AI Email Body Generator

AI Email Body Generator — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

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  Free Studio

  Free Studio is a freeware set of multimedia computer programs developed by DVDVideoSoft. The programs are available in one integrated package and also as separate downloads (Free Studio Manager is included in both). == Overview == The Free Studio software bundle consists of about 48 programs, grouped into several sections: YouTube, MP3 & Audio, CD-DVD-BD, DVD & Video, Photo & Images, Mobiles, Apple Devices, and 3D. The largest group is the DVD & Video section containing 14 different applications. Mobiles section is the second largest group with 13 programs. However, the YouTube section, particularly YouTube downloading programs, has gained more popularity among users. The programs have been tested and endorsed by a dozen of software portals and have won awards from these sites. Free Studio is most popular in Germany, Greece, Italy, and the United States. It is also popular in Japan, France, and the United Kingdom. Some of the programs in the package are free and open-source software. == History == DVDVideoSoft project was launched in 2006 by company Digital Wave Ltd., for software development to produce multimedia application software. The founders distributed paid software as an affiliate at the start, later their own products appeared on the site. Free YouTube Download was the first successful program, then DVDVideoSoft created and launched several other 'Free YouTube' applications. Later on upon users' requests DVDVideoSoft started developing other kinds of applications including media converters etc. Today DVDVideoSoft offers up to 49 different programs for video, audio and image processing individually or integrated into the Free Studio package. == Features == DVDVideoSoft YouTube programs can be used to download YouTube videos in their original format and convert them to AVI, DVD, MP4, WMV etc. or different audio formats. YouTube section contains Free Video Call Recorder for Skype button, but the program itself is not included into FS installation (it has to be downloaded and installed separately). The "MP3 & Audio" section consists of the programs which convert audio files between different formats, convert audio files to Flash for web, extract audio from video files, edit audio files (Free Audio Dub), rip and burn CDs. Enclosed in the CD-DVD-BD section are the applications that enable users to burn files and folders to discs, to convert videos to a DVD format and vice versa, to burn CDs, and to copy music from audio CDs into files. The "DVD and Video" section contains several desktop video and DVD converters. Some of the programs can flip, rotate and cut (Free Video Dub) videos. One of the most popular programs from the section is Free Video Dub. Converted videos are now, contrary to previous versions, watermarked if no paid membership is present. Free Studio includes several applications for Apple phones, iPods and other devices. The Mobiles section contains a dozen video converters for various mobile devices such as cell phones, Tablets and Game consoles. They convert videos to play them on (BlackBerry, HTC, LG phones, Sony/Sony Ericsson, Nintendo, Xbox, Motorola phones, etc.) The "Photo & Images" section incorporates the programs for image conversion and resizing, extracting JPEG frames from videos (Free Video To JPEG Converter), recording screen activities, making screenshots (Free Screen Recorder). The 3D section is composed of the programs to make 3D videos and 3D images. There are several algorithms which allow to create different types of 3D images. == Supported formats == === Video formats === Input: .avi; .ivf; .div; .divx; .mpg; .mpeg; .mpe; .mp4; .m4v; .wmv; .asf; .webm; .mkv; .mov; .qt; .ts; .mts; .m2t; .m2ts; .mod; .tod; .vro; .dat; .3gp2; .3gpp; .3gp; .3g2; .dvr-ms; .flv; .f4v; .amv; .rm; .rmm; .rv; .rmvb; .ogv; DVD video Output: .mp4; .wmv; .avi; .mkv; .webm; .flv; .swf; .mov; .3gp; .m2ts; DVD video === Audio formats === Input: .mp3 .wav; .aac; .m4a; .m4b; .wma; .ogg; .flac; .ra; .ram; .amr; .ape; .mka; .tta; .aiff; .au; .mpc; .spx; .ac3; audio cd Output: .mp3; .m4a; .aac; .wav; .wma; .ogg; .flac; .ape; audio CD === Image formats === Input: .jpg, .png, .bmp, .gif, .tga Output: .jpg, .png, .bmp, .gif, .tga, .pdf == Reception == The programs have been tested and endorsed by Chip Online, Tucows, SnapFiles, Brothersoft, and Softonic and have won awards from these sites. Free Studio is most popular in Germany, United States and Italy. It is also popular in Japan, France and the United Kingdom. The most popular applications, according to CNET statistics, include Free YouTube to MP3 Converter, Free Video to MP3 Converter, Free MP4 Video Converter and Free YouTube Download. Other programs with high rank: Free AVI Video Converter, Free Video Editor, Free Audio Converter and Free Studio in a whole. == Criticism == Free Studio (as can be common for freeware packages) is criticized for toolbar and Web search engine installation. Older versions have also included OpenCandy, which is loaded automatically, with no request for user approval. There can be difficulties installing only the programs needed without installing bundled extra programs. In March 2017, DVDVideoSoft announced that it had stopped showing other products' ads during installation and removed all toolbars, search engines, and OpenCandy.

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  Semidefinite embedding

  Maximum Variance Unfolding (MVU), also known as Semidefinite Embedding (SDE), is an algorithm in computer science that uses semidefinite programming to perform non-linear dimensionality reduction of high-dimensional vectorial input data. It is motivated by the observation that kernel Principal Component Analysis (kPCA) does not reduce the data dimensionality, as it leverages the Kernel trick to non-linearly map the original data into an inner-product space. == Algorithm == MVU creates a mapping from the high dimensional input vectors to some low dimensional Euclidean vector space in the following steps: A neighbourhood graph is created. Each input is connected with its k-nearest input vectors (according to Euclidean distance metric) and all k-nearest neighbors are connected with each other. If the data is sampled well enough, the resulting graph is a discrete approximation of the underlying manifold. The neighbourhood graph is "unfolded" with the help of semidefinite programming. Instead of learning the output vectors directly, the semidefinite programming aims to find an inner product matrix that maximizes the pairwise distances between any two inputs that are not connected in the neighbourhood graph while preserving the nearest neighbors distances. The low-dimensional embedding is finally obtained by application of multidimensional scaling on the learned inner product matrix. The steps of applying semidefinite programming followed by a linear dimensionality reduction step to recover a low-dimensional embedding into a Euclidean space were first proposed by Linial, London, and Rabinovich. == Optimization formulation == Let X {\displaystyle X\,\!} be the original input and Y {\displaystyle Y\,\!} be the embedding. If i , j {\displaystyle i,j\,\!} are two neighbors, then the local isometry constraint that needs to be satisfied is: | X i − X j | 2 = | Y i − Y j | 2 {\displaystyle |X_{i}-X_{j}|^{2}=|Y_{i}-Y_{j}|^{2}\,\!} Let G , K {\displaystyle G,K\,\!} be the Gram matrices of X {\displaystyle X\,\!} and Y {\displaystyle Y\,\!} (i.e.: G i j = X i ⋅ X j , K i j = Y i ⋅ Y j {\displaystyle G_{ij}=X_{i}\cdot X_{j},K_{ij}=Y_{i}\cdot Y_{j}\,\!} ). We can express the above constraint for every neighbor points i , j {\displaystyle i,j\,\!} in term of G , K {\displaystyle G,K\,\!} : G i i + G j j − G i j − G j i = K i i + K j j − K i j − K j i {\displaystyle G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji}\,\!} In addition, we also want to constrain the embedding Y {\displaystyle Y\,\!} to center at the origin: 0 = | ∑ i Y i | 2 ⇔ ( ∑ i Y i ) ⋅ ( ∑ i Y i ) ⇔ ∑ i , j Y i ⋅ Y j ⇔ ∑ i , j K i j {\displaystyle 0=|\sum _{i}Y_{i}|^{2}\Leftrightarrow (\sum _{i}Y_{i})\cdot (\sum _{i}Y_{i})\Leftrightarrow \sum _{i,j}Y_{i}\cdot Y_{j}\Leftrightarrow \sum _{i,j}K_{ij}} As described above, except the distances of neighbor points are preserved, the algorithm aims to maximize the pairwise distance of every pair of points. The objective function to be maximized is: T ( Y ) = 1 2 N ∑ i , j | Y i − Y j | 2 {\displaystyle T(Y)={\dfrac {1}{2N}}\sum _{i,j}|Y_{i}-Y_{j}|^{2}} Intuitively, maximizing the function above is equivalent to pulling the points as far away from each other as possible and therefore "unfold" the manifold. The local isometry constraint Let τ = m a x { η i j | Y i − Y j | 2 } {\displaystyle \tau =max\{\eta _{ij}|Y_{i}-Y_{j}|^{2}\}\,\!} where η i j := { 1 if i is a neighbour of j 0 otherwise . {\displaystyle \eta _{ij}:={\begin{cases}1&{\mbox{if}}\ i{\mbox{ is a neighbour of }}j\\0&{\mbox{otherwise}}.\end{cases}}} prevents the objective function from diverging (going to infinity). Since the graph has N points, the distance between any two points | Y i − Y j | 2 ≤ N τ {\displaystyle |Y_{i}-Y_{j}|^{2}\leq N\tau \,\!} . We can then bound the objective function as follows: T ( Y ) = 1 2 N ∑ i , j | Y i − Y j | 2 ≤ 1 2 N ∑ i , j ( N τ ) 2 = N 3 τ 2 2 {\displaystyle T(Y)={\dfrac {1}{2N}}\sum _{i,j}|Y_{i}-Y_{j}|^{2}\leq {\dfrac {1}{2N}}\sum _{i,j}(N\tau )^{2}={\dfrac {N^{3}\tau ^{2}}{2}}\,\!} The objective function can be rewritten purely in the form of the Gram matrix: T ( Y ) = 1 2 N ∑ i , j | Y i − Y j | 2 = 1 2 N ∑ i , j ( Y i 2 + Y j 2 − Y i ⋅ Y j − Y j ⋅ Y i ) = 1 2 N ( ∑ i , j Y i 2 + ∑ i , j Y j 2 − ∑ i , j Y i ⋅ Y j − ∑ i , j Y j ⋅ Y i ) = 1 2 N ( ∑ i , j Y i 2 + ∑ i , j Y j 2 − 0 − 0 ) = 1 N ( ∑ i Y i 2 ) = 1 N ( T r ( K ) ) {\displaystyle {\begin{aligned}T(Y)&{}={\dfrac {1}{2N}}\sum _{i,j}|Y_{i}-Y_{j}|^{2}\\&{}={\dfrac {1}{2N}}\sum _{i,j}(Y_{i}^{2}+Y_{j}^{2}-Y_{i}\cdot Y_{j}-Y_{j}\cdot Y_{i})\\&{}={\dfrac {1}{2N}}(\sum _{i,j}Y_{i}^{2}+\sum _{i,j}Y_{j}^{2}-\sum _{i,j}Y_{i}\cdot Y_{j}-\sum _{i,j}Y_{j}\cdot Y_{i})\\&{}={\dfrac {1}{2N}}(\sum _{i,j}Y_{i}^{2}+\sum _{i,j}Y_{j}^{2}-0-0)\\&{}={\dfrac {1}{N}}(\sum _{i}Y_{i}^{2})={\dfrac {1}{N}}(Tr(K))\\\end{aligned}}\,\!} Finally, the optimization can be formulated as: Maximize T r ( K ) subject to K ⪰ 0 , ∑ i j K i j = 0 and G i i + G j j − G i j − G j i = K i i + K j j − K i j − K j i , ∀ i , j where η i j = 1 , {\displaystyle {\begin{aligned}&{\text{Maximize}}&&Tr(\mathbf {K} )\\&{\text{subject to}}&&\mathbf {K} \succeq 0,\sum _{ij}\mathbf {K} _{ij}=0\\&{\text{and}}&&G_{ii}+G_{jj}-G_{ij}-G_{ji}=K_{ii}+K_{jj}-K_{ij}-K_{ji},\forall i,j{\mbox{ where }}\eta _{ij}=1,\end{aligned}}} After the Gram matrix K {\displaystyle K\,\!} is learned by semidefinite programming, the output Y {\displaystyle Y\,\!} can be obtained via Cholesky decomposition. In particular, the Gram matrix can be written as K i j = ∑ α = 1 N ( λ α V α i V α j ) {\displaystyle K_{ij}=\sum _{\alpha =1}^{N}(\lambda _{\alpha }V_{\alpha i}V_{\alpha j})\,\!} where V α i {\displaystyle V_{\alpha i}\,\!} is the i-th element of eigenvector V α {\displaystyle V_{\alpha }\,\!} of the eigenvalue λ α {\displaystyle \lambda _{\alpha }\,\!} . It follows that the α {\displaystyle \alpha \,\!} -th element of the output Y i {\displaystyle Y_{i}\,\!} is λ α V α i {\displaystyle {\sqrt {\lambda _{\alpha }}}V_{\alpha i}\,\!} .

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  T-distributed stochastic neighbor embedding

  t-distributed stochastic neighbor embedding (t-SNE) is a statistical method for visualizing high-dimensional data by giving each datapoint a location in a two or three-dimensional map. It is based on Stochastic Neighbor Embedding originally developed by Geoffrey Hinton and Sam Roweis, where Laurens van der Maaten and Hinton proposed the t-distributed variant. It is a nonlinear dimensionality reduction technique for embedding high-dimensional data for visualization in a low-dimensional space of two or three dimensions. Specifically, it models each high-dimensional object by a two- or three-dimensional point in such a way that similar objects are modeled by nearby points and dissimilar objects are modeled by distant points with high probability. The t-SNE algorithm comprises two main stages. First, t-SNE constructs a probability distribution over pairs of high-dimensional objects in such a way that similar objects are assigned a higher probability while dissimilar points are assigned a lower probability. Second, t-SNE defines a similar probability distribution over the points in the low-dimensional map, and it minimizes the Kullback–Leibler divergence (KL divergence) between the two distributions with respect to the locations of the points in the map. While the original algorithm uses the Euclidean distance between objects as the base of its similarity metric, this can be changed as appropriate. A Riemannian variant is UMAP. t-SNE has been used for visualization in a wide range of applications, including genomics, computer security research, natural language processing, music analysis, cancer research, bioinformatics, geological domain interpretation, and biomedical signal processing. For a data set with n {\displaystyle n} elements, t-SNE runs in O ( n 2 ) {\displaystyle O(n^{2})} time and requires O ( n 2 ) {\displaystyle O(n^{2})} space. == Details == Given a set of N {\displaystyle N} high-dimensional objects x 1 , … , x N {\displaystyle \mathbf {x} _{1},\dots ,\mathbf {x} _{N}} , t-SNE first computes probabilities p i j {\displaystyle p_{ij}} that are proportional to the similarity of objects x i {\displaystyle \mathbf {x} _{i}} and x j {\displaystyle \mathbf {x} _{j}} , as follows. For i ≠ j {\displaystyle i\neq j} , define p j ∣ i = exp ⁡ ( − ‖ x i − x j ‖ 2 / 2 σ i 2 ) ∑ k ≠ i exp ⁡ ( − ‖ x i − x k ‖ 2 / 2 σ i 2 ) {\displaystyle p_{j\mid i}={\frac {\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{j}\rVert ^{2}/2\sigma _{i}^{2})}{\sum _{k\neq i}\exp(-\lVert \mathbf {x} _{i}-\mathbf {x} _{k}\rVert ^{2}/2\sigma _{i}^{2})}}} and set p i ∣ i = 0 {\displaystyle p_{i\mid i}=0} . Note the above denominator ensures ∑ j p j ∣ i = 1 {\displaystyle \sum _{j}p_{j\mid i}=1} for all i {\displaystyle i} . As van der Maaten and Hinton explained: "The similarity of datapoint x j {\displaystyle x_{j}} to datapoint x i {\displaystyle x_{i}} is the conditional probability, p j | i {\displaystyle p_{j|i}} , that x i {\displaystyle x_{i}} would pick x j {\displaystyle x_{j}} as its neighbor if neighbors were picked in proportion to their probability density under a Gaussian centered at x i {\displaystyle x_{i}} ." Now define p i j = p j ∣ i + p i ∣ j 2 N {\displaystyle p_{ij}={\frac {p_{j\mid i}+p_{i\mid j}}{2N}}} This is motivated because p i {\displaystyle p_{i}} and p j {\displaystyle p_{j}} from the N samples are estimated as 1/N, so the conditional probability can be written as p i ∣ j = N p i j {\displaystyle p_{i\mid j}=Np_{ij}} and p j ∣ i = N p j i {\displaystyle p_{j\mid i}=Np_{ji}} . Since p i j = p j i {\displaystyle p_{ij}=p_{ji}} , you can obtain previous formula. Also note that p i i = 0 {\displaystyle p_{ii}=0} and ∑ i , j p i j = 1 {\displaystyle \sum _{i,j}p_{ij}=1} . The bandwidth of the Gaussian kernels σ i {\displaystyle \sigma _{i}} is set in such a way that the entropy of the conditional distribution equals a predefined entropy using the bisection method. As a result, the bandwidth is adapted to the density of the data: smaller values of σ i {\displaystyle \sigma _{i}} are used in denser parts of the data space. The entropy increases with the perplexity of this distribution P i {\displaystyle P_{i}} ; this relation is seen as P e r p ( P i ) = 2 H ( P i ) {\displaystyle Perp(P_{i})=2^{H(P_{i})}} where H ( P i ) {\displaystyle H(P_{i})} is the Shannon entropy H ( P i ) = − ∑ j p j | i log 2 ⁡ p j | i . {\displaystyle H(P_{i})=-\sum _{j}p_{j|i}\log _{2}p_{j|i}.} The perplexity is a hand-chosen parameter of t-SNE, and as the authors state, "perplexity can be interpreted as a smooth measure of the effective number of neighbors. The performance of SNE is fairly robust to changes in the perplexity, and typical values are between 5 and 50.". Since the Gaussian kernel uses the Euclidean distance ‖ x i − x j ‖ {\displaystyle \lVert x_{i}-x_{j}\rVert } , it is affected by the curse of dimensionality, and in high dimensional data when distances lose the ability to discriminate, the p i j {\displaystyle p_{ij}} become too similar (asymptotically, they would converge to a constant). It has been proposed to adjust the distances with a power transform, based on the intrinsic dimension of each point, to alleviate this. t-SNE aims to learn a d {\displaystyle d} -dimensional map y 1 , … , y N {\displaystyle \mathbf {y} _{1},\dots ,\mathbf {y} _{N}} (with y i ∈ R d {\displaystyle \mathbf {y} _{i}\in \mathbb {R} ^{d}} and d {\displaystyle d} typically chosen as 2 or 3) that reflects the similarities p i j {\displaystyle p_{ij}} as well as possible. To this end, it measures similarities q i j {\displaystyle q_{ij}} between two points in the map y i {\displaystyle \mathbf {y} _{i}} and y j {\displaystyle \mathbf {y} _{j}} , using a very similar approach. Specifically, for i ≠ j {\displaystyle i\neq j} , define q i j {\displaystyle q_{ij}} as q i j = ( 1 + ‖ y i − y j ‖ 2 ) − 1 ∑ k ∑ l ≠ k ( 1 + ‖ y k − y l ‖ 2 ) − 1 {\displaystyle q_{ij}={\frac {(1+\lVert \mathbf {y} _{i}-\mathbf {y} _{j}\rVert ^{2})^{-1}}{\sum _{k}\sum _{l\neq k}(1+\lVert \mathbf {y} _{k}-\mathbf {y} _{l}\rVert ^{2})^{-1}}}} and set q i i = 0 {\displaystyle q_{ii}=0} . Herein a heavy-tailed Student t-distribution (with one-degree of freedom, which is the same as a Cauchy distribution) is used to measure similarities between low-dimensional points in order to allow dissimilar objects to be modeled far apart in the map. The locations of the points y i {\displaystyle \mathbf {y} _{i}} in the map are determined by minimizing the (non-symmetric) Kullback–Leibler divergence of the distribution P {\displaystyle P} from the distribution Q {\displaystyle Q} , that is: K L ( P ∥ Q ) = ∑ i ≠ j p i j log ⁡ p i j q i j {\displaystyle \mathrm {KL} \left(P\parallel Q\right)=\sum _{i\neq j}p_{ij}\log {\frac {p_{ij}}{q_{ij}}}} The minimization of the Kullback–Leibler divergence with respect to the points y i {\displaystyle \mathbf {y} _{i}} is performed using gradient descent. The result of this optimization is a map that reflects the similarities between the high-dimensional inputs. == Output == While t-SNE plots often seem to display clusters, the visual clusters can be strongly influenced by the chosen parameterization (especially the perplexity) and so a good understanding of the parameters for t-SNE is needed. Such "clusters" can be shown to even appear in structured data with no clear clustering, and so may be false findings. Similarly, the size of clusters produced by t-SNE is not informative, and neither is the distance between clusters. Thus, interactive exploration may be needed to choose parameters and validate results. It has been shown that t-SNE can often recover well-separated clusters, and with special parameter choices, approximates a simple form of spectral clustering. == Software == A C++ implementation of Barnes-Hut is available on the github account of one of the original authors. The R package Rtsne implements t-SNE in R. ELKI contains tSNE, also with Barnes-Hut approximation scikit-learn, a popular machine learning library in Python implements t-SNE with both exact solutions and the Barnes-Hut approximation. Tensorboard, the visualization kit associated with TensorFlow, also implements t-SNE (online version) The Julia package TSne implements t-SNE

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  Induction of regular languages

  In computational learning theory, induction of regular languages refers to the task of learning a formal description (e.g. grammar) of a regular language from a given set of example strings. Although E. Mark Gold has shown that not every regular language can be learned this way (see language identification in the limit), approaches have been investigated for a variety of subclasses. They are sketched in this article. For learning of more general grammars, see Grammar induction. == Definitions == A regular language is defined as a (finite or infinite) set of strings that can be described by one of the mathematical formalisms called "finite automaton", "regular grammar", or "regular expression", all of which have the same expressive power. Since the latter formalism leads to shortest notations, it shall be introduced and used here. Given a set Σ of symbols (a.k.a. alphabet), a regular expression can be any of ∅ (denoting the empty set of strings), ε (denoting the singleton set containing just the empty string), a (where a is any character in Σ; denoting the singleton set just containing the single-character string a), r + s (where r and s are, in turn, simpler regular expressions; denoting their set's union) r ⋅ s (denoting the set of all possible concatenations of strings from r's and s's set), r + (denoting the set of n-fold repetitions of strings from r's set, for any n ≥ 1), or r (similarly denoting the set of n-fold repetitions, but also including the empty string, seen as 0-fold repetition). For example, using Σ = {0,1}, the regular expression (0+1+ε)⋅(0+1) denotes the set of all binary numbers with one or two digits (leading zero allowed), while 1⋅(0+1)⋅0 denotes the (infinite) set of all even binary numbers (no leading zeroes). Given a set of strings (also called "positive examples"), the task of regular language induction is to come up with a regular expression that denotes a set containing all of them. As an example, given {1, 10, 100}, a "natural" description could be the regular expression 1⋅0, corresponding to the informal characterization "a 1 followed by arbitrarily many (maybe even none) 0's". However, (0+1) and 1+(1⋅0)+(1⋅0⋅0) is another regular expression, denoting the largest (assuming Σ = {0,1}) and the smallest set containing the given strings, and called the trivial overgeneralization and undergeneralization, respectively. Some approaches work in an extended setting where also a set of "negative example" strings is given; then, a regular expression is to be found that generates all of the positive, but none of the negative examples. == Lattice of automata == Dupont et al. have shown that the set of all structurally complete finite automata generating a given input set of example strings forms a lattice, with the trivial undergeneralized and the trivial overgeneralized automaton as bottom and top element, respectively. Each member of this lattice can be obtained by factoring the undergeneralized automaton by an appropriate equivalence relation. For the above example string set {1, 10, 100}, the picture shows at its bottom the undergeneralized automaton Aa,b,c,d in grey, consisting of states a, b, c, and d. On the state set {a,b,c,d}, a total of 15 equivalence relations exist, forming a lattice. Mapping each equivalence E to the corresponding quotient automaton language L(Aa,b,c,d / E) obtains the partially ordered set shown in the picture. Each node's language is denoted by a regular expression. The language may be recognized by quotient automata w.r.t. different equivalence relations, all of which are shown below the node. An arrow between two nodes indicates that the lower node's language is a proper subset of the higher node's. If both positive and negative example strings are given, Dupont et al. build the lattice from the positive examples, and then investigate the separation border between automata that generate some negative example and such that do not. Most interesting are those automata immediately below the border. In the picture, separation borders are shown for the negative example strings 11 (green), 1001 (blue), 101 (cyan), and 0 (red). Coste and Nicolas present an own search method within the lattice, which they relate to Mitchell's version space paradigm. To find the separation border, they use a graph coloring algorithm on the state inequality relation induced by the negative examples. Later, they investigate several ordering relations on the set of all possible state fusions. Kudo and Shimbo use the representation by automaton factorizations to give a unique framework for the following approaches (sketched below): k-reversible languages and the "tail clustering" follow-up approach, Successor automata and the predecessor-successor method, and pumping-based approaches (framework-integration challenged by Luzeaux, however). Each of these approaches is shown to correspond to a particular kind of equivalence relations used for factorization. == Approaches == === k-reversible languages === Angluin considers so-called "k-reversible" regular automata, that is, deterministic automata in which each state can be reached from at most one state by following a transition chain of length k. Formally, if Σ, Q, and δ denote the input alphabet, the state set, and the transition function of an automaton A, respectively, then A is called k-reversible if: ∀a0, ..., ak ∈ Σ ∀s1, s2 ∈ Q: δ(s1, a0...ak) = δ(s2, a0...ak) ⇒ s1 = s2, where δ means the homomorphic extension of δ to arbitrary words. Angluin gives a cubic algorithm for learning of the smallest k-reversible language from a given set of input words; for k = 0, the algorithm has even almost linear complexity. The required state uniqueness after k + 1 given symbols forces unifying automaton states, thus leading to a proper generalization different from the trivial undergeneralized automaton. This algorithm has been used to learn simple parts of English syntax; later, an incremental version has been provided. Another approach based on k-reversible automata is the tail clustering method. === Successor automata === From a given set of input strings, Vernadat and Richetin build a so-called successor automaton, consisting of one state for each distinct character and a transition between each two adjacent characters' states. For example, the singleton input set {aabbaabb} leads to an automaton corresponding to the regular expression (a+⋅b+). An extension of this approach is the predecessor-successor method which generalizes each character repetition immediately to a Kleene + and then includes for each character the set of its possible predecessors in its state. Successor automata can learn exactly the class of local languages. Since each regular language is the homomorphic image of a local language, grammars from the former class can be learned by lifting, if an appropriate (depending on the intended application) homomorphism is provided. In particular, there is such a homomorphism for the class of languages learnable by the predecessor-successor method. The learnability of local languages can be reduced to that of k-reversible languages. === Early approaches === Chomsky and Miller (1957) used the pumping lemma: they guess a part v of an input string uvw and try to build a corresponding cycle into the automaton to be learned; using membership queries they ask, for appropriate k, which of the strings uw, uvvw, uvvvw, ..., uvkw also belongs to the language to be learned, thereby refining the structure of their automaton. In 1959, Solomonoff generalized this approach to context-free languages, which also obey a pumping lemma. === Cover automata === Câmpeanu et al. learn a finite automaton as a compact representation of a large finite language. Given such a language F, they search a so-called cover automaton A such that its language L(A) covers F in the following sense: L(A) ∩ Σ≤ l = F, where l is the length of the longest string in F, and Σ≤ l denotes the set of all strings not longer than l. If such a cover automaton exists, F is uniquely determined by A and l. For example, F = {ad, read, reread } has l = 6 and a cover automaton corresponding to the regular expression (r⋅e)⋅a⋅d. For two strings x and y, Câmpeanu et al. define x ~ y if xz ∈ F ⇔ yz ∈ F for all strings z of a length such that both xz and yz are not longer than l. Based on this relation, whose lack of transitivity causes considerable technical problems, they give an O(n4) algorithm to construct from F a cover automaton A of minimal state count. Moreover, for union, intersection, and difference of two finite languages they provide corresponding operations on their cover automata. Păun et al. improve the time complexity to O(n2). === Residual automata === For a set S of strings and a string u, the Brzozowski derivative u−1S is defined as the set of all rest-strings obtainable from a string in S by cutting off its prefix u (if possible), formally: u−1S = {v ∈ Σ: uv ∈ S}, cf. picture. Denis et al. define a

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  Security.txt

  security.txt is an accepted standard for website security information that allows security researchers to report security vulnerabilities easily. The standard prescribes a text file named security.txt in the well known location, similar in syntax to robots.txt but intended to be machine and human readable, for those wishing to contact a website's owner about security issues. security.txt files have been adopted by Google, GitHub, LinkedIn, and Facebook. == History == The Internet Draft was first submitted by Edwin Foudil in September 2017. At that time it covered four directives, "Contact", "Encryption", "Disclosure" and "Acknowledgement". Foudil expected to add further directives based on feedback. In addition, web security expert Scott Helme said he had seen positive feedback from the security community while use among the top 1 million websites was "as low as expected right now". In 2019, the Cybersecurity and Infrastructure Security Agency (CISA) published a draft binding operational directive that requires all US federal agencies to publish a security.txt file within 180 days. The Internet Engineering Steering Group (IESG) issued a Last Call for security.txt in December 2019 which ended on January 6, 2020. A study in 2021 found that over ten percent of top-100 websites published a security.txt file, with the percentage of sites publishing the file decreasing as more websites were considered. The study also noted a number of discrepancies between the standard and the content of the file. In April 2022 the security.txt file has been accepted by Internet Engineering Task Force (IETF) as RFC 9116. == File format == security.txt files can be served under the /.well-known/ directory (i.e. /.well-known/security.txt) or the top-level directory (i.e. /security.txt) of a website. The file must be served over HTTPS and in plaintext format.

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  Diffusion map

  Diffusion maps is a dimensionality reduction or feature extraction algorithm introduced by Coifman and Lafon which computes a family of embeddings of a data set into Euclidean space (often low-dimensional) whose coordinates can be computed from the eigenvectors and eigenvalues of a diffusion operator on the data. The Euclidean distance between points in the embedded space is equal to the "diffusion distance" between probability distributions centered at those points. Different from linear dimensionality reduction methods such as principal component analysis (PCA), diffusion maps are part of the family of nonlinear dimensionality reduction methods which focus on discovering the underlying manifold that the data has been sampled from. By integrating local similarities at different scales, diffusion maps give a global description of the data-set. Compared with other methods, the diffusion map algorithm is robust to noise perturbation and computationally inexpensive. == Definition of diffusion maps == Following and , diffusion maps can be defined in four steps. === Connectivity === Diffusion maps exploit the relationship between heat diffusion and random walk Markov chain. The basic observation is that if we take a random walk on the data, walking to a nearby data-point is more likely than walking to another that is far away. Let ( X , A , μ ) {\displaystyle (X,{\mathcal {A}},\mu )} be a measure space, where X {\displaystyle X} is the data set and μ {\displaystyle \mu } represents the distribution of the points on X {\displaystyle X} . Based on this, the connectivity k {\displaystyle k} between two data points, x {\displaystyle x} and y {\displaystyle y} , can be defined as the probability of walking from x {\displaystyle x} to y {\displaystyle y} in one step of the random walk. Usually, this probability is specified in terms of a kernel function of the two points: k : X × X → R {\displaystyle k:X\times X\rightarrow \mathbb {R} } . For example, the popular Gaussian kernel: k ( x , y ) = exp ⁡ ( − | | x − y | | 2 ϵ ) {\displaystyle k(x,y)=\exp \left(-{\frac {||x-y||^{2}}{\epsilon }}\right)} More generally, the kernel function has the following properties k ( x , y ) = k ( y , x ) {\displaystyle k(x,y)=k(y,x)} ( k {\displaystyle k} is symmetric) k ( x , y ) ≥ 0 ∀ x , y {\displaystyle k(x,y)\geq 0\,\,\forall x,y} ( k {\displaystyle k} is positivity preserving). The kernel constitutes the prior definition of the local geometry of the data-set. Since a given kernel will capture a specific feature of the data set, its choice should be guided by the application that one has in mind. This is a major difference with methods such as principal component analysis, where correlations between all data points are taken into account at once. Given ( X , k ) {\displaystyle (X,k)} , we can then construct a reversible discrete-time Markov chain on X {\displaystyle X} (a process known as the normalized graph Laplacian construction): d ( x ) = ∫ X k ( x , y ) d μ ( y ) {\displaystyle d(x)=\int _{X}k(x,y)d\mu (y)} and define: p ( x , y ) = k ( x , y ) d ( x ) {\displaystyle p(x,y)={\frac {k(x,y)}{d(x)}}} Although the new normalized kernel does not inherit the symmetric property, it does inherit the positivity-preserving property and gains a conservation property: ∫ X p ( x , y ) d μ ( y ) = 1 {\displaystyle \int _{X}p(x,y)d\mu (y)=1} === Diffusion process === From p ( x , y ) {\displaystyle p(x,y)} we can construct a transition matrix of a Markov chain ( M {\displaystyle M} ) on X {\displaystyle X} . In other words, p ( x , y ) {\displaystyle p(x,y)} represents the one-step transition probability from x {\displaystyle x} to y {\displaystyle y} , and M t {\displaystyle M^{t}} gives the t-step transition matrix. We define the diffusion matrix L {\displaystyle L} (it is also a version of graph Laplacian matrix) L i , j = k ( x i , x j ) {\displaystyle L_{i,j}=k(x_{i},x_{j})\,} We then define the new kernel L i , j ( α ) = k ( α ) ( x i , x j ) = L i , j ( d ( x i ) d ( x j ) ) α {\displaystyle L_{i,j}^{(\alpha )}=k^{(\alpha )}(x_{i},x_{j})={\frac {L_{i,j}}{(d(x_{i})d(x_{j}))^{\alpha }}}\,} or equivalently, L ( α ) = D − α L D − α {\displaystyle L^{(\alpha )}=D^{-\alpha }LD^{-\alpha }\,} where D is a diagonal matrix and D i , i = ∑ j L i , j . {\displaystyle D_{i,i}=\sum _{j}L_{i,j}.} We apply the graph Laplacian normalization to this new kernel: M = ( D ( α ) ) − 1 L ( α ) , {\displaystyle M=({D}^{(\alpha )})^{-1}L^{(\alpha )},\,} where D ( α ) {\displaystyle D^{(\alpha )}} is a diagonal matrix and D i , i ( α ) = ∑ j L i , j ( α ) . {\displaystyle {D}_{i,i}^{(\alpha )}=\sum _{j}L_{i,j}^{(\alpha )}.} p ( x j , t | x i ) = M i , j t {\displaystyle p(x_{j},t|x_{i})=M_{i,j}^{t}\,} One of the main ideas of the diffusion framework is that running the chain forward in time (taking larger and larger powers of M {\displaystyle M} ) reveals the geometric structure of X {\displaystyle X} at larger and larger scales (the diffusion process). Specifically, the notion of a cluster in the data set is quantified as a region in which the probability of escaping this region is low (within a certain time t). Therefore, t not only serves as a time parameter, but it also has the dual role of scale parameter. The eigendecomposition of the matrix M t {\displaystyle M^{t}} yields M i , j t = ∑ l λ l t ψ l ( x i ) ϕ l ( x j ) {\displaystyle M_{i,j}^{t}=\sum _{l}\lambda _{l}^{t}\psi _{l}(x_{i})\phi _{l}(x_{j})\,} where { λ l } {\displaystyle \{\lambda _{l}\}} is the sequence of eigenvalues of M {\displaystyle M} and { ψ l } {\displaystyle \{\psi _{l}\}} and { ϕ l } {\displaystyle \{\phi _{l}\}} are the biorthogonal left and right eigenvectors respectively. Due to the spectrum decay of the eigenvalues, only a few terms are necessary to achieve a given relative accuracy in this sum. ==== Parameter α and the diffusion operator ==== The reason to introduce the normalization step involving α {\displaystyle \alpha } is to tune the influence of the data point density on the infinitesimal transition of the diffusion. In some applications, the sampling of the data is generally not related to the geometry of the manifold we are interested in describing. In this case, we can set α = 1 {\displaystyle \alpha =1} and the diffusion operator approximates the Laplace–Beltrami operator. We then recover the Riemannian geometry of the data set regardless of the distribution of the points. To describe the long-term behavior of the point distribution of a system of stochastic differential equations, we can use α = 0.5 {\displaystyle \alpha =0.5} and the resulting Markov chain approximates the Fokker–Planck diffusion. With α = 0 {\displaystyle \alpha =0} , it reduces to the classical graph Laplacian normalization. === Diffusion distance === The diffusion distance at time t {\displaystyle t} between two points can be measured as the similarity of two points in the observation space with the connectivity between them. It is given by D t ( x i , x j ) 2 = ∑ y ( p ( y , t | x i ) − p ( y , t | x j ) ) 2 ϕ 0 ( y ) {\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{y}{\frac {(p(y,t|x_{i})-p(y,t|x_{j}))^{2}}{\phi _{0}(y)}}} where ϕ 0 ( y ) {\displaystyle \phi _{0}(y)} is the stationary distribution of the Markov chain, given by the first left eigenvector of M {\displaystyle M} . Explicitly: ϕ 0 ( y ) = d ( y ) ∑ z ∈ X d ( z ) {\displaystyle \phi _{0}(y)={\frac {d(y)}{\sum _{z\in X}d(z)}}} Intuitively, D t ( x i , x j ) {\displaystyle D_{t}(x_{i},x_{j})} is small if there is a large number of short paths connecting x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} . There are several interesting features associated with the diffusion distance, based on our previous discussion that t {\displaystyle t} also serves as a scale parameter: Points are closer at a given scale (as specified by D t ( x i , x j ) {\displaystyle D_{t}(x_{i},x_{j})} ) if they are highly connected in the graph, therefore emphasizing the concept of a cluster. This distance is robust to noise, since the distance between two points depends on all possible paths of length t {\displaystyle t} between the points. From a machine learning point of view, the distance takes into account all evidences linking x i {\displaystyle x_{i}} to x j {\displaystyle x_{j}} , allowing us to conclude that this distance is appropriate for the design of inference algorithms based on the majority of preponderance. === Diffusion process and low-dimensional embedding === The diffusion distance can be calculated using the eigenvectors by D t ( x i , x j ) 2 = ∑ l λ l 2 t ( ψ l ( x i ) − ψ l ( x j ) ) 2 {\displaystyle D_{t}(x_{i},x_{j})^{2}=\sum _{l}\lambda _{l}^{2t}(\psi _{l}(x_{i})-\psi _{l}(x_{j}))^{2}\,} So the eigenvectors can be used as a new set of coordinates for the data. The diffusion map is defined as: Ψ t ( x ) = ( λ 1 t ψ 1 ( x ) , λ 2 t ψ 2 ( x ) , … , λ k t ψ k ( x ) ) {\displaystyle \Psi _{t}(x)=(\lambda _{1}^{t}\psi _{1}(x),\lambda _{2}^{t}\psi _{2}(x),\ld

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  Curriculum learning

  Curriculum learning is a technique in machine learning in which a model is trained on examples of increasing difficulty, where the definition of "difficulty" may be provided externally or discovered as part of the training process. This is intended to attain good performance more quickly, or to converge to a better local optimum if the global optimum is not found. == Approach == Most generally, curriculum learning is the technique of successively increasing the difficulty of examples in the training set that is presented to a model over multiple training iterations. This can produce better results than exposing the model to the full training set immediately under some circumstances; most typically, when the model is able to learn general principles from easier examples, and then gradually incorporate more complex and nuanced information as harder examples are introduced, such as edge cases. This has been shown to work in many domains, most likely as a form of regularization. There are several major variations in how the technique is applied: A concept of "difficulty" must be defined. This may come from human annotation or an external heuristic; for example in language modeling, shorter sentences might be classified as easier than longer ones. Another approach is to use the performance of another model, with examples accurately predicted by that model being classified as easier (providing a connection to boosting). Difficulty can be increased steadily or in distinct epochs, and in a deterministic schedule or according to a probability distribution. This may also be moderated by a requirement for diversity at each stage, in cases where easier examples are likely to be disproportionately similar to each other. Applications must also decide the schedule for increasing the difficulty. Simple approaches may use a fixed schedule, such as training on easy examples for half of the available iterations and then all examples for the second half. Other approaches use self-paced learning to increase the difficulty in proportion to the performance of the model on the current set. Since curriculum learning only concerns the selection and ordering of training data, it can be combined with many other techniques in machine learning. The success of the method assumes that a model trained for an easier version of the problem can generalize to harder versions, so it can be seen as a form of transfer learning. Some authors also consider curriculum learning to include other forms of progressively increasing complexity, such as increasing the number of model parameters. It is frequently combined with reinforcement learning, such as learning a simplified version of a game first. Some domains have shown success with anti-curriculum learning: training on the most difficult examples first. One example is the ACCAN method for speech recognition, which trains on the examples with the lowest signal-to-noise ratio first. == History == The term "curriculum learning" was introduced by Yoshua Bengio et al in 2009, with reference to the psychological technique of shaping in animals and structured education for humans: beginning with the simplest concepts and then building on them. The authors also note that the application of this technique in machine learning has its roots in the early study of neural networks such as Jeffrey Elman's 1993 paper Learning and development in neural networks: the importance of starting small. Bengio et al showed good results for problems in image classification, such as identifying geometric shapes with progressively more complex forms, and language modeling, such as training with a gradually expanding vocabulary. They conclude that, for curriculum strategies, "their beneficial effect is most pronounced on the test set", suggesting good generalization. The technique has since been applied to many other domains: Natural language processing: Part-of-speech tagging Intent detection Sentiment analysis Machine translation Speech recognition Language model pre-training Image recognition: Facial recognition Object detection Reinforcement learning: Game-playing Graph learning Matrix factorization

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  Farthest-first traversal

  In computational geometry, the farthest-first traversal of a compact metric space is a sequence of points in the space, where the first point is selected arbitrarily and each successive point is as far as possible from the set of previously-selected points. The same concept can also be applied to a finite set of geometric points, by restricting the selected points to belong to the set or equivalently by considering the finite metric space generated by these points. For a finite metric space or finite set of geometric points, the resulting sequence forms a permutation of the points, also known as the greedy permutation. Every prefix of a farthest-first traversal provides a set of points that is widely spaced and close to all remaining points. More precisely, no other set of equally many points can be spaced more than twice as widely, and no other set of equally many points can be less than half as far to its farthest remaining point. In part because of these properties, farthest-point traversals have many applications, including the approximation of the traveling salesman problem and the metric k-center problem. They may be constructed in polynomial time, or (for low-dimensional Euclidean spaces) approximated in near-linear time. == Definition and properties == A farthest-first traversal is a sequence of points in a compact metric space, with each point appearing at most once. If the space is finite, each point appears exactly once, and the traversal is a permutation of all of the points in the space. The first point of the sequence may be any point in the space. Each point p after the first must have the maximum possible distance to the set of points earlier than p in the sequence, where the distance from a point to a set is defined as the minimum of the pairwise distances to points in the set. A given space may have many different farthest-first traversals, depending both on the choice of the first point in the sequence (which may be any point in the space) and on ties for the maximum distance among later choices. Farthest-point traversals may be characterized by the following properties. Fix a number k, and consider the prefix formed by the first k points of the farthest-first traversal of any metric space. Let r be the distance between the final point of the prefix and the other points in the prefix. Then this subset has the following two properties: All pairs of the selected points are at distance at least r from each other, and All points of the metric space are at distance at most r from the subset. Conversely any sequence having these properties, for all choices of k, must be a farthest-first traversal. These are the two defining properties of a Delone set, so each prefix of the farthest-first traversal forms a Delone set. == Applications == Rosenkrantz, Stearns & Lewis (1977) used the farthest-first traversal to define the farthest-insertion heuristic for the travelling salesman problem. This heuristic finds approximate solutions to the travelling salesman problem by building up a tour on a subset of points, adding one point at a time to the tour in the ordering given by a farthest-first traversal. To add each point to the tour, one edge of the previous tour is broken and replaced by a pair of edges through the added point, in the cheapest possible way. Although Rosenkrantz et al. prove only a logarithmic approximation ratio for this method, they show that in practice it often works better than other insertion methods with better provable approximation ratios. Later, the same sequence of points was popularized by Gonzalez (1985), who used it as part of greedy approximation algorithms for two problems in clustering, in which the goal is to partition a set of points into k clusters. One of the two problems that Gonzalez solve in this way seeks to minimize the maximum diameter of a cluster, while the other, known as the metric k-center problem, seeks to minimize the maximum radius, the distance from a chosen central point of a cluster to the farthest point from it in the same cluster. For instance, the k-center problem can be used to model the placement of fire stations within a city, in order to ensure that every address within the city can be reached quickly by a fire truck. For both clustering problems, Gonzalez chooses a set of k cluster centers by selecting the first k points of a farthest-first traversal, and then creates clusters by assigning each input point to the nearest cluster center. If r is the distance from the set of k selected centers to the next point at position k + 1 in the traversal, then with this clustering every point is within distance r of its center and every cluster has diameter at most 2r. However, the subset of k centers together with the next point are all at distance at least r from each other, and any k-clustering would put some two of these points into a single cluster, with one of them at distance at least r/2 from its center and with diameter at least r. Thus, Gonzalez's heuristic gives an approximation ratio of 2 for both clustering problems. Gonzalez's heuristic was independently rediscovered for the metric k-center problem by Dyer & Frieze (1985), who applied it more generally to weighted k-center problems. Another paper on the k-center problem from the same time, Hochbaum & Shmoys (1985), achieves the same approximation ratio of 2, but its techniques are different. Nevertheless, Gonzalez's heuristic, and the name "farthest-first traversal", are often incorrectly attributed to Hochbaum and Shmoys. For both the min-max diameter clustering problem and the metric k-center problem, these approximations are optimal: the existence of a polynomial-time heuristic with any constant approximation ratio less than 2 would imply that P = NP. As well as for clustering, the farthest-first traversal can also be used in another type of facility location problem, the max-min facility dispersion problem, in which the goal is to choose the locations of k different facilities so that they are as far apart from each other as possible. More precisely, the goal in this problem is to choose k points from a given metric space or a given set of candidate points, in such a way as to maximize the minimum pairwise distance between the selected points. Again, this can be approximated by choosing the first k points of a farthest-first traversal. If r denotes the distance of the kth point from all previous points, then every point of the metric space or the candidate set is within distance r of the first k − 1 points. By the pigeonhole principle, some two points of the optimal solution (whatever it is) must both be within distance r of the same point among these first k − 1 chosen points, and (by the triangle inequality) within distance 2r of each other. Therefore, the heuristic solution given by the farthest-first traversal is within a factor of two of optimal. Other applications of the farthest-first traversal include color quantization (clustering the colors in an image to a smaller set of representative colors), progressive scanning of images (choosing an order to display the pixels of an image so that prefixes of the ordering produce good lower-resolution versions of the whole image rather than filling in the image from top to bottom), point selection in the probabilistic roadmap method for motion planning, simplification of point clouds, generating masks for halftone images, hierarchical clustering, finding the similarities between polygon meshes of similar surfaces, choosing diverse and high-value observation targets for underwater robot exploration, fault detection in sensor networks, modeling phylogenetic diversity, matching vehicles in a heterogenous fleet to customer delivery requests, uniform distribution of geodetic observatories on the Earth's surface or of other types of sensor network, generation of virtual point lights in the instant radiosity computer graphics rendering method, and geometric range searching data structures. == Algorithms == === Greedy exact algorithm === The farthest-first traversal of a finite point set may be computed by a greedy algorithm that maintains the distance of each point from the previously selected points, performing the following steps: Initialize the sequence of selected points to the empty sequence, and the distances of each point to the selected points to infinity. While not all points have been selected, repeat the following steps: Scan the list of not-yet-selected points to find a point p that has the maximum distance from the selected points. Remove p from the not-yet-selected points and add it to the end of the sequence of selected points. For each remaining not-yet-selected point q, replace the distance stored for q by the minimum of its old value and the distance from p to q. For a set of n points, this algorithm takes O(n2) steps and O(n2) distance computations. === Approximations === A faster approximation algorithm, given by Har-Peled & Mendel (2006), applie

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  Software engine

  A software engine is a core component of a complex software system. The word "engine" is a metaphor of a car's engine. Thus a software engine is a complex subsystem; not unlike how a car engine functions. Software engines work in conjunction with other components of a process or system. They typically have an input and an output, and the productivity is usually linear to running speed. There is no formal guideline for what should be called an engine, but the term has become widespread in the software industry. == Notable examples == === Multi-engine systems === Mainstream web browsers have both a browser engine and a JavaScript engine. Video games are often based on a game engine. Some of these also have specialized physics or graphics engines.

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  Synaptic weight

  In neuroscience and computer science, synaptic weight refers to the strength or amplitude of a connection between two nodes, corresponding in biology to the amount of influence the firing of one neuron has on another. The term is typically used in artificial and biological neural network research. == Computation == In a computational neural network, a vector or set of inputs x {\displaystyle {\textbf {x}}} and outputs y {\displaystyle {\textbf {y}}} , or pre- and post-synaptic neurons respectively, are interconnected with synaptic weights represented by the matrix w {\displaystyle w} , where for a linear neuron y j = ∑ i w i j x i or y = w x {\displaystyle y_{j}=\sum _{i}w_{ij}x_{i}~~{\textrm {or}}~~{\textbf {y}}=w{\textbf {x}}} . where the rows of the synaptic matrix represent the vector of synaptic weights for the output indexed by j {\displaystyle j} . The synaptic weight is changed by using a learning rule, the most basic of which is Hebb's rule, which is usually stated in biological terms as Neurons that fire together, wire together. Computationally, this means that if a large signal from one of the input neurons results in a large signal from one of the output neurons, then the synaptic weight between those two neurons will increase. The rule is unstable, however, and is typically modified using such variations as Oja's rule, radial basis functions or the backpropagation algorithm. == Biology == For biological networks, the effect of synaptic weights is not as simple as for linear neurons or Hebbian learning. However, biophysical models such as BCM theory have seen some success in mathematically describing these networks. In the mammalian central nervous system, signal transmission is carried out by interconnected networks of nerve cells, or neurons. For the basic pyramidal neuron, the input signal is carried by the axon, which releases neurotransmitter chemicals into the synapse which is picked up by the dendrites of the next neuron, which can then generate an action potential which is analogous to the output signal in the computational case. The synaptic weight in this process is determined by several variable factors: How well the input signal propagates through the axon (see myelination), The amount of neurotransmitter released into the synapse and the amount that can be absorbed in the following cell (determined by the number of AMPA and NMDA receptors on the cell membrane and the amount of intracellular calcium and other ions), The number of such connections made by the axon to the dendrites, How well the signal propagates and integrates in the postsynaptic cell. The changes in synaptic weight that occur is known as synaptic plasticity, and the process behind long-term changes (long-term potentiation and depression) is still poorly understood. Hebb's original learning rule was originally applied to biological systems, but has had to undergo many modifications as a number of theoretical and experimental problems came to light.

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  SqueezeNet

  SqueezeNet is a deep neural network for image classification released in 2016. SqueezeNet was developed by researchers at DeepScale, University of California, Berkeley, and Stanford University. In designing SqueezeNet, the authors' goal was to create a smaller neural network with fewer parameters while achieving competitive accuracy. Their best-performing model achieved the same accuracy as AlexNet on ImageNet classification, but has a size 510x less than it. == Version history == SqueezeNet was originally released on February 22, 2016. This original version of SqueezeNet was implemented on top of the Caffe deep learning software framework. Shortly thereafter, the open-source research community ported SqueezeNet to a number of other deep learning frameworks. On February 26, 2016, Eddie Bell released a port of SqueezeNet for the Chainer deep learning framework. On March 2, 2016, Guo Haria released a port of SqueezeNet for the Apache MXNet framework. On June 3, 2016, Tammy Yang released a port of SqueezeNet for the Keras framework. In 2017, companies including Baidu, Xilinx, Imagination Technologies, and Synopsys demonstrated SqueezeNet running on low-power processing platforms such as smartphones, FPGAs, and custom processors. As of 2018, SqueezeNet ships "natively" as part of the source code of a number of deep learning frameworks such as PyTorch, Apache MXNet, and Apple CoreML. In addition, third party developers have created implementations of SqueezeNet that are compatible with frameworks such as TensorFlow. Below is a summary of frameworks that support SqueezeNet. == Relationship to other networks == === AlexNet === SqueezeNet was originally described in SqueezeNet: AlexNet-level accuracy with 50x fewer parameters and <0.5MB model size. AlexNet is a deep neural network that has 240 MB of parameters, and SqueezeNet has just 5 MB of parameters. This small model size can more easily fit into computer memory and can more easily be transmitted over a computer network. However, it's important to note that SqueezeNet is not a "squeezed version of AlexNet." Rather, SqueezeNet is an entirely different DNN architecture than AlexNet. What SqueezeNet and AlexNet have in common is that both of them achieve approximately the same level of accuracy when evaluated on the ImageNet image classification validation dataset. === Model compression === Model compression (e.g. quantization and pruning of model parameters) can be applied to a deep neural network after it has been trained. In the SqueezeNet paper, the authors demonstrated that a model compression technique called Deep Compression can be applied to SqueezeNet to further reduce the size of the parameter file from 5 MB to 500 KB. Deep Compression has also been applied to other DNNs, such as AlexNet and VGG. == Variants == Some of the members of the original SqueezeNet team have continued to develop resource-efficient deep neural networks for a variety of applications. A few of these works are noted in the following table. As with the original SqueezeNet model, the open-source research community has ported and adapted these newer "squeeze"-family models for compatibility with multiple deep learning frameworks. In addition, the open-source research community has extended SqueezeNet to other applications, including semantic segmentation of images and style transfer.

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  European Conference on Computer Vision

  The European Conference on Computer Vision (ECCV) is a biennial research conference with the proceedings published by Springer Science+Business Media. Similar to ICCV in scope and quality, it is held those years which ICCV is not. It is considered to be one of the top conferences in computer vision, alongside CVPR and ICCV, with an 'A' rating from the Australian Ranking of ICT Conferences and an 'A1' rating from the Brazilian ministry of education. The acceptance rate for ECCV 2010 was 24.4% for posters and 3.3% for oral presentations. Like other top computer vision conferences, ECCV has tutorial talks, technical sessions, and poster sessions. The conference is usually spread over five to six days with the main technical program occupying three days in the middle, and tutorial and workshops, focused on specific topics, being held in the beginning and at the end. The ECCV presents the Koenderink Prize annually to recognize fundamental contributions in computer vision. == Location == The conference is usually held in autumn in Europe.

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  Nolot

  Nolot is a chess test suite with 11 positions from real games. They were compiled by Pierre Nolot (French: [nɔ.lo]) for the French chess magazine Gambisco and posted on the rec.games.chess Usenet group in 1994. They were designed to be particularly hard to solve for chess engines to solve at the time, although modern engines can find a solution near-instantaneously. == Problem 1 == FEN: r3qb1k/1b4p1/p2pr2p/3n4/Pnp1N1N1/6RP/1B3PP1/1B1QR1K1 w - - 0 1 26.Nxh6!! c3 (26... Rxh6 27.Nxd6 Qh5 (best) 28.Rg5! Qxd1 29.Nf7+ Kg8 30.Nxh6+ Kh8 31.Rxd1 c3 32.Nf7+ Kg8 33.Bg6! Nf4 34.Bxc3 Nxg6 35.Bxb4 Kxf7 36.Rd7+ Kf6 37.Rxg6+ Kxg6 38.Rxb7 ±) 27.Nf5! cxb2 28.Qg4 Bc8 (28... g6!? 29.Kh2! 29.Qd7 30.Nh4 Bc6 31.Nc5! dxc 32.Rxe6 Nf6 33.Nxg6+ Kg7 34.Qg5 Nbd5 35.Ne5 Kh8 36.Nxd7 ±) 29.Qh4+ Rh6 30.Nxh6 gxh6 31.Kh2! Qe5 32.Ng5 Qf6 33.Re8 Bf5 34.Qxh6 (missing a mate in 6: 34.Nf7+ Qxf7 35.Qxh6+ Bh7 36.Rxa8 Nf6 37.Rxf8 Qxf8 38.Qxf8+ Ng8 39.Qg7#) 34...Qxh6 35.Nf7+ Kh7 36.Bxf5+ Qg6 37.Bxg6+ Kg7 38.Rxa8 Be7 39.Rb8 a5 40.Be4+ Kxf7 41.Bxd5+ 1–0 The best Novag computer, the Diablo 68000, finds 26. Nxh6 after seven and a half months (Pierre Nolot has let it run on the position for 14 months and one day, until a power failure stopped an analysis of over 80,000,000,000 nodes.) but for wrong reasons: it evaluates white's position as inferior and thinks this move would enable it to draw. Today Gambit Tiger 2.0 for example can find it quite quickly: Most free engines running on 64-bit processors in 2010 could solve this problem and the others in a few seconds. 1.Qd4 c3 2.Bxc3 Nxc3 3.Qxb4 Nxe4 4.Qxb7 Rb8 5.Qxb8 Qxb8 6.Bxe4 d5 7.Rb1 μ (-1.20) Depth: 12 00:00:09 6055 kN 1.Nxh6 c3 2.Nf5 cxb2 3.Qg4 Rb8 4.Nxg7 Rg6 5.Qxg6 Qxg6 6.Rxg6 Bxg7 7.Nxd6 ³ (-0.48) Depth: 12 00:00:21 14368 kN 1.Nxh6 c3 2.Nf5 cxb2 3.Qg4 Rc8 4.Nxg7 Rg6 5.Nxe8 Rxg4 6.Rxg4 Rxe8 7.Rg6 μ (-0.74) Depth: 13 00:00:55 38455 kN 1.Ne3 Rxe4 2.Bxe4 Qxe4 3.Nxd5 Qxd5 4.Qc1 Qf5 5.Qxh6+ Qh7 6.Qe6 Nd3 7.Re2 Nxb2 8.Rxb2 ³ (-0.58) Depth: 13 00:01:30 62979 kN 1.Ne3 Rxe4 ³ (-0.58) Depth: 14 00:02:02 84941 kN 1.Ne3 Nxe3 2.Rexe3 Bxe4 3.Qg4 Rg6 4.Qxe4 Qxe4 5.Bxe4 Rxg3 6.Rxg3 d5 7.Bf5 Re8 8.Bc3 ³ (-0.30) Depth: 15 00:03:05 128968 kN 1.Nxh6 ² (0.32) Depth: 15 00:07:58 350813 kN With the next ply showing a clear advantage. Stockfish 14dev 64bit 4CPU running on 2020 hardware recognises the significance of Nxh6!! in 1 second. Stockfish_21092606_x64_avx2: NNUE evaluation using nn-13406b1dcbe0.nnue enabled. 19/32 00:01 7708k 4882k +3,00 Nxh6 Rxh6 Nxd6 Qh5 Bg6 Qxd1 Nf7+ Kg8 Nxh6+ gxh6 Bh5+ Kh7 Rxd1 c3 Bxc3 Nxc3 Rd7+ Kh8 Rxb7 Ne4 Re3 Nxf2 Kxf2 Bc5 Ke2 Bxe3 Kxe3 Nd5+ Kf2 49/73 15:02 5118270k 5673k +6,15 Nxh6 Rxh6 Nxd6 Qh5 Rg5 Qxd1 Nf7+ Kg8 Nxh6+ Kh8 Rxd1 c3 Nf7+ Kg8 Bg6 Nf4 Bxc3 Nbd5 Rb1 Bc6 Bd2 Nxg6 Rxg6 Ne7 Rxc6 Nxc6 Rb6 Rc8 Ng5 a5 Ra6 Bb4 Be3 Ne5 Bd4 Nc6 Bb6 Bd2 h4 Kf8 Bc5+ Kg8 Be3 Bxe3 fxe3 Kf8 Kf2 Ke7 Nf3 Kd7 Rb6 Ne7 Rb5 Kd6 Rxa5 Rc2+ Kg3 Re2 Nd4 Rxe3+ Kf4 Rd3 Nf5+ Kc7 Nxe7 == Problem 2 == FEN: r4rk1/pp1n1p1p/1nqP2p1/2b1P1B1/4NQ2/1B3P2/PP2K2P/2R5 w - - 0 1 22.Rxc5!! Nxc5 23.Nf6+ Kh8 24.Qh4 Qb5+ (computers think there is perpetual check here, but...) 25.Ke3! 25... h5 26.Nxh5 Qxb3+ (26... d5+ 27.Bxd5 Qd3 28.Kf2 Ne4+ 29.Bxe4 Qd4+ 30.Kg2 Qxb2+ 31.Kh3 ±) and White won in 41 moves. Today Deep Junior 8.ZX for example finds it very quickly (around 1 minute): 1.Kd1 Rac8 2.Bh6 Qb5 3.Rc3 Qf1+ 4.Kc2 Rc6 5.Bxf8 −+ (-2.11) Depth: 12 00:00:04 10422 kN 1.Nxc5 Nxc5 2.Rxc5 Qxc5 3.e6 Rae8 4.e7 Nc8 5.Kf1 Nxd6 6.Bf6 b5 −+ (-2.10) Depth: 12 00:00:14 25054 kN 1.Bf6! μ (-1.35) Depth: 12 00:00:17 34601 kN 1.Bf6 Qb5+ 2.Ke1 Bb4+ 3.Kf2 Bc5+ = (0.00) Depth: 12 00:00:20 34601 kN 1.Bf6 Qb5+ 2.Ke1 Nxf6 3.Nxf6+ Kg7 4.Nh5+ gxh5 5.Qf6+ Kg8 6.Qg5+ Kh8 7.Qf6+ = (0.00) Depth: 15 00:01:01 130544 kN 1.Rxc5! = (0.15) Depth: 15 00:01:12 145875 kN 1.Rxc5 Nxc5 2.Nf6+ Kh8 3.Qh4 Qb5+ 4.Ke3 h5 5.Nxh5 Qd3+ 6.Kf2 Ne4+ 7.fxe4 Qd4+ 8.Kf1 Qd3+ 9.Ke1 Qb1+ 10.Bd1 ± (2.18) Depth: 15 00:01:18 145875 kN Stockfish 14dev 64bit 4CPU running on 2020 hardware recognises the significance of Rxc5!! in 1 second. Stockfish_21092606_x64_avx2: NNUE evaluation using nn-13406b1dcbe0.nnue enabled. 21/25 00:01 5822k 5545k +6,61 Rxc5 Qxc5 Nxc5 Nxc5 Bh6 Nbd7 Bxf8 Rxf8 Qe3 Rc8 f4 Nxe5 Qxe5 Ne6 Bxe6 Rc2+ Kd3 Rxh2 46/86 11:27 5057055k 7355k +7,61 Rxc5 Qxc5 Nxc5 Nxc5 Bf6 Ne6 Qh6 Nd4+ Kf2 Nf5 Qg5 Nd7 h4 Nxf6 Qxf6 Ng7 d7 b5 Bd5 Rab8 b4 Nh5 Bxf7+ Rxf7 d8R+ Rxd8 Qxd8+ Rf8 Qd5+ Kg7 e6 Kf6 Qd7 Ng7 Qd4+ Kxe6 Qxg7 Rf7 Qc3 Ke7 Qc5+ Ke8 Qc8+ Ke7 h5 gxh5 Kg3 h4+ Kh2 h6 Qc5+ Kf6 Qxb5 Kg7 f4 Rxf4 Qe5+ Rf6 b5 h3 Qd4 Kg8 Qxf6 h5 Blacks 22. .. Nxc5 is suboptimal and leads faster mate 77/44 09:18 6987714k 12518k +M22 Nf6+ Kh8 Qh4 Qb5+ Ke3 Qxb3+ axb3 h5 Nxh5 Nd5+ Kd4 Ne6+ Kxd5 Nxg5 Qxg5 gxh5 f4 Rad8 f5 f6 Qxh5+ Kg7 Qg6+ Kh8 e6 b6 e7 Rb8 exf8Q+ Rxf8 Ke6 b5 Ke7 Rb8 Qh5+ Kg7 Qf7+ Kh8 Kxf6 Rf8 Qxf8+ Kh7 Qg7+ == Problem 3 == FEN: r2qk2r/ppp1b1pp/2n1p3/3pP1n1/3P2b1/2PB1NN1/PP4PP/R1BQK2R w KQkq - 0 1 12.Nxg5!! Bxd1 13.Nxe6 Qb8 14.Nxg7+!! Kf8 15.Bh6! Bg4 16.0-0+ Kg8 17.Rf4 ± White wins with a queen sac but black has defensive resources. Stockfish 8 64bit 3CPU running on 2016 hardware recognizes the significance of Nxg5!! in 55 seconds. Stockfish 14 dev (Stockfish_21092606_x64_avx2) 64bit 4CPU running on 2020 hardware recognizes the significance of Nxg5!! in 1 second. NNUE evaluation using nn-13406b1dcbe0.nnue enabled. 21/34 00:01 8291k 4530k +2,78 Nxg5 Bxd1 Nxe6 Qb8 Nxg7+ Kd8 Kxd1 b5 N3f5 Bf8 Rf1 Kc8 Nh5 Kb7 Bxb5 Ne7 g4 a6 Ba4 Nxf5 gxf5 Ka7 Nf4 c5 47/59 37:49 10390430k 4578k +3,16 Nxg5 Bxd1 Nxe6 Qb8 Nxg7+ Kd8 Kxd1 b5 Rf1 Kc8 N3f5 Bf8 Ne6 Kd7 Nf4 Ne7 g4 a5 Ke2 Qb7 h4 Ra6 a3 Kc8 Be3 Kb8 Kf3 Rb6 Bd2 Qc8 Kg3 c5 Be3 c4 Nxe7 Bxe7 Bf5 Qd8 h5 Qg8 Kh3 Bg5 Rf3 Ra6 Raf1 b4 Nxd5 Qxd5 Bxg5 bxc3 bxc3 Rb6 Be3 Rb3 Blacks 14 .. Kf8 is suboptimal and leads loss fast 41/68 06:31 3269727k 8350k +9,28 Bh6 Kg8 Rxd1 Bf8 N3h5 Bxg7 Nxg7 Qf8 Nf5 Ne7 Bxf8 Nxf5 Bxf5 Rxf8 Be6+ Kg7 Rd3 Rf4 Bxd5 c6 Rg3+ Kf8 Rf3 Rxf3 Bxf3 Kg7 Rf1 Re8 Be4 Re6 Ke2 a5 Ke3 Rh6 h3 a4 Kf4 Re6 h4 Re8 Ke3 h6 h5 Rf8 Rxf8 Kxf8 == Problem 4 == FEN: r1b1kb1r/1p1n1ppp/p2ppn2/6BB/2qNP3/2N5/PPP2PPP/R2Q1RK1 w kq - 0 1 10.Nxe6!! Qxe6 11.Nd5 Kd8 12.Bg4 Qe5 13.f4 Qxe4 (13...Qxb2 stronger but not sufficient: 14.Bxd7 Bxd7 15.Rb1 Qa3 16.Nxf6 Bb5 17.Qd4 Qc5 18.Rfd1 ±) 14.Bxd7 Bxd7 15.Nxf6 gxf6 16.Bxf6+ Kc7 17.Bxh8 and Black resigned on move 27. Stockfish 14dev 64bit 4CPU running on 2020 hardware recognises the significance of 10.Nxe6 in 1 second. Stockfish_21092606_x64_avx2: NNUE evaluation using nn-13406b1dcbe0.nnue enabled. 22/37 00:01 6955k 5367k +4,00 Nxe6 Qxe6 Nd5 Kd8 Bg4 Qe5 f4 Qxb2 Rb1 Qa3 Bxd7 Bxd7 Nxf6 Bb5 Rf3 Qxa2 c4 Bxc4 Rf2 Qa5 Nd5+ f6 Nxf6 Kc7 Rc1 b5 Qd5 gxf6 Bxf6 Kb8 Rxc4 Qe1+ Rf1 51/70 47:10 14538911k 5137k +5,76 Nxe6 Qxe6 Nd5 Kd8 Bg4 Qe5 f4 Qxe4 Bxd7 Bxd7 Nxf6 Qf5 Qd4 Kc8 Nd5 Bc6 c4 f6 Nb6+ Kb8 Bh4 Be7 Rae1 Bd8 Nxa8 Kxa8 Bf2 Kb8 Qxd6+ Bc7 Ba7+ Kc8 Qe6+ Qxe6 Rxe6 h5 h4 Rd8 Re7 g6 Be3 Ba5 Kf2 Rd6 Rc1 Bd8 Rg7 Be4 Rg8 Kd7 c5 Rd3 Rc4 Bd5 Rg7+ Ke6 Rd4 Rxd4 Bxd4 Kf5 Rd7 Bc6 Rxd8 Kxf4 Bxf6 == Problem 5 == FEN: r2qrb1k/1p1b2p1/p2ppn1p/8/3NP3/1BN5/PPP3QP/1K3RR1 w - - 0 1 21.e5!! dxe5 22.Ne4! Nh5 23.Qg6!? (stronger is 23.Qg4!! Nf4 24.Nf3 Qc7 25.Nh4 ± ) 23...exd4? (23...Nf4 24.Rxf4! exf4 25.Nf3! Qb6 26.Rg5!! covering b5 and threatening Nf6 or Ne5-f7+) 24.Ng5 1−0 Stockfish 8 64bit 3CPU running on 2016 hardware recognises the significance of 21.e5 in 5 seconds. Stockfish 12 dev (Stockfish_20062212_x64_modern) 64bit 1CPU running on 2016 hardware recognizes the significance of 21.e5 in 11 seconds. 25/42 00:06 7 963k 1309k +6,93 e5 Nh5 Ne4 dxe5 Nf3 Nf4 Qg4 Qc7 Nh4 Bc6 Nf6 g5 Rxf4 exf4 Qh5 Qe7 Ng6+ Kg7 Nxe7 Rxe7 Ng4 37/62 03:12 298 083k 1545k +10,70 e5 Ng4 Qxg4 Qg5 Qh3 Qxe5 Nde2 g5 Rxf8+ Kg7 Rff1 Rf8 Re1 Qf5 Qg3 Rad8 Nd4 Qf4 Nxe6+ Bxe6 Rxe6 Qxg3 == Problem 6 == FEN: rnbqk2r/1p3ppp/p7/1NpPp3/QPP1P1n1/P4N2/4KbPP/R1B2B1R b kq - 0 1 13... axb5!! offers an exchange to keep the white queen out of play. 14.Qxa8 Bd4 15.Nxd4 cxd4 16.Qxb8 0-0! 17.Ke1 Qh4 18.g3 Qf6 19.Bf4 g5? (Ivanchuk found 19...d3! during post-game analysis.) 20.Rc1 exf4 21.Qxf4 Qd4 22.Rd1 bxc4 23.e5 Qc3+ 24.Rd2 Re8 25.Bxd3 cxd3 −+ Tasc R30 finds 19... d3! in 2 1/2 hours. 19... Bf5!! is even stronger than 19... d3. Position is already lost at 19... d3 +8.00 for black, ... Bf5 not much better Stockfish 14dev 64bit 4CPU running on 2020 hardware recognises the significance of axb5!! in 1 second. Stockfish_21092606_x64_avx2: NNUE evaluation using nn-13406b1dcbe0.nnue enabled. 21/28 00:01 9264k 4714k -1,22 axb5 Qxa8 Bd4 Nxd4 cxd4 h3 Nf6 Bg5 0-0 cxb5 h6 Bxf6 Qxf6 Re1 Nd7 Kd1 Qg6 Qa4 Qg3 Qc2 Qxa3 Bd3 Qxb4 Qb1 46/67 1:05:00 18113493k 4644k -2,40 axb5 Qxa8 Bd4 h3 Nf6 Nxd4 exd4 Kf2 Nxe4+ Kg1 Nd7 Bg5 Qxg5 Qxc8+ Ke7 Qc7 Qe5 d6+ Qxd6 Qxd6+ Kxd6 bxc5+ Ndxc5 cxb5 d3 h4 d2 Rh3 Ke5 Be2 f5 Ra2 Rd8 Bd1 Rd4 Re3 f4 Re2 b6 a4 Kd6 Rc2 Kd5 Ra2 h6 Rb2 Nxa4 Bxa4 Rxa4 Rexd2+ Nxd2 Rxd2+ Kc4 Rd7 g6 == Problem 7 == FEN 1r1bk2r/2R2ppp/p3p3/1b2P2q/4QP2/4N3/1B4PP/3R2K1 w k - 0 1 1.Rxd8+!! Rxd8 (1...Kxd8 2.Ra7! Qe2 3.Qd4+ Ke8 4.h3 Qe1+ 5.Kh2 Rd8 6.Qc5 Qh4 7.Ba3 Rd7 8.Ra8+ Rd8 9.g3 1−0)

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  Evolutionary algorithm

  Evolutionary algorithms (EA) reproduce essential elements of biological evolution in a computer algorithm in order to solve "difficult" problems, at least approximately, for which no exact or satisfactory solution methods are known. They are metaheuristics and population-based bio-inspired algorithms and evolutionary computation, which itself are part of the field of computational intelligence. The mechanisms of biological evolution that an EA mainly imitates are reproduction, mutation, recombination and selection. Candidate solutions to the optimization problem play the role of individuals in a population, and the fitness function determines the quality of the solutions (see also loss function). Evolution of the population then takes place after the repeated application of the above operators. Evolutionary algorithms often perform well approximating solutions to all types of problems because they ideally do not make any assumption about the underlying fitness landscape. Techniques from evolutionary algorithms applied to the modeling of biological evolution are generally limited to explorations of microevolution (microevolutionary processes) and planning models based upon cellular processes. In most real applications of EAs, computational complexity is a prohibiting factor. In fact, this computational complexity is due to fitness function evaluation. Fitness approximation is one of the solutions to overcome this difficulty. However, seemingly simple EA can solve often complex problems; therefore, there may be no direct link between algorithm complexity and problem complexity. == Generic definition == The following is an example of a generic evolutionary algorithm: Randomly generate the initial population of individuals, the first generation. Evaluate the fitness of each individual in the population. Check, if the goal is reached and the algorithm can be terminated. Select individuals as parents, preferably of higher fitness. Produce offspring with optional crossover (mimicking reproduction). Apply mutation operations on the offspring. Select individuals preferably of lower fitness for replacement with new individuals (mimicking natural selection). Return to 2 == Types == Similar techniques differ in genetic representation and other implementation details, and the nature of the particular applied problem. Genetic algorithm – This is the most popular type of EA. One seeks the solution of a problem in the form of strings of numbers (traditionally binary, although the best representations are usually those that reflect something about the problem being solved), by applying operators such as recombination and mutation (sometimes one, sometimes both). This type of EA is often used in optimization problems. Genetic programming – Here the solutions are in the form of computer programs, and their fitness is determined by their ability to solve a computational problem. There are many variants of Genetic Programming: Cartesian genetic programming Gene expression programming Grammatical evolution Linear genetic programming Multi expression programming Evolutionary programming – Similar to evolution strategy, but with a deterministic selection of all parents. Evolution strategy (ES) – Works with vectors of real numbers as representations of solutions, and typically uses self-adaptive mutation rates. The method is mainly used for numerical optimization, although there are also variants for combinatorial tasks. CMA-ES Natural evolution strategy Differential evolution – Based on vector differences and is therefore primarily suited for numerical optimization problems. Coevolutionary algorithm – Similar to genetic algorithms and evolution strategies, but the created solutions are compared on the basis of their outcomes from interactions with other solutions. Solutions can either compete or cooperate during the search process. Coevolutionary algorithms are often used in scenarios where the fitness landscape is dynamic, complex, or involves competitive interactions. Neuroevolution – Similar to genetic programming but the genomes represent artificial neural networks by describing structure and connection weights. The genome encoding can be direct or indirect. Learning classifier system – Here the solution is a set of classifiers (rules or conditions). A Michigan-LCS evolves at the level of individual classifiers whereas a Pittsburgh-LCS uses populations of classifier-sets. Initially, classifiers were only binary, but now include real, neural net, or S-expression types. Fitness is typically determined with either a strength or accuracy based reinforcement learning or supervised learning approach. Quality–Diversity algorithms – QD algorithms simultaneously aim for high-quality and diverse solutions. Unlike traditional optimization algorithms that solely focus on finding the best solution to a problem, QD algorithms explore a wide variety of solutions across a problem space and keep those that are not just high performing, but also diverse and unique. == Theoretical background == The following theoretical principles apply to all or almost all EAs. === No free lunch theorem === The no free lunch theorem of optimization states that all optimization strategies are equally effective when the set of all optimization problems is considered. Under the same condition, no evolutionary algorithm is fundamentally better than another. This can only be the case if the set of all problems is restricted. This is exactly what is inevitably done in practice. Therefore, to improve an EA, it must exploit problem knowledge in some form (e.g. by choosing a certain mutation strength or a problem-adapted coding). Thus, if two EAs are compared, this constraint is implied. In addition, an EA can use problem specific knowledge by, for example, not randomly generating the entire start population, but creating some individuals through heuristics or other procedures. Another possibility to tailor an EA to a given problem domain is to involve suitable heuristics, local search procedures or other problem-related procedures in the process of generating the offspring. This form of extension of an EA is also known as a memetic algorithm. Both extensions play a major role in practical applications, as they can speed up the search process and make it more robust. === Convergence === For EAs in which, in addition to the offspring, at least the best individual of the parent generation is used to form the subsequent generation (so-called elitist EAs), there is a general proof of convergence under the condition that an optimum exists. Without loss of generality, a maximum search is assumed for the proof: From the property of elitist offspring acceptance and the existence of the optimum it follows that per generation k {\displaystyle k} an improvement of the fitness F {\displaystyle F} of the respective best individual x ′ {\displaystyle x'} will occur with a probability P > 0 {\displaystyle P>0} . Thus: F ( x 1 ′ ) ≤ F ( x 2 ′ ) ≤ F ( x 3 ′ ) ≤ ⋯ ≤ F ( x k ′ ) ≤ ⋯ {\displaystyle F(x'_{1})\leq F(x'_{2})\leq F(x'_{3})\leq \cdots \leq F(x'_{k})\leq \cdots } I.e., the fitness values represent a monotonically non-decreasing sequence, which is bounded due to the existence of the optimum. From this follows the convergence of the sequence against the optimum. Since the proof makes no statement about the speed of convergence, it is of little help in practical applications of EAs. But it does justify the recommendation to use elitist EAs. However, when using the usual panmictic population model, elitist EAs tend to converge prematurely more than non-elitist ones. In a panmictic population model, mate selection (see step 4 of the generic definition) is such that every individual in the entire population is eligible as a mate. In non-panmictic populations, selection is suitably restricted, so that the dispersal speed of better individuals is reduced compared to panmictic ones. Thus, the general risk of premature convergence of elitist EAs can be significantly reduced by suitable population models that restrict mate selection. === Virtual alphabets === With the theory of virtual alphabets, David E. Goldberg showed in 1990 that by using a representation with real numbers, an EA that uses classical recombination operators (e.g. uniform or n-point crossover) cannot reach certain areas of the search space, in contrast to a coding with binary numbers. This results in the recommendation for EAs with real representation to use arithmetic operators for recombination (e.g. arithmetic mean or intermediate recombination). With suitable operators, real-valued representations are more effective than binary ones, contrary to earlier opinion. == Comparison to other concepts == === Biological processes === A possible limitation of many evolutionary algorithms is their lack of a clear genotype–phenotype distinction. In nature, the fertilized egg cell undergoes a complex process known as embryogenesis to become a mature p

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  Variable-order Bayesian network

  Variable-order Bayesian network (VOBN) models provide an important extension of both the Bayesian network models and the variable-order Markov models. VOBN models are used in machine learning in general and have shown great potential in bioinformatics applications. These models extend the widely used position weight matrix (PWM) models, Markov models, and Bayesian network (BN) models. In contrast to the BN models, where each random variable depends on a fixed subset of random variables, in VOBN models these subsets may vary based on the specific realization of observed variables. The observed realizations are often called the context and, hence, VOBN models are also known as context-specific Bayesian networks. The flexibility in the definition of conditioning subsets of variables turns out to be a real advantage in classification and analysis applications, as the statistical dependencies between random variables in a sequence of variables (not necessarily adjacent) may be taken into account efficiently, and in a position-specific and context-specific manner.

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