial by Devin Coldewey, he described the ads as "huge" in the space they occupy, "intolerable" in the way they appear in the user interface, and "irrelevant" due to the lack of context. Coldewey finished by writing "Advertising is how things get paid for on the internet, including TechCrunch, so I'm not an advocate of eliminating it or blocking it altogether. But bad advertising experiences can spoil a perfectly good app like (for the purposes of argument) Messenger. Messaging is a personal, purposeful use case and these ads are a bad way to monetize it." == Reception == In November 2014, the Electronic Frontier Foundation (EFF) listed Messenger (Facebook chat) on its Secure Messaging Scorecard. It received a score of 2 out of 7 points on the scorecard. It received points for having communications encrypted in transit and for having recently completed an independent security audit. It missed points because the communications were not encrypted with keys the provider didn't have access to, users could not verify contacts' identities, past messages were not secure if the encryption keys were stolen, the source code was not open to independent review, and the security design was not properly documented. As stated by Facebook in its Help Center, there is no way to log out of the Messenger application. Instead, users can choose between different availability statuses, including "Appear as inactive", "S

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  Distribution learning theory

  The distributional learning theory or learning of probability distribution is a framework in computational learning theory. It has been proposed from Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert Schapire and Linda Sellie in 1994 and it was inspired from the PAC-framework introduced by Leslie Valiant. In this framework the input is a number of samples drawn from a distribution that belongs to a specific class of distributions. The goal is to find an efficient algorithm that, based on these samples, determines with high probability the distribution from which the samples have been drawn. Because of its generality, this framework has been used in a large variety of different fields like machine learning, approximation algorithms, applied probability and statistics. This article explains the basic definitions, tools and results in this framework from the theory of computation point of view. == Definitions == Let X {\displaystyle \textstyle X} be the support of the distributions of interest. As in the original work of Kearns et al. if X {\displaystyle \textstyle X} is finite it can be assumed without loss of generality that X = { 0 , 1 } n {\displaystyle \textstyle X=\{0,1\}^{n}} where n {\displaystyle \textstyle n} is the number of bits that have to be used in order to represent any y ∈ X {\displaystyle \textstyle y\in X} . We focus in probability distributions over X {\displaystyle \textstyle X} . There are two possible representations of a probability distribution D {\displaystyle \textstyle D} over X {\displaystyle \textstyle X} . probability distribution function (or evaluator) an evaluator E D {\displaystyle \textstyle E_{D}} for D {\displaystyle \textstyle D} takes as input any y ∈ X {\displaystyle \textstyle y\in X} and outputs a real number E D [ y ] {\displaystyle \textstyle E_{D}[y]} which denotes the probability that of y {\displaystyle \textstyle y} according to D {\displaystyle \textstyle D} , i.e. E D [ y ] = Pr [ Y = y ] {\displaystyle \textstyle E_{D}[y]=\Pr[Y=y]} if Y ∼ D {\displaystyle \textstyle Y\sim D} . generator a generator G D {\displaystyle \textstyle G_{D}} for D {\displaystyle \textstyle D} takes as input a string of truly random bits y {\displaystyle \textstyle y} and outputs G D [ y ] ∈ X {\displaystyle \textstyle G_{D}[y]\in X} according to the distribution D {\displaystyle \textstyle D} . Generator can be interpreted as a routine that simulates sampling from the distribution D {\displaystyle \textstyle D} given a sequence of fair coin tosses. A distribution D {\displaystyle \textstyle D} is called to have a polynomial generator (respectively evaluator) if its generator (respectively evaluator) exists and can be computed in polynomial time. Let C X {\displaystyle \textstyle C_{X}} a class of distribution over X, that is C X {\displaystyle \textstyle C_{X}} is a set such that every D ∈ C X {\displaystyle \textstyle D\in C_{X}} is a probability distribution with support X {\displaystyle \textstyle X} . The C X {\displaystyle \textstyle C_{X}} can also be written as C {\displaystyle \textstyle C} for simplicity. In order to evaluate learnability, it is necessary to have a way to measure how well an approximated distribution D ′ {\displaystyle \textstyle D'} fits the sampled distribution D {\displaystyle \textstyle D} . There are several ways to measure the divergence between two distributions. Three common possibilities are Kullback–Leibler divergence Total variation distance of probability measures Kolmogorov distance Total variation and Kolmogorov distance are true metrics, while KL divergence is not (it lacks symmetry). These measures are ordered by convergence strength: closeness in KL divergence implies closeness in total variation (via Pinsker's inequality), which in turn implies closeness in Kolmogorov distance. Therefore, a learnability result proven under KL divergence automatically holds under the weaker measures, but not vice versa. Since certain measures may be more appropriate in specific applications, we will use d ( D , D ′ ) {\displaystyle \textstyle d(D,D')} to denote a selected divergence between the distribution D {\displaystyle \textstyle D} and the distribution D ′ {\displaystyle \textstyle D'} . The basic input that we use in order to learn a distribution is a number of samples drawn by this distribution. For the computational point of view the assumption is that such a sample is given in a constant amount of time. So it's like having access to an oracle G E N ( D ) {\displaystyle \textstyle GEN(D)} that returns a sample from the distribution D {\displaystyle \textstyle D} . Sometimes the interest is, apart from measuring the time complexity, to measure the number of samples that have to be used in order to learn a specific distribution D {\displaystyle \textstyle D} in class of distributions C {\displaystyle \textstyle C} . This quantity is called sample complexity of the learning algorithm. In order for the problem of distribution learning to be more clear consider the problem of supervised learning as defined in. In this framework of statistical learning theory a training set S = { ( x 1 , y 1 ) , … , ( x n , y n ) } {\displaystyle \textstyle S=\{(x_{1},y_{1}),\dots ,(x_{n},y_{n})\}} and the goal is to find a target function f : X → Y {\displaystyle \textstyle f:X\rightarrow Y} that minimizes some loss function, e.g. the square loss function. More formally f = arg ⁡ min g ∫ V ( y , g ( x ) ) d ρ ( x , y ) {\displaystyle f=\arg \min _{g}\int V(y,g(x))d\rho (x,y)} , where V ( ⋅ , ⋅ ) {\displaystyle V(\cdot ,\cdot )} is the loss function, e.g. V ( y , z ) = ( y − z ) 2 {\displaystyle V(y,z)=(y-z)^{2}} and ρ ( x , y ) {\displaystyle \rho (x,y)} the probability distribution according to which the elements of the training set are sampled. If the conditional probability distribution ρ x ( y ) {\displaystyle \rho _{x}(y)} is known then the target function has the closed form f ( x ) = ∫ y y d ρ x ( y ) {\displaystyle f(x)=\int _{y}yd\rho _{x}(y)} . So the set S {\displaystyle S} is a set of samples from the probability distribution ρ ( x , y ) {\displaystyle \rho (x,y)} . Now the goal of distributional learning theory if to find ρ {\displaystyle \rho } given S {\displaystyle S} which can be used to find the target function f {\displaystyle f} . Definition of learnability A class of distributions C {\displaystyle \textstyle C} is called efficiently learnable if for every ϵ > 0 {\displaystyle \textstyle \epsilon >0} and 0 < δ ≤ 1 {\displaystyle \textstyle 0<\delta \leq 1} given access to G E N ( D ) {\displaystyle \textstyle GEN(D)} for an unknown distribution D ∈ C {\displaystyle \textstyle D\in C} , there exists a polynomial time algorithm A {\displaystyle \textstyle A} , called learning algorithm of C {\displaystyle \textstyle C} , that outputs a generator or an evaluator of a distribution D ′ {\displaystyle \textstyle D'} such that Pr [ d ( D , D ′ ) ≤ ϵ ] ≥ 1 − δ {\displaystyle \Pr[d(D,D')\leq \epsilon ]\geq 1-\delta } If we know that D ′ ∈ C {\displaystyle \textstyle D'\in C} then A {\displaystyle \textstyle A} is called proper learning algorithm, otherwise is called improper learning algorithm. In some settings the class of distributions C {\displaystyle \textstyle C} is a class with well known distributions which can be described by a set of parameters. For instance C {\displaystyle \textstyle C} could be the class of all the Gaussian distributions N ( μ , σ 2 ) {\displaystyle \textstyle N(\mu ,\sigma ^{2})} . In this case the algorithm A {\displaystyle \textstyle A} should be able to estimate the parameters μ , σ {\displaystyle \textstyle \mu ,\sigma } . In this case A {\displaystyle \textstyle A} is called parameter learning algorithm. Obviously the parameter learning for simple distributions is a very well studied field that is called statistical estimation and there is a very long bibliography on different estimators for different kinds of simple known distributions. But distributions learning theory deals with learning class of distributions that have more complicated description. == First results == In their seminal work, Kearns et al. deal with the case where A {\displaystyle \textstyle A} is described in term of a finite polynomial sized circuit and they proved the following for some specific classes of distribution. O R {\displaystyle \textstyle OR} gate distributions for this kind of distributions there is no polynomial-sized evaluator, unless # P ⊆ P / poly {\displaystyle \textstyle \#P\subseteq P/{\text{poly}}} . On the other hand, this class is efficiently learnable with generator. Parity gate distributions this class is efficiently learnable with both generator and evaluator. Mixtures of Hamming Balls this class is efficiently learnable with both generator and evaluator. Probabilistic Finite Automata this class is not efficiently learnable with evaluator under the Noisy Parity Assumption which is an impossibility assumption in the PAC learning fram

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  Vapnik–Chervonenkis dimension

  In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the function class can shatter—that is, for which all possible binary labelings can be realized by some function in the class. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high-degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. Such a polynomial has a high capacity. A much simpler alternative is to threshold a linear function. This function may not fit the training set well, because it has a low capacity. This notion of capacity is made rigorous below. == Definitions == === VC dimension of a set-family === Let C = { C } C ∈ C {\displaystyle {\mathcal {C}}=\{C\}_{C\in {\mathcal {C}}}} be a family of sets (also called set family, collection of sets or set of sets) and X {\displaystyle X} a set. Their intersection is defined as the following set family: C ∩ X := { C ∩ X ∣ C ∈ C } . {\displaystyle {\mathcal {C}}\cap X:=\{C\cap X\mid C\in {\mathcal {C}}\}.} Here typically X {\displaystyle X} and each C ∈ C {\displaystyle C\in {\mathcal {C}}} are subsets of a big "universe" of possibilities U {\displaystyle U} where intersection takes place. We say that a set X {\displaystyle X} is shattered by C {\displaystyle {\mathcal {C}}} if P ( X ) = C ∩ X {\displaystyle {\mathcal {P}}(X)={\mathcal {C}}\cap X} i.e. the set of intersections contains (hence is equal to) all the subsets of X {\displaystyle X} . For finite sets X {\displaystyle X} this is equivalent to | C ∩ X | = 2 | X | . {\displaystyle |{\mathcal {C}}\cap X|=2^{|X|}.} The VC dimension D {\displaystyle D} of C {\displaystyle {\mathcal {C}}} is the cardinality of the largest set that is shattered by C {\displaystyle {\mathcal {C}}} . If arbitrarily large sets can be shattered, the VC dimension of C {\displaystyle {\mathcal {C}}} is ∞ {\displaystyle \infty } . === VC dimension of a classification model === A binary classification model f {\displaystyle f} with some parameter vector θ {\displaystyle \theta } is said to shatter a set of generally positioned data points ( x 1 , x 2 , … , x n ) {\displaystyle (x_{1},x_{2},\ldots ,x_{n})} if, for every assignment of labels to those points, there exists a θ {\displaystyle \theta } such that the model f {\displaystyle f} makes no errors when evaluating that set of data points. The VC dimension of a model f {\displaystyle f} is the maximum number of points that can be arranged so that f {\displaystyle f} shatters them. More formally, it is the maximum cardinal D {\displaystyle D} such that there exists a generally positioned data point set of cardinality D {\displaystyle D} that can be shattered by f {\displaystyle f} . == Examples == f {\displaystyle f} is a constant classifier (with no parameters); Its VC dimension is 0 since it cannot shatter even a single point. In general, the VC dimension of a finite classification model, which can return at most 2 d {\displaystyle 2^{d}} different classifiers, is at most d {\displaystyle d} (this is an upper bound on the VC dimension; the Sauer–Shelah lemma gives a lower bound on the dimension). f {\displaystyle f} is a single-parametric threshold classifier on real numbers; i.e., for a certain threshold θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number is larger than θ {\displaystyle \theta } and 0 otherwise. The VC dimension of f {\displaystyle f} is 1 because: (a) It can shatter a single point. For every point x {\displaystyle x} , a classifier f θ {\displaystyle f_{\theta }} labels it as 0 if θ > x {\displaystyle \theta >x} and labels it as 1 if θ < x {\displaystyle \theta x + 2 {\displaystyle \theta >x+2} , as (1,0) if θ ∈ [ x − 4 , x − 2 ) {\displaystyle \theta \in [x-4,x-2)} , as (1,1) if θ ∈ [ x − 2 , x ] {\displaystyle \theta \in [x-2,x]} , and as (0,1) if θ ∈ ( x , x + 2 ] {\displaystyle \theta \in (x,x+2]} . (b) It cannot shatter any set of three points. For every set of three numbers, if the smallest and the largest are labeled 1, then the middle one must also be labeled 1, so not all labelings are possible. f {\displaystyle f} is a straight line as a classification model on points in a two-dimensional plane (this is the model used by a perceptron). The line should separate positive data points from negative data points. There exist sets of 3 points that can indeed be shattered using this model (any 3 points that are not collinear can be shattered). However, no set of 4 points can be shattered: by Radon's theorem, any four points can be partitioned into two subsets with intersecting convex hulls, so it is not possible to separate one of these two subsets from the other. Thus, the VC dimension of this particular classifier is 3. It is important to remember that while one can choose any arrangement of points, the arrangement of those points cannot change when attempting to shatter for some label assignment. Note, only 3 of the 23 = 8 possible label assignments are shown for the three points. f {\displaystyle f} is a single-parametric sine classifier, i.e., for a certain parameter θ {\displaystyle \theta } , the classifier f θ {\displaystyle f_{\theta }} returns 1 if the input number x {\displaystyle x} has sin ⁡ ( θ x ) > 0 {\displaystyle \sin(\theta x)>0} and 0 otherwise. The VC dimension of f {\displaystyle f} is infinite, since it can shatter any finite subset of the set { 2 − m ∣ m ∈ N } {\displaystyle \{2^{-m}\mid m\in \mathbb {N} \}} . == Uses == === In statistical learning theory === The VC dimension can predict a probabilistic upper bound on the test error of a classification model. Vapnik proved that the probability of the test error (i.e., risk with 0–1 loss function) distancing from an upper bound (on data that is drawn i.i.d. from the same distribution as the training set) is given by: Pr ( test error ⩽ training error + 1 N [ D ( log ⁡ ( 2 N D ) + 1 ) − log ⁡ ( η 4 ) ] ) = 1 − η , {\displaystyle \Pr \left({\text{test error}}\leqslant {\text{training error}}+{\sqrt {{\frac {1}{N}}\left[D\left(\log \left({\tfrac {2N}{D}}\right)+1\right)-\log \left({\tfrac {\eta }{4}}\right)\right]}}\,\right)=1-\eta ,} where D {\displaystyle D} is the VC dimension of the classification model, 0 < η ⩽ 1 {\displaystyle 0<\eta \leqslant 1} , and N {\displaystyle N} is the size of the training set (restriction: this formula is valid when D ≪ N {\displaystyle D\ll N} . When D {\displaystyle D} is larger, the test-error may be much higher than the training-error. This is due to overfitting). The VC dimension also appears in sample-complexity bounds. A space of binary functions with VC dimension D {\displaystyle D} can be learned with: N = Θ ( D + ln ⁡ 1 δ ε 2 ) {\displaystyle N=\Theta \left({\frac {D+\ln {1 \over \delta }}{\varepsilon ^{2}}}\right)} samples, where ε {\displaystyle \varepsilon } is the learning error and δ {\displaystyle \delta } is the failure probability. Thus, the sample-complexity is a linear function of the VC dimension of the hypothesis space. === In computational geometry === The VC dimension is one of the critical parameters in the size of ε-nets, which determines the complexity of approximation algorithms based on them; range sets without finite VC dimension may not have finite ε-nets at all. == Bounds == The VC dimension of the dual set-family of C {\displaystyle {\mathcal {C}}} is strictly less than 2 vc ⁡ ( C ) + 1 {\displaystyle 2^{\operatorname {vc} ({\mathcal {C}})+1}} , and this is best possible. The VC dimension of a finite set-family C {\displaystyle {\mathcal {C}}} is at most log 2 ⁡ | C | {\displaystyle \log _{2}|{\mathcal {C}}|} . This is because | C ∩ X | ≤ | X | {\displaystyle |{\mathcal {C}}\cap X|\leq |X|} by definition. Given a set-fa

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  Softplus

  In mathematics and machine learning, the softplus function is f ( x ) = ln ⁡ ( 1 + e x ) . {\displaystyle f(x)=\ln(1+e^{x}).} It is a smooth approximation (in fact, an analytic function) to the ramp function, which is known as the rectifier or ReLU (rectified linear unit) in machine learning. For large negative x {\displaystyle x} it is ln ⁡ ( 1 + e x ) = ln ⁡ ( 1 + ϵ ) ⪆ ln ⁡ 1 = 0 {\displaystyle \ln(1+e^{x})=\ln(1+\epsilon )\gtrapprox \ln 1=0} , so just above 0, while for large positive x {\displaystyle x} it is ln ⁡ ( 1 + e x ) ⪆ ln ⁡ ( e x ) = x {\displaystyle \ln(1+e^{x})\gtrapprox \ln(e^{x})=x} , so just above x {\displaystyle x} . The names softplus and SmoothReLU are used in machine learning. The name "softplus" (2000), by analogy with the earlier softmax (1989) is presumably because it is a smooth (soft) approximation of the positive part of x, which is sometimes denoted with a superscript plus, x + := max ( 0 , x ) {\displaystyle x^{+}:=\max(0,x)} . == Alternative forms == This function can be approximated as: ln ⁡ ( 1 + e x ) ≈ { ln ⁡ 2 , x = 0 , x 1 − e − x / ln ⁡ 2 , x ≠ 0 {\displaystyle \ln \left(1+e^{x}\right)\approx {\begin{cases}\ln 2,&x=0,\\[6pt]{\frac {x}{1-e^{-x/\ln 2}}},&x\neq 0\end{cases}}} By making the change of variables x = y ln ⁡ ( 2 ) {\displaystyle x=y\ln(2)} , this is equivalent to log 2 ⁡ ( 1 + 2 y ) ≈ { 1 , y = 0 , y 1 − e − y , y ≠ 0. {\displaystyle \log _{2}(1+2^{y})\approx {\begin{cases}1,&y=0,\\[6pt]{\frac {y}{1-e^{-y}}},&y\neq 0.\end{cases}}} A sharpness parameter k {\displaystyle k} may be included: f ( x ) = ln ⁡ ( 1 + e k x ) k , f ′ ( x ) = e k x 1 + e k x = 1 1 + e − k x . {\displaystyle f(x)={\frac {\ln(1+e^{kx})}{k}},\qquad \qquad f'(x)={\frac {e^{kx}}{1+e^{kx}}}={\frac {1}{1+e^{-kx}}}.} Additionally, the softplus function is equivalent to the log of the sigmoid function in the following way: − ln ⁡ ( sigmoid ( − x ) ) = − ln ⁡ ( 1 1 + e x ) = ln ⁡ ( 1 + e x ) = softplus ( x ) {\displaystyle -\ln({\text{sigmoid}}(-x))=-\ln \left({\frac {1}{1+e^{x}}}\right)=\ln \left(1+e^{x}\right)={\text{softplus}}(x)} == Related functions == The derivative of softplus is the standard logistic function: f ′ ( x ) = e x 1 + e x = 1 1 + e − x {\displaystyle f'(x)={\frac {e^{x}}{1+e^{x}}}={\frac {1}{1+e^{-x}}}} The logistic function or the sigmoid function is a smooth approximation of the rectifier, the Heaviside step function. === LogSumExp === The multivariable generalization of single-variable softplus is the LogSumExp with the first argument set to zero: L S E 0 + ⁡ ( x 1 , … , x n ) := LSE ⁡ ( 0 , x 1 , … , x n ) = ln ⁡ ( 1 + e x 1 + ⋯ + e x n ) . {\displaystyle \operatorname {LSE_{0}} ^{+}(x_{1},\dots ,x_{n}):=\operatorname {LSE} (0,x_{1},\dots ,x_{n})=\ln(1+e^{x_{1}}+\cdots +e^{x_{n}}).} The LogSumExp function is LSE ⁡ ( x 1 , … , x n ) = ln ⁡ ( e x 1 + ⋯ + e x n ) , {\displaystyle \operatorname {LSE} (x_{1},\dots ,x_{n})=\ln(e^{x_{1}}+\cdots +e^{x_{n}}),} and its gradient is the softmax; the softmax with the first argument set to zero is the multivariable generalization of the logistic function. Both LogSumExp and softmax are used in machine learning. === Convex conjugate === The convex conjugate (specifically, the Legendre transformation) of the softplus function is the negative binary entropy function (with base e). This is because (following the definition of the Legendre transformation: the derivatives are inverse functions) the derivative of softplus is the logistic function, whose inverse function is the logit, which is the derivative of negative binary entropy. Softplus can be interpreted as logistic loss (as a positive number), so, by duality, minimizing logistic loss corresponds to maximizing entropy. This justifies the principle of maximum entropy as loss minimization.

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  Evolvability (computer science)

  The term evolvability is a framework of computational learning introduced by Leslie Valiant in his paper of the same name. The aim of this theory is to model biological evolution and categorize which types of mechanisms are evolvable. Evolution is an extension of PAC learning and learning from statistical queries. == General framework == Let F n {\displaystyle F_{n}\,} and R n {\displaystyle R_{n}\,} be collections of functions on n {\displaystyle n\,} variables. Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , the goal is to find by local search a representation r ∈ R n {\displaystyle r\in R_{n}} that closely approximates f {\displaystyle f\,} . This closeness is measured by the performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} of r {\displaystyle r\,} with respect to f {\displaystyle f\,} . As is the case in the biological world, there is a difference between genotype and phenotype. In general, there can be multiple representations (genotypes) that correspond to the same function (phenotype). That is, for some r , r ′ ∈ R n {\displaystyle r,r'\in R_{n}} , with r ≠ r ′ {\displaystyle r\neq r'\,} , still r ( x ) = r ′ ( x ) {\displaystyle r(x)=r'(x)\,} for all x ∈ X n {\displaystyle x\in X_{n}} . However, this need not be the case. The goal then, is to find a representation that closely matches the phenotype of the ideal function, and the spirit of the local search is to allow only small changes in the genotype. Let the neighborhood N ( r ) {\displaystyle N(r)\,} of a representation r {\displaystyle r\,} be the set of possible mutations of r {\displaystyle r\,} . For simplicity, consider Boolean functions on X n = { − 1 , 1 } n {\displaystyle X_{n}=\{-1,1\}^{n}\,} , and let D n {\displaystyle D_{n}\,} be a probability distribution on X n {\displaystyle X_{n}\,} . Define the performance in terms of this. Specifically, Perf ⁡ ( f , r ) = ∑ x ∈ X n f ( x ) r ( x ) D n ( x ) . {\displaystyle \operatorname {Perf} (f,r)=\sum _{x\in X_{n}}f(x)r(x)D_{n}(x).} Note that Perf ⁡ ( f , r ) = Prob ⁡ ( f ( x ) = r ( x ) ) − Prob ⁡ ( f ( x ) ≠ r ( x ) ) . {\displaystyle \operatorname {Perf} (f,r)=\operatorname {Prob} (f(x)=r(x))-\operatorname {Prob} (f(x)\neq r(x)).} In general, for non-Boolean functions, the performance will not correspond directly to the probability that the functions agree, although it will have some relationship. Throughout an organism's life, it will only experience a limited number of environments, so its performance cannot be determined exactly. The empirical performance is defined by Perf s ⁡ ( f , r ) = 1 s ∑ x ∈ S f ( x ) r ( x ) , {\displaystyle \operatorname {Perf} _{s}(f,r)={\frac {1}{s}}\sum _{x\in S}f(x)r(x),} where S {\displaystyle S\,} is a multiset of s {\displaystyle s\,} independent selections from X n {\displaystyle X_{n}\,} according to D n {\displaystyle D_{n}\,} . If s {\displaystyle s\,} is large enough, evidently Perf s ⁡ ( f , r ) {\displaystyle \operatorname {Perf} _{s}(f,r)} will be close to the actual performance Perf ⁡ ( f , r ) {\displaystyle \operatorname {Perf} (f,r)} . Given an ideal function f ∈ F n {\displaystyle f\in F_{n}} , initial representation r ∈ R n {\displaystyle r\in R_{n}} , sample size s {\displaystyle s\,} , and tolerance t {\displaystyle t\,} , the mutator Mut ⁡ ( f , r , s , t ) {\displaystyle \operatorname {Mut} (f,r,s,t)} is a random variable defined as follows. Each r ′ ∈ N ( r ) {\displaystyle r'\in N(r)} is classified as beneficial, neutral, or deleterious, depending on its empirical performance. Specifically, r ′ {\displaystyle r'\,} is a beneficial mutation if Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) ≥ t {\displaystyle \operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r)\geq t} ; r ′ {\displaystyle r'\,} is a neutral mutation if − t < Perf s ⁡ ( f , r ′ ) − Perf s ⁡ ( f , r ) < t {\displaystyle -t<\operatorname {Perf} _{s}(f,r')-\operatorname {Perf} _{s}(f,r) 0 {\displaystyle \epsilon >0\,} , for all ideal functions f ∈ F n {\displaystyle f\in F_{n}} and representations r 0 ∈ R n {\displaystyle r_{0}\in R_{n}} , with probability at least 1 − ϵ {\displaystyle 1-\epsilon \,} , Perf ⁡ ( f , r g ( n , 1 / ϵ ) ) ≥ 1 − ϵ , {\displaystyle \operatorname {Perf} (f,r_{g(n,1/\epsilon )})\geq 1-\epsilon ,} where the sizes of neighborhoods N ( r ) {\displaystyle N(r)\,} for r ∈ R n {\displaystyle r\in R_{n}\,} are at most p ( n , 1 / ϵ ) {\displaystyle p(n,1/\epsilon )\,} , the sample size is s ( n , 1 / ϵ ) {\displaystyle s(n,1/\epsilon )\,} , the tolerance is t ( 1 / n , ϵ ) {\displaystyle t(1/n,\epsilon )\,} , and the generation size is g ( n , 1 / ϵ ) {\displaystyle g(n,1/\epsilon )\,} . F {\displaystyle F\,} is evolvable over D {\displaystyle D\,} if it is evolvable by some R {\displaystyle R\,} over D {\displaystyle D\,} . F {\displaystyle F\,} is evolvable if it is evolvable over all distributions D {\displaystyle D\,} . == Results == The class of conjunctions and the class of disjunctions are evolvable over the uniform distribution for short conjunctions and disjunctions, respectively. The class of parity functions (which evaluate to the parity of the number of true literals in a given subset of literals) are not evolvable, even for the uniform distribution. Evolvability implies PAC learnability.

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  Bioz

  Bioz is a search engine for life science experimentation. == History == Bioz was founded by Karin Lachmi and Daniel Levitt. Lachmi is a scientist who completed her postdoc in molecular and cellular biology at the Stanford University School of Medicine. During her lab work she found little available data regarding preferable lab tools, reagents and related products for experimentation. There are 50,000 vendors selling 300 million scientific products. She decided to start the company in order to provide researchers with adequate information for that purpose. Co-founder Daniel Levitt is an entrepreneur who sold his company WebAppoint to Microsoft in the year 2000. He also co-founded the company StemRad. At Bioz, Lachmi serves as the Chief Scientific Officer and Levitt serves as the chief executive officer. Bioz claims to have over a million researcher-users from 196 countries. Among the investors are Esther Dyson and the Stanford-StartX Fund. The company's advisory board includes Nobel Laureates in Chemistry Michael Levitt, Roger Kornberg, and Ada Yonath. == Technology == The company uses artificial intelligence, machine learning and natural language processing in order to extract experimentation data from scientific articles, such as the products that researchers used, the companies that supply the products, the protocol conditions that researchers selected, and the types of experiments and techniques. The algorithm ranks products based on how frequently they were used by researchers in their experiments, how recently a product was used, and the impact factor of the journal. The algorithm's output is a Bioz stars score for each product that was mentioned in an article. Bioz is a data-driven platform for product recommendations, which is contrary to platforms such as TripAdvisor and OpenTable that are based on user-generated reviews and ratings. The recommendations and scoring system that the company has developed are meant to assist researchers with the process of developing future medications and finding cures for diseases. They are guided towards products and techniques that were previously used by other researchers when planning and performing experiments. The company's revenue is based on selling SaaS subscriptions to researchers in biopharma companies. They also charge product suppliers for content syndication.

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  Cross-entropy

  In information theory, the cross-entropy between two probability distributions p {\displaystyle p} and q {\displaystyle q} , over the same underlying set of events, measures the average number of bits needed to identify an event drawn from the set when the coding scheme used for the set is optimized for an estimated probability distribution q {\displaystyle q} , rather than the true distribution p {\displaystyle p} . == Definition == The cross-entropy of the distribution q {\displaystyle q} relative to a distribution p {\displaystyle p} over a given set is defined as follows: H ( p , q ) = − E p ⁡ [ log ⁡ q ] , {\displaystyle H(p,q)=-\operatorname {E} _{p}[\log q],} where E p ⁡ [ ⋅ ] {\displaystyle \operatorname {E} _{p}[\cdot ]} is the expected value operator with respect to the distribution p {\displaystyle p} . The definition may be formulated using the Kullback–Leibler divergence D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , divergence of p {\displaystyle p} from q {\displaystyle q} (also known as the relative entropy of p {\displaystyle p} with respect to q {\displaystyle q} ). H ( p , q ) = H ( p ) + D K L ( p ∥ q ) , {\displaystyle H(p,q)=H(p)+D_{\mathrm {KL} }(p\parallel q),} where H ( p ) {\displaystyle H(p)} is the entropy of p {\displaystyle p} . For discrete probability distributions p {\displaystyle p} and q {\displaystyle q} with the same support X {\displaystyle {\mathcal {X}}} , this means The situation for continuous distributions is analogous. We have to assume that p {\displaystyle p} and q {\displaystyle q} are absolutely continuous with respect to some reference measure r {\displaystyle r} (usually r {\displaystyle r} is a Lebesgue measure on a Borel σ-algebra). Let P {\displaystyle P} and Q {\displaystyle Q} be probability density functions of p {\displaystyle p} and q {\displaystyle q} with respect to r {\displaystyle r} . Then − ∫ X P ( x ) log ⁡ Q ( x ) d x = E p ⁡ [ − log ⁡ Q ] , {\displaystyle -\int _{\mathcal {X}}P(x)\,\log Q(x)\,\mathrm {d} x=\operatorname {E} _{p}[-\log Q],} and therefore NB: The notation H ( p , q ) {\displaystyle H(p,q)} is also used for a different concept, the joint entropy of p {\displaystyle p} and q {\displaystyle q} . == Motivation == In information theory, the Kraft–McMillan theorem establishes that any directly decodable coding scheme for coding a message to identify one value x i {\displaystyle x_{i}} out of a set of possibilities { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} can be seen as representing an implicit probability distribution q ( x i ) = ( 1 2 ) ℓ i {\displaystyle q(x_{i})=\left({\frac {1}{2}}\right)^{\ell _{i}}} over { x 1 , … , x n } {\displaystyle \{x_{1},\ldots ,x_{n}\}} , where ℓ i {\displaystyle \ell _{i}} is the length of the code for x i {\displaystyle x_{i}} in bits. Therefore, cross-entropy can be interpreted as the expected message-length per datum when a wrong distribution q {\displaystyle q} is assumed while the data actually follows a distribution p {\displaystyle p} . That is why the expectation is taken over the true probability distribution p {\displaystyle p} and not q . {\displaystyle q.} Indeed the expected message-length under the true distribution p {\displaystyle p} is E p ⁡ [ ℓ ] = − E p ⁡ [ ln ⁡ q ( x ) ln ⁡ ( 2 ) ] = − E p ⁡ [ log 2 ⁡ q ( x ) ] = − ∑ x i p ( x i ) log 2 ⁡ q ( x i ) = − ∑ x p ( x ) log 2 ⁡ q ( x ) = H ( p , q ) . {\displaystyle {\begin{aligned}\operatorname {E} _{p}[\ell ]&=-\operatorname {E} _{p}\left[{\frac {\ln {q(x)}}{\ln(2)}}\right]\\[1ex]&=-\operatorname {E} _{p}\left[\log _{2}{q(x)}\right]\\[1ex]&=-\sum _{x_{i}}p(x_{i})\,\log _{2}q(x_{i})\\[1ex]&=-\sum _{x}p(x)\,\log _{2}q(x)=H(p,q).\end{aligned}}} == Estimation == There are many situations where cross-entropy needs to be measured but the distribution of p {\displaystyle p} is unknown. An example is language modeling, where a model is created based on a training set T {\displaystyle T} , and then its cross-entropy is measured on a test set to assess how accurate the model is in predicting the test data. In this example, p {\displaystyle p} is the true distribution of words in any corpus, and q {\displaystyle q} is the distribution of words as predicted by the model. Since the true distribution is unknown, cross-entropy cannot be directly calculated. In these cases, an estimate of cross-entropy is calculated using the following formula: H ( T , q ) = − ∑ i = 1 N 1 N log 2 ⁡ q ( x i ) {\displaystyle H(T,q)=-\sum _{i=1}^{N}{\frac {1}{N}}\log _{2}q(x_{i})} where N {\displaystyle N} is the size of the test set, and q ( x ) {\displaystyle q(x)} is the probability of event x {\displaystyle x} estimated from the training set. In other words, q ( x i ) {\displaystyle q(x_{i})} is the probability estimate of the model that the i-th word of the text is x i {\displaystyle x_{i}} . The sum is averaged over the N {\displaystyle N} words of the test. This is a Monte Carlo estimate of the true cross-entropy, where the test set is treated as samples from p ( x ) {\displaystyle p(x)} . == Relation to maximum likelihood == The cross entropy arises in classification problems when introducing a logarithm in the guise of the log-likelihood function. This section concerns the estimation of the probabilities of different discrete outcomes. To this end, denote a parametrized family of distributions by q θ {\displaystyle q_{\theta }} , with θ {\displaystyle \theta } subject to the optimization effort. Consider a given finite sequence of N {\displaystyle N} values x i {\displaystyle x_{i}} from a training set, obtained from conditionally independent sampling. The likelihood assigned to any considered parameter θ {\displaystyle \theta } of the model is then given by the product over all probabilities q θ ( X = x i ) {\displaystyle q_{\theta }(X=x_{i})} . Repeated occurrences are possible, leading to equal factors in the product. If the count of occurrences of the value equal to x {\displaystyle x} is denoted by # x {\displaystyle \#x} , then the frequency of that value equals # x / N {\displaystyle \#x/N} . If p ( X = x ) {\displaystyle p(X=x)} is the underlying probability distribution, for large N {\displaystyle N} we expect p ( X = x ) ≈ # x / N {\displaystyle p(X=x)\approx \#x/N} , by the law of large numbers. Writing our likelihood function as the product of observations from the distribution q θ {\displaystyle q_{\theta }} : L ( θ ; x ) = ∏ i q θ ( X = x i ) = ∏ x q θ ( X = x ) # x ≈ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) = exp ⁡ log ⁡ [ ∏ x q θ ( X = x ) N ⋅ p ( X = x ) ] = exp ⁡ ( ∑ x N ⋅ p ( X = x ) log ⁡ q θ ( X = x ) ) , {\displaystyle {\begin{aligned}{\mathcal {L}}(\theta ;{\mathbf {x} })&=\prod _{i}q_{\theta }(X=x_{i})=\prod _{x}q_{\theta }(X=x)^{\#x}\\&\approx \prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}=\exp \log \left[\prod _{x}q_{\theta }(X=x)^{N\cdot p(X=x)}\right]\\&=\exp \left(\sum _{x}N\cdot p(X=x)\log q_{\theta }(X=x)^{}\right),\end{aligned}}} where we have used the calculation rules for the logarithm in the final line. Notice how the exponent contains a − H ( p , q θ ) {\displaystyle -H(p,q_{\theta })} term. Taking the logarithm of both sides gives: log ⁡ L ( θ ; x ) = − N ⋅ H ( p , q θ ) . {\displaystyle \log {\mathcal {L}}(\theta ;{\mathbf {x} })=-N\cdot H(p,q_{\theta }).} Since the logarithm is a monotonically increasing function, the maximizing value of θ {\displaystyle \theta } is unaffected by this final step. Similarly, the maximizing value of θ {\displaystyle \theta } is unaffected by the factor of N {\displaystyle N} . So we observe that the likelihood maximization amounts to minimization of the cross-entropy. == Cross-entropy minimization == Cross-entropy minimization is frequently used in optimization and rare-event probability estimation. When comparing a distribution q {\displaystyle q} against a fixed reference distribution p {\displaystyle p} , cross-entropy and KL divergence are identical up to an additive constant (since p {\displaystyle p} is fixed): According to the Gibbs' inequality, both take on their minimal values when p = q {\displaystyle p=q} , which is 0 {\displaystyle 0} for KL divergence, and H ( p ) {\displaystyle \mathrm {H} (p)} for cross-entropy. In the engineering literature, the principle of minimizing KL divergence (Kullback's "Principle of Minimum Discrimination Information") is often called the Principle of Minimum Cross-Entropy (MCE), or Minxent. However, as discussed in the article Kullback–Leibler divergence, sometimes the distribution q {\displaystyle q} is the fixed prior reference distribution, and the distribution p {\displaystyle p} is optimized to be as close to q {\displaystyle q} as possible, subject to some constraint. In this case the two minimizations are not equivalent. This has led to some ambiguity in the literature, with some authors attempting to resolve the inconsistency by restating cross-entropy to be D K L ( p ∥ q ) {\displaystyle D_{\mathrm {KL} }(p\parallel q)} , rather than H (

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