A Gödel machine is a hypothetical self-improving computer program that solves problems in an optimal way. It uses a recursive self-improvement protocol in which it rewrites its own code when it can prove the new code provides a better strategy. The machine was invented by Jürgen Schmidhuber (first proposed in 2003), but is named after Kurt Gödel who inspired the mathematical theories. The Gödel machine is often discussed when dealing with issues of meta-learning, also known as "learning to learn." Applications include automating human design decisions and transfer of knowledge between multiple related tasks, and may lead to design of more robust and general learning architectures. Though theoretically possible, no full implementation has been created. The Gödel machine is often compared with Marcus Hutter's AIXI, another formal specification for an artificial general intelligence. Schmidhuber points out that the Gödel machine could start out by implementing AIXItl as its initial sub-program, and self-modify after it finds proof that another algorithm for its search code will be better. == Limitations == Traditional problems solved by a computer only require one input and provide some output. Computers of this sort had their initial algorithm hardwired. This does not take into account the dynamic natural environment, and thus was a goal for the Gödel machine to overcome. The Gödel machine has limitations of its own, however. According to Gödel's First Incompleteness Theorem, any formal system that encompasses arithmetic is either flawed or allows for statements that cannot be proved in the system. Hence even a Gödel machine with unlimited computational resources must ignore those self-improvements whose effectiveness it cannot prove. == Variables of interest == There are three variables that are particularly useful in the run time of the Gödel machine. At some time t {\displaystyle t} , the variable time {\displaystyle {\text{time}}} will have the binary equivalent of t {\displaystyle t} . This is incremented steadily throughout the run time of the machine. Any input meant for the Gödel machine from the natural environment is stored in variable x {\displaystyle x} . It is likely the case that x {\displaystyle x} will hold different values for different values of variable time {\displaystyle {\text{time}}} . The outputs of the Gödel machine are stored in variable y {\displaystyle y} , where y ( t ) {\displaystyle y(t)} would be the output bit-string at some time t {\displaystyle t} . At any given time t {\displaystyle t} , where ( 1 ≤ t ≤ T ) {\displaystyle (1\leq t\leq T)} , the goal is to maximize future success or utility. A typical utility function follows the pattern u ( s , E n v ) : S × E → R {\displaystyle u(s,\mathrm {Env} ):S\times E\rightarrow \mathbb {R} } : u ( s , E n v ) = E μ [ ∑ τ = time T r ( τ ) ∣ s , E n v ] {\displaystyle u(s,\mathrm {Env} )=E_{\mu }{\Bigg [}\sum _{\tau ={\text{time}}}^{T}r(\tau )\mid s,\mathrm {Env} {\Bigg ]}} where r ( t ) {\displaystyle r(t)} is a real-valued reward input (encoded within s ( t ) {\displaystyle s(t)} ) at time t {\displaystyle t} , E μ [ ⋅ ∣ ⋅ ] {\displaystyle E_{\mu }[\cdot \mid \cdot ]} denotes the conditional expectation operator with respect to some possibly unknown distribution μ {\displaystyle \mu } from a set M {\displaystyle M} of possible distributions ( M {\displaystyle M} reflects whatever is known about the possibly probabilistic reactions of the environment), and the above-mentioned time = time ( s ) {\displaystyle {\text{time}}=\operatorname {time} (s)} is a function of state s {\displaystyle s} which uniquely identifies the current cycle. Note that we take into account the possibility of extending the expected lifespan through appropriate actions. == Instructions used by proof techniques == The nature of the six proof-modifying instructions below makes it impossible to insert an incorrect theorem into proof, thus trivializing proof verification. === get-axiom(n) === Appends the n-th axiom as a theorem to the current theorem sequence. Below is the initial axiom scheme: Hardware Axioms formally specify how components of the machine could change from one cycle to the next. Reward Axioms define the computational cost of hardware instruction and the physical cost of output actions. Related Axioms also define the lifetime of the Gödel machine as scalar quantities representing all rewards/costs. Environment Axioms restrict the way new inputs x are produced from the environment, based on previous sequences of inputs y. Uncertainty Axioms/String Manipulation Axioms are standard axioms for arithmetic, calculus, probability theory, and string manipulation that allow for the construction of proofs related to future variable values within the Gödel machine. Initial State Axioms contain information about how to reconstruct parts or all of the initial state. Utility Axioms describe the overall goal in the form of utility function u. === apply-rule(k, m, n) === Takes in the index k of an inference rule (such as Modus tollens, Modus ponens), and attempts to apply it to the two previously proved theorems m and n. The resulting theorem is then added to the proof. === delete-theorem(m) === Deletes the theorem stored at index m in the current proof. This helps to mitigate storage constraints caused by redundant and unnecessary theorems. Deleted theorems can no longer be referenced by the above apply-rule function. === set-switchprog(m, n) === Replaces switchprog S pm:n, provided it is a non-empty substring of S p. === check() === Verifies whether the goal of the proof search has been reached. A target theorem states that given the current axiomatized utility function u (Item 1f), the utility of a switch from p to the current switchprog would be higher than the utility of continuing the execution of p (which would keep searching for alternative switchprogs). === state2theorem(m, n) === Takes in two arguments, m and n, and attempts to convert the contents of Sm:n into a theorem. == Example applications == === Time-limited NP-hard optimization === The initial input to the Gödel machine is the representation of a connected graph with a large number of nodes linked by edges of various lengths. Within given time T it should find a cyclic path connecting all nodes. The only real-valued reward will occur at time T. It equals 1 divided by the length of the best path found so far (0 if none was found). There are no other inputs. The by-product of maximizing expected reward is to find the shortest path findable within the limited time, given the initial bias. === Fast theorem proving === Prove or disprove as quickly as possible that all even integers > 2 are the sum of two primes (Goldbach’s conjecture). The reward is 1/t, where t is the time required to produce and verify the first such proof. === Maximizing expected reward with bounded resources === A cognitive robot that needs at least 1 liter of gasoline per hour interacts with a partially unknown environment, trying to find hidden, limited gasoline depots to occasionally refuel its tank. It is rewarded in proportion to its lifetime, and dies after at most 100 years or as soon as its tank is empty or it falls off a cliff, and so on. The probabilistic environmental reactions are initially unknown but assumed to be sampled from the axiomatized Speed Prior, according to which hard-to-compute environmental reactions are unlikely. This permits a computable strategy for making near-optimal predictions. One by-product of maximizing expected reward is to maximize expected lifetime.
Plant Nanny
Plant Nanny is a water tracker mobile application which reminds users to drink water. It was developed by Taiwanese app maker Fourdesire. The app was first released in 2013 and is available on the Apple App Store for iPhones and the Google Play Store for Android devices. == Description == Play Nanny uses a game method that allows users to turn their virtual selves into plants, which grows and thrives as the user drinks more water. The app sends occasional push notifications to remind users to drink water throughout the day. Users can choose from a wide range of plants, including cacti and carnations, and track their water intake. The app uses two resources, How to calculate how much water you should drink by Jennifer Stone (2018) and Human energy requirements by the Food and Agriculture Organization (2004), to calculate the recommended daily water intake for its users. Upon downloading the app, users are prompted to input basic personal information which is then used to calculate the recommended daily water intake and prompts them to drink the appropriate amount. == Accolades ==
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
Persian Speech Corpus
The Persian Speech Corpus is a Modern Persian speech corpus for speech synthesis. The corpus contains phonetic and orthographic transcriptions of about 2.5 hours of Persian speech aligned with recorded speech on the phoneme level, including annotations of word boundaries. Previous spoken corpora of Persian include FARSDAT, which consists of read aloud speech from newspaper texts from 100 Persian speakers and the Telephone FARsi Spoken language DATabase (TFARSDAT) which comprises seven hours of read and spontaneous speech produced by 60 native speakers of Persian from ten regions of Iran. The Persian Speech Corpus was built using the same methodologies laid out in the doctoral project on Modern Standard Arabic of Nawar Halabi at the University of Southampton. The work was funded by MicroLinkPC, who own an exclusive license to commercialise the corpus, though the corpus is available for non-commercial use through the corpus' website. It is distributed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. The corpus was built for speech synthesis purposes, but has been used for building HMM based voices in Persian. It can also be used to automatically align other speech corpora with their phonetic transcript and could be used as part of a larger corpus for training speech recognition systems. == Contents == The corpus is downloadable from its website, and contains the following: 396 .wav files containing spoken utterances 396 .lab files containing text utterances 396 .TextGrid files containing the phoneme labels with time stamps of the boundaries where these occur in the .wav files. phonetic-transcript.txt which has the form "[wav_filename]" "[Phoneme Sequence]" in every line orthographic-transcript.txt which has the form "[wav_filename]" "[Orthographic Transcript]" in every line
Gremlin (query language)
Gremlin is a graph traversal language and virtual machine developed by Apache TinkerPop of the Apache Software Foundation. Gremlin works for both OLTP-based graph databases as well as OLAP-based graph processors. Gremlin's automata and functional language foundation enable Gremlin to naturally support imperative and declarative querying, host language agnosticism, user-defined domain specific languages, an extensible compiler/optimizer, single- and multi-machine execution models, and hybrid depth- and breadth-first evaluation with Turing completeness. As an explanatory analogy, Apache TinkerPop and Gremlin are to graph databases what the JDBC and SQL are to relational databases. Likewise, the Gremlin traversal machine is to graph computing as what the Java virtual machine is to general purpose computing. == History == 2009-10-30 the project is born, and immediately named "TinkerPop" 2009-12-25 v0.1 is the first release 2011-05-21 v1.0 is released 2012-05-24 v2.0 is released 2015-01-16 TinkerPop becomes an Apache Incubator project 2015-07-09 v3.0.0-incubating is released 2016-05-23 Apache TinkerPop becomes a top-level project 2016-07-18 v3.1.3 and v3.2.1 are first releases as Apache TinkerPop 2017-12-17 v3.3.1 is released 2018-05-08 v3.3.3 is released 2019-08-05 v3.4.3 is released 2020-02-20 v3.4.6 is released 2021-05-01 v3.5.0 is released 2022-04-04 v3.6.0 is released 2023-07-31 v3.7.0 is released 2025-11-12 v3.8.0 is released == Vendor integration == Gremlin is an Apache2-licensed graph traversal language that can be used by graph system vendors. There are typically two types of graph system vendors: OLTP graph databases and OLAP graph processors. The table below outlines those graph vendors that support Gremlin. == Traversal examples == The following examples of Gremlin queries and responses in a Gremlin-Groovy environment are relative to a graph representation of the MovieLens dataset. The dataset includes users who rate movies. Users each have one occupation, and each movie has one or more categories associated with it. The MovieLens graph schema is detailed below. === Simple traversals === For each vertex in the graph, emit its label, then group and count each distinct label. What year was the oldest movie made? What is Die Hard's average rating? === Projection traversals === For each category, emit a map of its name and the number of movies it represents. For each movie with at least 11 ratings, emit a map of its name and average rating. Sort the maps in decreasing order by their average rating. Emit the first 10 maps (i.e. top 10). === Declarative pattern matching traversals === Gremlin supports declarative graph pattern matching similar to SPARQL. For instance, the following query below uses Gremlin's match()-step. What 80's action movies do 30-something programmers like? Group count the movies by their name and sort the group count map in decreasing order by value. Clip the map to the top 10 and emit the map entries. === OLAP traversal === Which movies are most central in the implicit 5-stars graph? == Gremlin graph traversal machine == Gremlin is a virtual machine composed of an instruction set as well as an execution engine. An analogy is drawn between Gremlin and Java. === Gremlin steps (instruction set) === The following traversal is a Gremlin traversal in the Gremlin-Java8 dialect. The Gremlin language (i.e. the fluent-style of expressing a graph traversal) can be represented in any host language that supports function composition and function nesting. Due to this simple requirement, there exists various Gremlin dialects including Gremlin-Groovy, Gremlin-Scala, Gremlin-Clojure, etc. The above Gremlin-Java8 traversal is ultimately compiled down to a step sequence called a traversal. A string representation of the traversal above provided below. The steps are the primitives of the Gremlin graph traversal machine. They are the parameterized instructions that the machine ultimately executes. The Gremlin instruction set is approximately 30 steps. These steps are sufficient to provide general purpose computing and what is typically required to express the common motifs of any graph traversal query. Given that Gremlin is a language, an instruction set, and a virtual machine, it is possible to design another traversal language that compiles to the Gremlin traversal machine (analogous to how Scala compiles to the JVM). For instance, the popular SPARQL graph pattern match language can be compiled to execute on the Gremlin machine. The following SPARQL query would compile to In Gremlin-Java8, the SPARQL query above would be represented as below and compile to the identical Gremlin step sequence (i.e. traversal). === Gremlin Machine (virtual machine) === The Gremlin graph traversal machine can execute on a single machine or across a multi-machine compute cluster. Execution agnosticism allows Gremlin to run over both graph databases (OLTP) and graph processors (OLAP).
List of COBOL software and tools
This is a list of software and programming tools for the COBOL programming language, which includes compilers, IDEs, build tools, testing, frameworks, and related projects. == Compilers and runtimes == Fujitsu NetCOBOL — COBOL compiler for Windows, Linux, and mainframes GnuCOBOL — open-source COBOL compiler translating COBOL to C and then compiling with GCC IBM COBOL — mainframe COBOL compiler for IBM z/OS and IBM i platforms Micro Focus COBOL — commercial COBOL compiler and runtime for enterprise systems FairCom RTG – A commercial real-time database and runtime solution developed by FairCom Corporation. It provides integration with COBOL applications for transaction processing and modernization projects, and is used in enterprise environments requiring high-performance data management. == Integrated development environments == Eclipse IDE — with COBOL plugin support, Micro Focus or Bitlang extensions. IBM Developer for z/OS — IDE for COBOL and PL/I mainframe development Micro Focus Visual COBOL — IDE integration for Visual Studio, Visual Studio Code, and Eclipse OpenCOBOLIDE — open-source lightweight IDE for GnuCOBOL Visual Studio Code — with COBOL extensions via Bitlang COBOL and GnuCOBOL Language Server == Frameworks, libraries, and APIs == ACUCOBOL-GT — runtime and API library suite from Micro Focus CICS — IBM middleware for transaction processing in COBOL applications DB2 and IMS APIs — database access libraries commonly used with COBOL applications == Build tools and package managers == Apache Ant — scripting and build automation for COBOL/Java hybrid systems GNU Make — common build tool for compiling COBOL via GnuCOBOL Jenkins — used for CI/CD automation with COBOL builds == Testing and quality assurance == COBOL Check — open-source unit testing framework for COBOL IBM Rational Performance Tester — automated performance testing of web and server-based applications from the Rational Software division of IBM Micro Focus Unit Testing Framework — integrated COBOL unit testing tool == Debugging and profiling tools == GnuCOBOL debug mode — command-line debugging integrated in GnuCOBOL compiler IBM Debug Tool for z/OS — mainframe debugging for COBOL and PL/I Micro Focus Animator — step-through debugger for COBOL code
Elastic map
Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998. == Energy of elastic map == Let S {\displaystyle {\mathcal {S}}} be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes w j {\displaystyle {\bf {w}}_{j}} in the same space. Each datapoint s ∈ S {\displaystyle s\in {\mathcal {S}}} has a host node, namely the closest node w j {\displaystyle {\bf {w}}_{j}} (if there are several closest nodes then one takes the node with the smallest number). The data set S {\displaystyle {\mathcal {S}}} is divided into classes K j = { s | w j is a host of s } {\displaystyle K_{j}=\{s\ |\ {\bf {w}}_{j}{\mbox{ is a host of }}s\}} . The approximation energy D is the distortion D = 1 2 ∑ j = 1 k ∑ s ∈ K j ‖ s − w j ‖ 2 {\displaystyle D={\frac {1}{2}}\sum _{j=1}^{k}\sum _{s\in K_{j}}\|s-{\bf {w}}_{j}\|^{2}} , which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points { s i } {\displaystyle \{s_{i}\}} . On the set of nodes an additional structure is defined. Some pairs of nodes, ( w i , w j ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j})} , are connected by elastic edges. Call this set of pairs E {\displaystyle E} . Some triplets of nodes, ( w i , w j , w k ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})} , form bending ribs. Call this set of triplets G {\displaystyle G} . The stretching energy is U E = 1 2 λ ∑ ( w i , w j ) ∈ E ‖ w i − w j ‖ 2 {\displaystyle U_{E}={\frac {1}{2}}\lambda \sum _{({\bf {w}}_{i},{\bf {w}}_{j})\in E}\|{\bf {w}}_{i}-{\bf {w}}_{j}\|^{2}} , The bending energy is U G = 1 2 μ ∑ ( w i , w j , w k ) ∈ G ‖ w i − 2 w j + w k ‖ 2 {\displaystyle U_{G}={\frac {1}{2}}\mu \sum _{({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})\in G}\|{\bf {w}}_{i}-2{\bf {w}}_{j}+{\bf {w}}_{k}\|^{2}} , where λ {\displaystyle \lambda } and μ {\displaystyle \mu } are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. The total energy of the elastic map is thus U = D + U E + U G . {\displaystyle U=D+U_{E}+U_{G}.} The position of the nodes { w j } {\displaystyle \{{\bf {w}}_{j}\}} is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy U {\displaystyle U} . == Expectation-maximization algorithm == For a given splitting of dataset S {\displaystyle {\mathcal {S}}} in classes K j {\displaystyle K_{j}} , minimization of the quadratic functional U {\displaystyle U} is a linear problem with the sparse matrix of coefficients. Therefore, similar to principal component analysis or k-means, a splitting method is used: For given { w j } {\displaystyle \{{\bf {w}}_{j}\}} find { K j } {\displaystyle \{K_{j}\}} ; For given { K j } {\displaystyle \{K_{j}\}} minimize U {\displaystyle U} and find { w j } {\displaystyle \{{\bf {w}}_{j}\}} ; If no change, terminate. This expectation-maximization algorithm guarantees a local minimum of U {\displaystyle U} . For improving the approximation various additional methods are proposed. For example, the softening strategy is used. This strategy starts with a rigid grids (small length, small bending and large elasticity modules λ {\displaystyle \lambda } and μ {\displaystyle \mu } coefficients) and finishes with soft grids (small λ {\displaystyle \lambda } and μ {\displaystyle \mu } ). The training goes in several epochs, each epoch with its own grid rigidness. Another adaptive strategy is growing net: one starts from a small number of nodes and gradually adds new nodes. Each epoch goes with its own number of nodes. == Applications == Most important applications of the method and free software are in bioinformatics for exploratory data analysis and visualisation of multidimensional data, for data visualisation in economics, social and political sciences, as an auxiliary tool for data mapping in geographic informational systems and for visualisation of data of various nature. The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images. This reconstruction is used for quantifying the geodesic distances between trichomes and their patterning, which is a marker of the capability of a plant to resist to pathogenes. Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of financial portfolios. The method of elastic maps has been systematically tested and compared with several machine learning methods on the applied problem of identification of the flow regime of a gas-liquid flow in a pipe. There are various regimes: Single phase water or air flow, Bubbly flow, Bubbly-slug flow, Slug flow, Slug-churn flow, Churn flow, Churn-annular flow, and Annular flow. The simplest and most common method used to identify the flow regime is visual observation. This approach is, however, subjective and unsuitable for relatively high gas and liquid flow rates. Therefore, the machine learning methods are proposed by many authors. The methods are applied to differential pressure data collected during a calibration process. The method of elastic maps provided a 2D map, where the area of each regime is represented. The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure. Here, ANN stands for the backpropagation artificial neural networks, SVM stands for the support vector machine, SOM for the self-organizing maps. The hybrid technology was developed for engineering applications. In this technology, elastic maps are used in combination with Principal Component Analysis (PCA), Independent Component Analysis (ICA) and backpropagation ANN. The textbook provides a systematic comparison of elastic maps and self-organizing maps (SOMs) in applications to economic and financial decision-making.