Gene expression programming (GEP) in computer programming is an evolutionary algorithm that creates computer programs or models. These computer programs are complex tree structures that learn and adapt by changing their sizes, shapes, and composition, much like a living organism. And like living organisms, the computer programs of GEP are also encoded in simple linear chromosomes of fixed length. Thus, GEP is a genotype–phenotype system, benefiting from a simple genome to keep and transmit the genetic information and a complex phenotype to explore the environment and adapt to it. == Background == Evolutionary algorithms use populations of individuals, select individuals according to fitness, and introduce genetic variation using one or more genetic operators. Their use in artificial computational systems dates back to the 1950s where they were used to solve optimization problems (e.g. Box 1957 and Friedman 1959). But it was with the introduction of evolution strategies by Rechenberg in 1965 that evolutionary algorithms gained popularity. A good overview text on evolutionary algorithms is the book "An Introduction to Genetic Algorithms" by Mitchell (1996). Gene expression programming belongs to the family of evolutionary algorithms and is closely related to genetic algorithms and genetic programming. From genetic algorithms it inherited the linear chromosomes of fixed length; and from genetic programming it inherited the expressive parse trees of varied sizes and shapes. In gene expression programming the linear chromosomes work as the genotype and the parse trees as the phenotype, creating a genotype/phenotype system. This genotype/phenotype system is multigenic, thus encoding multiple parse trees in each chromosome. This means that the computer programs created by GEP are composed of multiple parse trees. Because these parse trees are the result of gene expression, in GEP they are called expression trees. Masood Nekoei, et al. utilized this expression programming style in ABC optimization to conduct ABCEP as a method that outperformed other evolutionary algorithms.ABCEP == Encoding: the genotype == The genome of gene expression programming consists of a linear, symbolic string or chromosome of fixed length composed of one or more genes of equal size. These genes, despite their fixed length, code for expression trees of different sizes and shapes. An example of a chromosome with two genes, each of size 9, is the string (position zero indicates the start of each gene): 012345678012345678 L+a-baccdcLabacd where “L” represents the natural logarithm function and “a”, “b”, “c”, and “d” represent the variables and constants used in a problem. == Expression trees: the phenotype == As shown above, the genes of gene expression programming have all the same size. However, these fixed length strings code for expression trees of different sizes. This means that the size of the coding regions varies from gene to gene, allowing for adaptation and evolution to occur smoothly. For example, the mathematical expression: ( a − b ) ( c + d ) {\displaystyle {\sqrt {(a-b)(c+d)}}\,} can also be represented as an expression tree: where "Q” represents the square root function. This kind of expression tree consists of the phenotypic expression of GEP genes, whereas the genes are linear strings encoding these complex structures. For this particular example, the linear string corresponds to: 01234567 Q-+abcd which is the straightforward reading of the expression tree from top to bottom and from left to right. These linear strings are called k-expressions (from Karva notation). Going from k-expressions to expression trees is also very simple. For example, the following k-expression: 01234567890 Qb+baQba is composed of two different terminals (the variables “a” and “b”), two different functions of two arguments (“” and “+”), and a function of one argument (“Q”). Its expression gives: == K-expressions and genes == The k-expressions of gene expression programming correspond to the region of genes that gets expressed. This means that there might be sequences in the genes that are not expressed, which is indeed true for most genes. The reason for these noncoding regions is to provide a buffer of terminals so that all k-expressions encoded in GEP genes correspond always to valid programs or expressions. The genes of gene expression programming are therefore composed of two different domains – a head and a tail – each with different properties and functions. The head is used mainly to encode the functions and variables chosen to solve the problem at hand, whereas the tail, while also used to encode the variables, provides essentially a reservoir of terminals to ensure that all programs are error-free. For GEP genes the length of the tail is given by the formula: t = h ( n max − 1 ) + 1 {\displaystyle t=h(n_{\max }-1)+1} where h is the head's length and nmax is maximum arity. For example, for a gene created using the set of functions F = {Q, +, −, ∗, /} and the set of terminals T = {a, b}, nmax = 2. And if we choose a head length of 15, then t = 15 (2–1) + 1 = 16, which gives a gene length g of 15 + 16 = 31. The randomly generated string below is an example of one such gene: 0123456789012345678901234567890 b+a-aQab+//+b+babbabbbababbaaa It encodes the expression tree: which, in this case, only uses 8 of the 31 elements that constitute the gene. It's not hard to see that, despite their fixed length, each gene has the potential to code for expression trees of different sizes and shapes, with the simplest composed of only one node (when the first element of a gene is a terminal) and the largest composed of as many nodes as there are elements in the gene (when all the elements in the head are functions with maximum arity). It's also not hard to see that it is trivial to implement all kinds of genetic modification (mutation, inversion, insertion, recombination, and so on) with the guarantee that all resulting offspring encode correct, error-free programs. == Multigenic chromosomes == The chromosomes of gene expression programming are usually composed of more than one gene of equal length. Each gene codes for a sub-expression tree (sub-ET) or sub-program. Then the sub-ETs can interact with one another in different ways, forming a more complex program. The figure shows an example of a program composed of three sub-ETs. In the final program the sub-ETs could be linked by addition or some other function, as there are no restrictions to the kind of linking function one might choose. Some examples of more complex linkers include taking the average, the median, the midrange, thresholding their sum to make a binomial classification, applying the sigmoid function to compute a probability, and so on. These linking functions are usually chosen a priori for each problem, but they can also be evolved elegantly and efficiently by the cellular system of gene expression programming. == Cells and code reuse == In gene expression programming, homeotic genes control the interactions of the different sub-ETs or modules of the main program. The expression of such genes results in different main programs or cells, that is, they determine which genes are expressed in each cell and how the sub-ETs of each cell interact with one another. In other words, homeotic genes determine which sub-ETs are called upon and how often in which main program or cell and what kind of connections they establish with one another. === Homeotic genes and the cellular system === Homeotic genes have exactly the same kind of structural organization as normal genes and they are built using an identical process. They also contain a head domain and a tail domain, with the difference that the heads contain now linking functions and a special kind of terminals – genic terminals – that represent the normal genes. The expression of the normal genes results as usual in different sub-ETs, which in the cellular system are called ADFs (automatically defined functions). As for the tails, they contain only genic terminals, that is, derived features generated on the fly by the algorithm. For example, the chromosome in the figure has three normal genes and one homeotic gene and encodes a main program that invokes three different functions a total of four times, linking them in a particular way. From this example it is clear that the cellular system not only allows the unconstrained evolution of linking functions but also code reuse. And it shouldn't be hard to implement recursion in this system. === Multiple main programs and multicellular systems === Multicellular systems are composed of more than one homeotic gene. Each homeotic gene in this system puts together a different combination of sub-expression trees or ADFs, creating multiple cells or main programs. For example, the program shown in the figure was created using a cellular system with two cells and three normal genes. The applications of these multicellular systems are mu
Showbox.com
Showbox is an online video streaming platform that enables users to stream and download many videos, commonly movies and TV shows, for free. == History == The company opened the platforms to users who registered from its beta in late 2015. The platform was officially launched in February 2016, enabling any visitor to sign up and create videos online. In April 2016, Showbox was featured on the Product Hunt website, coming to the top of the website's lists for that day and week with over 1400 upvotes from the Product Hunt community. Also in April 2016, Showbox partnered with YouTube's leading multi-channel networks, including Fullscreen, BroadbandTV, StyleHaul, AwesomenessTV, and BuzzMyVideos, to enable their communities of creators to access the platform. In June 2016, the company launched Showbox For Brands, a business-oriented video creation platform, enabling companies to create video content in-house and with their communities and influencers. In March 2017, the company launched Showbox Engage, a use case of its B2B product launched in 2016, enabling companies to launch user-generated content campaigns with their communities. In April 2017, Showbox and the United Nations announced a partnership around the 70th anniversary of the declaration of human rights, with an annual, ongoing global campaign in 135 languages, inviting people worldwide to create their part of the declaration in a video from anywhere around the world. In November 2017, Showbox partnered with the Ad:tech and Digital Marketing World Forum conferences (DMWF) in New York to provide their users and communities with a User Generated Content video solution. == Technology == Showbox's video creation technology includes an online green screen feature, proprietary computer vision algorithms, deep learning technology to support the automatic creation of videos in the cloud, and advanced video composition, including special effects. == Coverage and awards == In March 2015, Showbox was nominated as one of the 10 Israeli startups to take over our TV screens this year. In July 2016, Showbox won the Publicis90 award as part of Publicis' "global initiative to foster digital entrepreneurship". In March 2017, Showbox was chosen as one of The Culture Trip's 10 startups to watch for in 2017.
Multiple sequence alignment
Multiple sequence alignment (MSA) is the process or the result of sequence alignment of three or more biological sequences, generally protein, DNA, or RNA. These alignments are used to infer evolutionary relationships via phylogenetic analysis and can highlight homologous features between sequences. Alignments highlight mutation events such as point mutations (single amino acid or nucleotide changes), insertion mutations and deletion mutations, and alignments are used to assess sequence conservation and infer the presence and activity of protein domains, tertiary structures, secondary structures, and individual amino acids or nucleotides. Multiple sequence alignments require more sophisticated methodologies than pairwise alignments, as they are more computationally complex. Most multiple sequence alignment programs use heuristic methods rather than global optimization because identifying the optimal alignment between more than a few sequences of moderate length is prohibitively computationally expensive. However, heuristic methods generally cannot guarantee high-quality solutions and have been shown to fail to yield near-optimal solutions on benchmark test cases. == Problem statement == Given m {\displaystyle m} sequences S i {\displaystyle S_{i}} , i = 1 , ⋯ , m {\displaystyle i=1,\cdots ,m} similar to the form below: S := { S 1 = ( S 11 , S 12 , … , S 1 n 1 ) S 2 = ( S 21 , S 22 , ⋯ , S 2 n 2 ) ⋮ S m = ( S m 1 , S m 2 , … , S m n m ) {\displaystyle S:={\begin{cases}S_{1}=(S_{11},S_{12},\ldots ,S_{1n_{1}})\\S_{2}=(S_{21},S_{22},\cdots ,S_{2n_{2}})\\\,\,\,\,\,\,\,\,\,\,\vdots \\S_{m}=(S_{m1},S_{m2},\ldots ,S_{mn_{m}})\end{cases}}} A multiple sequence alignment is taken of this set of sequences S {\displaystyle S} by inserting any amount of gaps needed into each of the S i {\displaystyle S_{i}} sequences of S {\displaystyle S} until the modified sequences, S i ′ {\displaystyle S'_{i}} , all conform to length L ≥ max { n i ∣ i = 1 , … , m } {\displaystyle L\geq \max\{n_{i}\mid i=1,\ldots ,m\}} and no values in the sequences of S {\displaystyle S} of the same column consists of only gaps. The mathematical form of an MSA of the above sequence set is shown below: S ′ := { S 1 ′ = ( S 11 ′ , S 12 ′ , … , S 1 L ′ ) S 2 ′ = ( S 21 ′ , S 22 ′ , … , S 2 L ′ ) ⋮ S m ′ = ( S m 1 ′ , S m 2 ′ , … , S m L ′ ) {\displaystyle S':={\begin{cases}S'_{1}=(S'_{11},S'_{12},\ldots ,S'_{1L})\\S'_{2}=(S'_{21},S'_{22},\ldots ,S'_{2L})\\\,\,\,\,\,\,\,\,\,\,\vdots \\S'_{m}=(S'_{m1},S'_{m2},\ldots ,S'_{mL})\end{cases}}} To return from each particular sequence S i ′ {\displaystyle S'_{i}} to S i {\displaystyle S_{i}} , remove all gaps. == Graphing approach == A general approach when calculating multiple sequence alignments is to use graphs to identify all of the different alignments. When finding alignments via graph, a complete alignment is created in a weighted graph that contains a set of vertices and a set of edges. Each of the graph edges has a weight based on a certain heuristic that helps to score each alignment or subset of the original graph. === Tracing alignments === When determining the best suited alignments for each MSA, a trace is usually generated. A trace is a set of realized, or corresponding and aligned, vertices that has a specific weight based on the edges that are selected between corresponding vertices. When choosing traces for a set of sequences it is necessary to choose a trace with a maximum weight to get the best alignment of the sequences. == Alignment methods == There are various alignment methods used within multiple sequence to maximize scores and correctness of alignments. Each is usually based on a certain heuristic with an insight into the evolutionary process. Most try to replicate evolution to get the most realistic alignment possible to best predict relations between sequences. === Dynamic programming === A direct method for producing an MSA uses the dynamic programming technique to identify the globally optimal alignment solution. For proteins, this method usually involves two sets of parameters: a gap penalty and a substitution matrix assigning scores or probabilities to the alignment of each possible pair of amino acids based on the similarity of the amino acids' chemical properties and the evolutionary probability of the mutation. For nucleotide sequences, a similar gap penalty is used, but a much simpler substitution matrix, wherein only identical matches and mismatches are considered, is typical. The scores in the substitution matrix may be either all positive or a mix of positive and negative in the case of a global alignment, but must be both positive and negative, in the case of a local alignment. For n individual sequences, the naive method requires constructing the n-dimensional equivalent of the matrix formed in standard pairwise sequence alignment. The search space thus increases exponentially with increasing n and is also strongly dependent on sequence length. Expressed with the big O notation commonly used to measure computational complexity, a naïve MSA takes O(LengthNseqs) time to produce. To find the global optimum for n sequences this way has been shown to be an NP-complete problem. In 1989, based on Carrillo-Lipman Algorithm, Altschul introduced a practical method that uses pairwise alignments to constrain the n-dimensional search space. In this approach pairwise dynamic programming alignments are performed on each pair of sequences in the query set, and only the space near the n-dimensional intersection of these alignments is searched for the n-way alignment. The MSA program optimizes the sum of all of the pairs of characters at each position in the alignment (the so-called sum of pair score) and has been implemented in a software program for constructing multiple sequence alignments. In 2019, Hosseininasab and van Hoeve showed that by using decision diagrams, MSA may be modeled in polynomial space complexity. === Progressive alignment construction === The most widely used approach to multiple sequence alignments uses a heuristic search known as progressive technique (also known as the hierarchical or tree method) developed by Da-Fei Feng and Doolittle in 1987. Progressive alignment builds up a final MSA by combining pairwise alignments beginning with the most similar pair and progressing to the most distantly related. All progressive alignment methods require two stages: a first stage in which the relationships between the sequences are represented as a phylogenetic tree, called a guide tree, and a second step in which the MSA is built by adding the sequences sequentially to the growing MSA according to the guide tree. The initial guide tree is determined by an efficient clustering method such as neighbor-joining or unweighted pair group method with arithmetic mean (UPGMA), and may use distances based on the number of identical two-letter sub-sequences (as in FASTA rather than a dynamic programming alignment). Progressive alignments are not guaranteed to be globally optimal. The primary problem is that when errors are made at any stage in growing the MSA, these errors are then propagated through to the final result. Performance is also particularly bad when all of the sequences in the set are rather distantly related. Most modern progressive methods modify their scoring function with a secondary weighting function that assigns scaling factors to individual members of the query set in a nonlinear fashion based on their phylogenetic distance from their nearest neighbors. This corrects for non-random selection of the sequences given to the alignment program. Progressive alignment methods are efficient enough to implement on a large scale for many (100s to 1000s) sequences. A popular progressive alignment method has been the Clustal family. ClustalW is used extensively for phylogenetic tree construction, in spite of the author's explicit warnings that unedited alignments should not be used in such studies and as input for protein structure prediction by homology modeling. European Bioinformatics Institute (EMBL-EBI) announced that CLustalW2 will expire in August 2015. They recommend Clustal Omega which performs based on seeded guide trees and HMM profile-profile techniques for protein alignments. An alternative tool for progressive DNA alignments is multiple alignment using fast Fourier transform (MAFFT). Another common progressive alignment method named T-Coffee is slower than Clustal and its derivatives but generally produces more accurate alignments for distantly related sequence sets. T-Coffee calculates pairwise alignments by combining the direct alignment of the pair with indirect alignments that aligns each sequence of the pair to a third sequence. It uses the output from Clustal as well as another local alignment program LALIGN, which finds multiple regions of local alignment between two sequences. The resulting alignment and phylogenetic tree are used as a guide to produce new and more accurate w
AI Essay Writers: Free vs Paid (2026)
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Quotient automaton
In computer science, in particular in formal language theory, a quotient automaton can be obtained from a given nondeterministic finite automaton by joining some of its states. The quotient recognizes a superset of the given automaton; in some cases, handled by the Myhill–Nerode theorem, both languages are equal. == Formal definition == A (nondeterministic) finite automaton is a quintuple A = ⟨Σ, S, s0, δ, Sf⟩, where: Σ is the input alphabet (a finite, non-empty set of symbols), S is a finite, non-empty set of states, s0 is the initial state, an element of S, δ is the state-transition relation: δ ⊆ S × Σ × S, and Sf is the set of final states, a (possibly empty) subset of S. A string a1...an ∈ Σ is recognized by A if there exist states s1, ..., sn ∈ S such that ⟨si-1,ai,si⟩ ∈ δ for i=1,...,n, and sn ∈ Sf. The set of all strings recognized by A is called the language recognized by A; it is denoted as L(A). For an equivalence relation ≈ on the set S of A’s states, the quotient automaton A/≈ = ⟨Σ, S/≈, [s0], δ/≈, Sf/≈⟩ is defined by the input alphabet Σ being the same as that of A, the state set S/≈ being the set of all equivalence classes of states from S, the start state [s0] being the equivalence class of A’s start state, the state-transition relation δ/≈ being defined by δ/≈([s],a,[t]) if δ(s,a,t) for some s ∈ [s] and t ∈ [t], and the set of final states Sf/≈ being the set of all equivalence classes of final states from Sf. The process of computing A/≈ is also called factoring A by ≈. == Example == For example, the automaton A shown in the first row of the table is formally defined by ΣA = {0,1}, SA = {a,b,c,d}, sA0 = a, δA = { ⟨a,1,b⟩, ⟨b,0,c⟩, ⟨c,0,d⟩ }, and SAf = { b,c,d }. It recognizes the finite set of strings { 1, 10, 100 }; this set can also be denoted by the regular expression "1+10+100". The relation (≈) = { ⟨a,a⟩, ⟨a,b⟩, ⟨b,a⟩, ⟨b,b⟩, ⟨c,c⟩, ⟨c,d⟩, ⟨d,c⟩, ⟨d,d⟩ }, more briefly denoted as a≈b,c≈d, is an equivalence relation on the set {a,b,c,d} of automaton A’s states. Building the quotient of A by that relation results in automaton C in the third table row; it is formally defined by ΣC = {0,1}, SC = {a,c}, sC0 = a, δC = { ⟨a,1,a⟩, ⟨a,0,c⟩, ⟨c,0,c⟩ }, and SCf = { a,c }. It recognizes the finite set of all strings composed of arbitrarily many 1s, followed by arbitrarily many 0s, i.e. { ε, 1, 10, 100, 1000, ..., 11, 110, 1100, 11000, ..., 111, ... }; this set can also be denoted by the regular expression "10". Informally, C can be thought of resulting from A by glueing state a onto state b, and glueing state c onto state d. The table shows some more quotient relations, such as B = A/a≈b, and D = C/a≈c. == Properties == For every automaton A and every equivalence relation ≈ on its states set, L(A/≈) is a superset of (or equal to) L(A). Given a finite automaton A over some alphabet Σ, an equivalence relation ≈ can be defined on Σ by x ≈ y if ∀ z ∈ Σ: xz ∈ L(A) ↔ yz ∈ L(A). By the Myhill–Nerode theorem, A/≈ is a deterministic automaton that recognizes the same language as A. As a consequence, the quotient of A by every refinement of ≈ also recognizes the same language as A.
Similarity learning
Similarity learning is an area of supervised machine learning in artificial intelligence. It is closely related to regression and classification, but the goal is to learn a similarity function that measures how similar or related two objects are. It has applications in ranking, in recommendation systems, visual identity tracking, face verification, and speaker verification. == Learning setup == There are four common setups for similarity and metric distance learning. Regression similarity learning In this setup, pairs of objects are given ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} together with a measure of their similarity y i ∈ R {\displaystyle y_{i}\in R} . The goal is to learn a function that approximates f ( x i 1 , x i 2 ) ∼ y i {\displaystyle f(x_{i}^{1},x_{i}^{2})\sim y_{i}} for every new labeled triplet example ( x i 1 , x i 2 , y i ) {\displaystyle (x_{i}^{1},x_{i}^{2},y_{i})} . This is typically achieved by minimizing a regularized loss min W ∑ i l o s s ( w ; x i 1 , x i 2 , y i ) + r e g ( w ) {\displaystyle \min _{W}\sum _{i}loss(w;x_{i}^{1},x_{i}^{2},y_{i})+reg(w)} . Classification similarity learning Given are pairs of similar objects ( x i , x i + ) {\displaystyle (x_{i},x_{i}^{+})} and non similar objects ( x i , x i − ) {\displaystyle (x_{i},x_{i}^{-})} . An equivalent formulation is that every pair ( x i 1 , x i 2 ) {\displaystyle (x_{i}^{1},x_{i}^{2})} is given together with a binary label y i ∈ { 0 , 1 } {\displaystyle y_{i}\in \{0,1\}} that determines if the two objects are similar or not. The goal is again to learn a classifier that can decide if a new pair of objects is similar or not. Ranking similarity learning Given are triplets of objects ( x i , x i + , x i − ) {\displaystyle (x_{i},x_{i}^{+},x_{i}^{-})} whose relative similarity obey a predefined order: x i {\displaystyle x_{i}} is known to be more similar to x i + {\displaystyle x_{i}^{+}} than to x i − {\displaystyle x_{i}^{-}} . The goal is to learn a function f {\displaystyle f} such that for any new triplet of objects ( x , x + , x − ) {\displaystyle (x,x^{+},x^{-})} , it obeys f ( x , x + ) > f ( x , x − ) {\displaystyle f(x,x^{+})>f(x,x^{-})} (contrastive learning). This setup assumes a weaker form of supervision than in regression, because instead of providing an exact measure of similarity, one only has to provide the relative order of similarity. For this reason, ranking-based similarity learning is easier to apply in real large-scale applications. Locality sensitive hashing (LSH) Hashes input items so that similar items map to the same "buckets" in memory with high probability (the number of buckets being much smaller than the universe of possible input items). It is often applied in nearest neighbor search on large-scale high-dimensional data, e.g., image databases, document collections, time-series databases, and genome databases. A common approach for learning similarity is to model the similarity function as a bilinear form. For example, in the case of ranking similarity learning, one aims to learn a matrix W that parametrizes the similarity function f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . When data is abundant, a common approach is to learn a siamese network – a deep network model with parameter sharing. == Metric learning == Similarity learning is closely related to distance metric learning. Metric learning is the task of learning a distance function over objects. A metric or distance function has to obey four axioms: non-negativity, identity of indiscernibles, symmetry and subadditivity (or the triangle inequality). In practice, metric learning algorithms ignore the condition of identity of indiscernibles and learn a pseudo-metric. When the objects x i {\displaystyle x_{i}} are vectors in R d {\displaystyle R^{d}} , then any matrix W {\displaystyle W} in the symmetric positive semi-definite cone S + d {\displaystyle S_{+}^{d}} defines a distance pseudo-metric of the space of x through the form D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ W ( x 1 − x 2 ) {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }W(x_{1}-x_{2})} . When W {\displaystyle W} is a symmetric positive definite matrix, D W {\displaystyle D_{W}} is a metric. Moreover, as any symmetric positive semi-definite matrix W ∈ S + d {\displaystyle W\in S_{+}^{d}} can be decomposed as W = L ⊤ L {\displaystyle W=L^{\top }L} where L ∈ R e × d {\displaystyle L\in R^{e\times d}} and e ≥ r a n k ( W ) {\displaystyle e\geq rank(W)} , the distance function D W {\displaystyle D_{W}} can be rewritten equivalently D W ( x 1 , x 2 ) 2 = ( x 1 − x 2 ) ⊤ L ⊤ L ( x 1 − x 2 ) = ‖ L ( x 1 − x 2 ) ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=(x_{1}-x_{2})^{\top }L^{\top }L(x_{1}-x_{2})=\|L(x_{1}-x_{2})\|_{2}^{2}} . The distance D W ( x 1 , x 2 ) 2 = ‖ x 1 ′ − x 2 ′ ‖ 2 2 {\displaystyle D_{W}(x_{1},x_{2})^{2}=\|x_{1}'-x_{2}'\|_{2}^{2}} corresponds to the Euclidean distance between the transformed feature vectors x 1 ′ = L x 1 {\displaystyle x_{1}'=Lx_{1}} and x 2 ′ = L x 2 {\displaystyle x_{2}'=Lx_{2}} . Many formulations for metric learning have been proposed. Some well-known approaches for metric learning include learning from relative comparisons, which is based on the triplet loss, large margin nearest neighbor, and information theoretic metric learning (ITML). In statistics, the covariance matrix of the data is sometimes used to define a distance metric called Mahalanobis distance. == Applications == Similarity learning is used in information retrieval for learning to rank, in face verification or face identification, and in recommendation systems. Also, many machine learning approaches rely on some metric. This includes unsupervised learning such as clustering, which groups together close or similar objects. It also includes supervised approaches like K-nearest neighbor algorithm which rely on labels of nearby objects to decide on the label of a new object. Metric learning has been proposed as a preprocessing step for many of these approaches. == Scalability == Metric and similarity learning scale quadratically with the dimension of the input space, as can easily see when the learned metric has a bilinear form f W ( x , z ) = x T W z {\displaystyle f_{W}(x,z)=x^{T}Wz} . Scaling to higher dimensions can be achieved by enforcing a sparseness structure over the matrix model, as done with HDSL, and with COMET. == Software == metric-learn is a free software Python library which offers efficient implementations of several supervised and weakly-supervised similarity and metric learning algorithms. The API of metric-learn is compatible with scikit-learn. OpenMetricLearning is a Python framework to train and validate the models producing high-quality embeddings. == Further information == For further information on this topic, see the surveys on metric and similarity learning by Bellet et al. and Kulis.
Heikki Mannila
Heikki Olavi Mannila (born 4 January 1960 in Espoo) is a Finnish computer scientist, the president of the Academy of Finland. Mannila earned his Ph.D. in 1985 from the University of Helsinki under the supervision of Esko Ukkonen and for many years he was a professor at the University of Helsinki himself. From 2004 to 2008 he was Academy Professor at the Academy of Finland. He became Vice President for Academic Affairs at Aalto University in 2009, and was appointed by the Finnish government as president of the Academy of Finland for a term lasting from 2012 to 2017. The appointment was renewed for the period 2017–2022. Mannila is known for his research in data mining, and has published highly cited papers on association rule learning and sequence mining. With David Hand and Padhraic Smyth, he is the co-author of the book Principles of Data Mining (MIT Press, 2001). Heikki Mannila is son to the professor Elina Haavio-Mannila.