A finite-state transducer (FST) is a finite-state machine with two memory tapes, following the terminology for Turing machines: an input tape and an output tape. This contrasts with an ordinary finite-state automaton, which has a single tape. An FST is a type of finite-state automaton (FSA) that maps between two sets of symbols. An FST is more general than an FSA. An FSA defines a formal language by defining a set of accepted strings, while an FST defines a relation between sets of strings. An FST will read a set of strings on the input tape and generate a set of relations on the output tape. An FST can be thought of as a translator or relater between strings in a set. In morphological parsing, an example would be inputting a string of letters into the FST, the FST would then output a string of morphemes. == Overview == An automaton can be said to recognize a string if we view the content of its tape as input. In other words, the automaton computes a function that maps strings into the set {0,1}. Alternatively, we can say that an automaton generates strings, which means viewing its tape as an output tape. On this view, the automaton generates a formal language, which is a set of strings. The two views of automata are equivalent: the function that the automaton computes is precisely the indicator function of the set of strings it generates. The class of languages generated by finite automata is known as the class of regular languages. The two tapes of a transducer are typically viewed as an input tape and an output tape. On this view, a transducer is said to transduce (i.e., translate) the contents of its input tape to its output tape, by accepting a string on its input tape and generating another string on its output tape. It may do so nondeterministically and it may produce more than one output for each input string. A transducer may also produce no output for a given input string, in which case it is said to reject the input. In general, a transducer computes a relation between two formal languages. Each string-to-string finite-state transducer relates the input alphabet Σ to the output alphabet Γ. Relations R on Σ×Γ that can be implemented as finite-state transducers are called rational relations. Rational relations that are partial functions, i.e. that relate every input string from Σ to at most one Γ, are called rational functions. Finite-state transducers are often used for phonological and morphological analysis in natural language processing research and applications. Pioneers in this field include Ronald Kaplan, Lauri Karttunen, Martin Kay and Kimmo Koskenniemi. A common way of using transducers is in a so-called "cascade", where transducers for various operations are combined into a single transducer by repeated application of the composition operator (defined below). == Formal construction == Formally, a finite transducer T is a 6-tuple (Q, Σ, Γ, I, F, δ) such that: Q is a finite set, the set of states; Σ is a finite set, called the input alphabet; Γ is a finite set, called the output alphabet; I is a subset of Q, the set of initial states; F is a subset of Q, the set of final states; and δ ⊆ Q × ( Σ ∪ { ϵ } ) × ( Γ ∪ { ϵ } ) × Q {\displaystyle \delta \subseteq Q\times (\Sigma \cup \{\epsilon \})\times (\Gamma \cup \{\epsilon \})\times Q} (where ε is the empty string) is the transition relation. We can view (Q, δ) as a labeled directed graph, known as the transition graph of T: the set of vertices is Q, and ( q , a , b , r ) ∈ δ {\displaystyle (q,a,b,r)\in \delta } means that there is a labeled edge going from vertex q to vertex r. We also say that a is the input label and b the output label of that edge. NOTE: This definition of finite transducer is also called letter transducer (Roche and Schabes 1997); alternative definitions are possible, but can all be converted into transducers following this one. Define the extended transition relation δ ∗ {\displaystyle \delta ^{}} as the smallest set such that: δ ⊆ δ ∗ {\displaystyle \delta \subseteq \delta ^{}} ; ( q , ϵ , ϵ , q ) ∈ δ ∗ {\displaystyle (q,\epsilon ,\epsilon ,q)\in \delta ^{}} for all q ∈ Q {\displaystyle q\in Q} ; and whenever ( q , x , y , r ) ∈ δ ∗ {\displaystyle (q,x,y,r)\in \delta ^{}} and ( r , a , b , s ) ∈ δ {\displaystyle (r,a,b,s)\in \delta } then ( q , x a , y b , s ) ∈ δ ∗ {\displaystyle (q,xa,yb,s)\in \delta ^{}} . The extended transition relation is essentially the reflexive transitive closure of the transition graph that has been augmented to take edge labels into account. The elements of δ ∗ {\displaystyle \delta ^{}} are known as paths. The edge labels of a path are obtained by concatenating the edge labels of its constituent transitions in order. The behavior of the transducer T is the rational relation [T] defined as follows: x [ T ] y {\displaystyle x[T]y} if and only if there exists i ∈ I {\displaystyle i\in I} and f ∈ F {\displaystyle f\in F} such that ( i , x , y , f ) ∈ δ ∗ {\displaystyle (i,x,y,f)\in \delta ^{}} . This is to say that T transduces a string x ∈ Σ ∗ {\displaystyle x\in \Sigma ^{}} into a string y ∈ Γ ∗ {\displaystyle y\in \Gamma ^{}} if there exists a path from an initial state to a final state whose input label is x and whose output label is y. === Weighted automata === Finite State Transducers can be weighted, where each transition is labelled with a weight in addition to the input and output labels. A Weighted Finite State Transducer (WFST) over a set K of weights can be defined similarly to an unweighted one as an 8-tuple T=(Q, Σ, Γ, I, F, E, λ, ρ), where: Q, Σ, Γ, I, F are defined as above; E ⊆ Q × ( Σ ∪ { ϵ } ) × ( Γ ∪ { ϵ } ) × Q × K {\displaystyle E\subseteq Q\times (\Sigma \cup \{\epsilon \})\times (\Gamma \cup \{\epsilon \})\times Q\times K} (where ε is the empty string) is the finite set of transitions; λ : I → K {\displaystyle \lambda :I\rightarrow K} maps initial states to weights; ρ : F → K {\displaystyle \rho :F\rightarrow K} maps final states to weights. In order to make certain operations on WFSTs well-defined, it is convenient to require the set of weights to form a semiring. Two typical semirings used in practice are the log semiring and tropical semiring: nondeterministic automata may be regarded as having weights in the Boolean semiring. Two weighted FST can be composed. == Operations on finite-state transducers == The following operations defined on finite automata also apply to finite transducers: Union. Given transducers T and S, there exists a transducer T ∪ S {\displaystyle T\cup S} such that x [ T ∪ S ] y {\displaystyle x[T\cup S]y} if and only if x [ T ] y {\displaystyle x[T]y} or x [ S ] y {\displaystyle x[S]y} . Concatenation. Given transducers T and S, there exists a transducer T ⋅ S {\displaystyle T\cdot S} such that x [ T ⋅ S ] y {\displaystyle x[T\cdot S]y} if and only if there exist x 1 , x 2 , y 1 , y 2 {\displaystyle x_{1},x_{2},y_{1},y_{2}} with x = x 1 x 2 , y = y 1 y 2 , x 1 [ T ] y 1 {\displaystyle x=x_{1}x_{2},y=y_{1}y_{2},x_{1}[T]y_{1}} and x 2 [ S ] y 2 . {\displaystyle x_{2}[S]y_{2}.} Kleene closure. Given a transducer T, there might exist a transducer T ∗ {\displaystyle T^{}} with the following properties: and x [ T ∗ ] y {\displaystyle x[T^{}]y} does not hold unless mandated by (k1) or (k2). Composition. Given a transducer T on alphabets Σ and Γ and a transducer S on alphabets Γ and Δ, there exists a transducer T ∘ S {\displaystyle T\circ S} on Σ and Δ such that x [ T ∘ S ] z {\displaystyle x[T\circ S]z} if and only if there exists a string y ∈ Γ ∗ {\displaystyle y\in \Gamma ^{}} such that x [ T ] y {\displaystyle x[T]y} and y [ S ] z {\displaystyle y[S]z} . This operation extends to the weighted case. This definition uses the same notation used in mathematics for relation composition. However, the conventional reading for relation composition is the other way around: given two relations T and S, ( x , z ) ∈ T ∘ S {\displaystyle (x,z)\in T\circ S} when there exist some y such that ( x , y ) ∈ S {\displaystyle (x,y)\in S} and ( y , z ) ∈ T . {\displaystyle (y,z)\in T.} Projection to an automaton. There are two projection functions: π 1 {\displaystyle \pi _{1}} preserves the input tape, and π 2 {\displaystyle \pi _{2}} preserves the output tape. The first projection, π 1 {\displaystyle \pi _{1}} is defined as follows: Given a transducer T, there exists a finite automaton π 1 T {\displaystyle \pi _{1}T} such that π 1 T {\displaystyle \pi _{1}T} accepts x if and only if there exists a string y for which x [ T ] y . {\displaystyle x[T]y.} :The second projection, π 2 {\displaystyle \pi _{2}} is defined similarly. Determinization. Given a transducer T, we want to build an equivalent transducer that has a unique initial state and such that no two transitions leaving any state share the same input label. The powerset construction can be extended to transducers, or even weighted transducers, but sometimes fails to halt; indeed, some non-deterministic transducers do not admit equivalent
Butler in a Box
Butler in a Box was an early voice-controlled home automation device developed in 1983 by magician Gus Searcy and programmer Franz Kavan. The device allowed users to control various home electronics, such as lights and phones, using voice commands. It predated modern smart speakers and virtual assistants by several decades. == History == The idea for the Butler in a Box originated in 1983 when Searcy was asked by friends why he couldn't simply command lights to turn on and off if he could pull rabbits out of hats, given his background as a professional magician. Searcy partnered with former IBM programmer Kavan to develop the device, with their first prototype being named "Sidney". The Butler in a Box combined remote control technology with voice recognition to enable control of home devices. However, it faced challenges due to the technological limitations of the era and its high price point of nearly $1,500 (equivalent to around $3,700 in 2021). == Features and functionality == Users could activate the Butler in a Box by speaking a wake word, typically a traditional butler name, and the device would address the user as "boss". It was capable of performing tasks such as: Turning lights on and off, controlling individual zones if lights were connected to remote control modules Making and receiving phone calls Setting timers Pairing with sensors to function as a security alarm system However, the device required extensive voice training for each user, a time-consuming process compared to modern voice recognition. Additionally, settings and trained commands would be lost if power was out for over 3 hours due to the volatile memory technology used at the time. == Reception and legacy == While innovative for its time, the Butler in a Box did not achieve widespread commercial success due to its high price and the technical limitations of the 1980s. Nevertheless, it served as an important early step in the development of home automation and showcased the potential for voice-controlled technology to enhance accessibility and convenience in the home. Decades later, products like Amazon Alexa, Google Home, and Apple's Siri would make voice-controlled smart home devices commonplace and affordable, building on the groundwork laid by early attempts like the Butler in a Box.
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
Apache Giraph
Apache Giraph is an Apache project to perform graph processing on big data. Giraph utilizes Apache Hadoop's MapReduce implementation to process graphs. Facebook used Giraph with some performance improvements to analyze one trillion edges using 200 machines in 4 minutes. Giraph is based on a paper published by Google about its own graph processing system called Pregel. It can be compared to other Big Graph processing libraries such as Cassovary. As of September 2023, it is no longer actively developed.
NETtalk (artificial neural network)
NETtalk is an artificial neural network that learns to pronounce written English text by supervised learning. It takes English text as input, and produces a matching phonetic transcriptions as output. It is the result of research carried out in the mid-1980s by Terrence Sejnowski and Charles Rosenberg. The intent behind NETtalk was to construct simplified models that might shed light on the complexity of learning human level cognitive tasks, and their implementation as a connectionist model that could also learn to perform a comparable task. The authors trained it by backpropagation. The network was trained on a large amount of English words and their corresponding pronunciations, and is able to generate pronunciations for unseen words with a high level of accuracy. The output of the network was a stream of phonemes, which fed into DECtalk to produce audible speech. It achieved popular success, appearing on the Today show. From the point of view of modeling human cognition, NETtalk does not specifically model the image processing stages and letter recognition of the visual cortex. Rather, it assumes that the letters have been pre-classified and recognized. It is NETtalk's task to learn proper associations between the correct pronunciation with a given sequence of letters based on the context in which the letters appear. A similar architecture was subsequently used for the opposite task, that of converting continuous speech signal to a phoneme sequence. == Training == The training dataset was a 20,008-word subset of the Brown Corpus, with manually annotated phoneme and stress for each letter. The development process was described in a 1993 interview. It took three months -- 250 person-hours -- to create the training dataset, but only a few days to train the network. After it was run successfully on this, the authors tried it on a phonological transcription of an interview with a young Latino boy from a barrio in Los Angeles. This resulted in a network that reproduced his Spanish accent. The original NETtalk was implemented on a Ridge 32, which took 0.275 seconds per learning step (one forward and one backward pass). Training NETtalk became a benchmark to test for the efficiency of backpropagation programs. For example, an implementation on Connection Machine-1 (with 16384 processors) ran at 52x speedup. An implementation on a 10-cell Warp ran at 340x speedup. The following table compiles the benchmark scores as of 1988. Speed is measured in "millions of connections per second" (MCPS). For example, the original NETtalk on Ridge 32 took 0.275 seconds per forward-backward pass, giving 18629 / 10 6 0.275 = 0.068 {\displaystyle {\frac {18629/10^{6}}{0.275}}=0.068} MCPS. Relative times are normalized to the MicroVax. == Architecture == The network had three layers and 18,629 adjustable weights, large by the standards of 1986. There were worries that it would overfit the dataset, but it was trained successfully. The input of the network has 203 units, divided into 7 groups of 29 units each. Each group is a one-hot encoding of one character. There are 29 possible characters: 26 letters, comma, period, and word boundary (whitespace). To produce the pronunciation of a single character, the network takes the character itself, as well as 3 characters before and 3 characters after it. The hidden layer has 80 units. The output has 26 units. 21 units encode for articulatory features (point of articulation, voicing, vowel height, etc.) of phonemes, and 5 units encode for stress and syllable boundaries. Sejnowski studied the learned representation in the network, and found that phonemes that sound similar are clustered together in representation space. The output of the network degrades, but remains understandable, when some hidden neurons are removed.
Halloween Problem
In computing, the Halloween Problem refers to a phenomenon in databases in which an update operation causes a change in the physical location of a row, potentially allowing the row to be visited again later in the same update operation. This could even cause an infinite loop in some cases where updates continually place the updated record ahead of the scan performing the update operation. The potential for this database error was first discovered by Don Chamberlin, Pat Selinger, and Morton Astrahan in the mid-1970s, on Halloween day, while working on query optimization. They wrote a SQL query supposed to give a ten percent raise to every employee who earned less than $25,000. This query would run successfully, with no errors, but when finished all the employees in the database earned at least $25,000, because it kept giving them a raise until they reached that level. The expectation was that the query would iterate over each of the employee records with a salary less than $25,000 precisely once. In fact, because even updated records were visible to the query execution engine and so continued to match the query's criteria, salary records were matching multiple times and each time being given a 10% raise until they were all greater than $25,000. Contrary to what some believe, the name is not descriptive of the nature of the problem but rather was given due to the day it was discovered on. As recounted by Don Chamberlin: Pat and Morton discovered this problem on Halloween... I remember they came into my office and said, "Chamberlin, look at this. We have to make sure that when the optimizer is making a plan for processing an update, it doesn't use an index that is based on the field that is being updated. How are we going to do that?" It happened to be on a Friday, and we said, "Listen, we are not going to be able to solve this problem this afternoon. Let's just give it a name. We'll call it the Halloween Problem and we'll work on it next week." And it turns out it has been called that ever since.
Radial basis function
In mathematics a radial basis function (RBF) is a real-valued function φ {\textstyle \varphi } whose value depends only on the distance between the input and some fixed point, either the origin, so that φ ( x ) = φ ^ ( ‖ x ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)} , or some other fixed point c {\textstyle \mathbf {c} } , called a center, so that φ ( x ) = φ ^ ( ‖ x − c ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} -\mathbf {c} \right\|)} . Any function φ {\textstyle \varphi } that satisfies the property φ ( x ) = φ ^ ( ‖ x ‖ ) {\textstyle \varphi (\mathbf {x} )={\hat {\varphi }}(\left\|\mathbf {x} \right\|)} is a radial function. The distance is usually Euclidean distance, although other metrics are sometimes used. They are often used as a collection { φ k } k {\displaystyle \{\varphi _{k}\}_{k}} which forms a basis for some function space of interest, hence the name. Sums of radial basis functions are typically used to approximate given functions. This approximation process can also be interpreted as a simple kind of neural network; this was the context in which they were originally applied to machine learning, in work by David Broomhead and David Lowe in 1988, which stemmed from Michael J. D. Powell's seminal research from 1977. RBFs are also used as a kernel in support vector classification. The technique has proven effective and flexible enough that radial basis functions are now applied in a variety of engineering applications. == Definition == A radial function is a function φ : [ 0 , ∞ ) → R {\textstyle \varphi :[0,\infty )\to \mathbb {R} } . When paired with a norm ‖ ⋅ ‖ : V → [ 0 , ∞ ) {\textstyle \|\cdot \|:V\to [0,\infty )} on a vector space, a function of the form φ c = φ ( ‖ x − c ‖ ) {\textstyle \varphi _{\mathbf {c} }=\varphi (\|\mathbf {x} -\mathbf {c} \|)} is said to be a radial kernel centered at c ∈ V {\textstyle \mathbf {c} \in V} . A radial function and the associated radial kernels are said to be radial basis functions if, for any finite set of nodes { x k } k = 1 n ⊆ V {\displaystyle \{\mathbf {x} _{k}\}_{k=1}^{n}\subseteq V} , all of the following conditions are true: === Examples === Commonly used types of radial basis functions include (writing r = ‖ x − x i ‖ {\textstyle r=\left\|\mathbf {x} -\mathbf {x} _{i}\right\|} and using ε {\textstyle \varepsilon } to indicate a shape parameter that can be used to scale the input of the radial kernel): == Approximation == Radial basis functions are typically used to build up function approximations of the form where the approximating function y ( x ) {\textstyle y(\mathbf {x} )} is represented as a sum of N {\displaystyle N} radial basis functions, each associated with a different center x i {\textstyle \mathbf {x} _{i}} , and weighted by an appropriate coefficient w i . {\textstyle w_{i}.} The weights w i {\textstyle w_{i}} can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights w i {\textstyle w_{i}} . Approximation schemes of this kind have been particularly used in time series prediction and control of nonlinear systems exhibiting sufficiently simple chaotic behaviour and 3D reconstruction in computer graphics (for example, hierarchical RBF and Pose Space Deformation). == RBF Network == The sum can also be interpreted as a rather simple single-layer type of artificial neural network called a radial basis function network, with the radial basis functions taking on the role of the activation functions of the network. It can be shown that any continuous function on a compact interval can in principle be interpolated with arbitrary accuracy by a sum of this form, if a sufficiently large number N {\textstyle N} of radial basis functions is used. The approximant y ( x ) {\textstyle y(\mathbf {x} )} is differentiable with respect to the weights w i {\textstyle w_{i}} . The weights could thus be learned using any of the standard iterative methods for neural networks. Using radial basis functions in this manner yields a reasonable interpolation approach provided that the fitting set has been chosen such that it covers the entire range systematically (equidistant data points are ideal). However, without a polynomial term that is orthogonal to the radial basis functions, estimates outside the fitting set tend to perform poorly. == RBFs for PDEs == Radial basis functions are used to approximate functions and so can be used to discretize and numerically solve Partial Differential Equations (PDEs). This was first done in 1990 by E. J. Kansa who developed the first RBF based numerical method. It is called the Kansa method and was used to solve the elliptic Poisson equation and the linear advection-diffusion equation. The function values at points x {\displaystyle \mathbf {x} } in the domain are approximated by the linear combination of RBFs: The derivatives are approximated as such: where N {\displaystyle N} are the number of points in the discretized domain, d {\displaystyle d} the dimension of the domain and λ {\displaystyle \lambda } the scalar coefficients that are unchanged by the differential operator. Different numerical methods based on Radial Basis Functions were developed thereafter. Some methods are the RBF-FD method, the RBF-QR method and the RBF-PUM method.