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  History of machine translation

  Machine translation is a sub-field of computational linguistics that investigates the use of software to translate text or speech from one natural language to another. In the 1950s, machine translation became a reality in research, although references to the subject can be found as early as the 17th century. The Georgetown experiment, which involved successful fully automatic translation of more than sixty Russian sentences into English in 1954, was one of the earliest recorded projects. Researchers of the Georgetown experiment asserted their belief that machine translation would be a solved problem within a few years. In the Soviet Union, similar experiments were performed shortly after. Consequently, the success of the experiment ushered in an era of significant funding for machine translation research in the United States. The achieved progress was much slower than expected; in 1966, the ALPAC report found that ten years of research had not fulfilled the expectations of the Georgetown experiment and resulted in dramatically reduced funding. Interest grew in statistical models for machine translation, which became more common and also less expensive in the 1980s as available computational power increased. Although there exists no autonomous system of "fully automatic high quality translation of unrestricted text," there are many programs now available that are capable of providing useful output within strict constraints. Several of these programs are available online, such as Google Translate and the SYSTRAN system that powers AltaVista's BabelFish (which was replaced by Microsoft Bing translator in May 2012). == The beginning == The origins of machine translation can be traced back to the work of Al-Kindi, a 9th-century Arabic cryptographer who developed techniques for systemic language translation, including cryptanalysis, frequency analysis, and probability and statistics, which are used in modern machine translation. The idea of machine translation later appeared in the 17th century. In 1629, René Descartes proposed a universal language, with equivalent ideas in different tongues sharing one symbol. In the mid-1930s the first patents for "translating machines" were applied for by Georges Artsrouni, for an automatic bilingual dictionary using punched tape. Russian Peter Troyanskii submitted a more detailed proposal that included both the bilingual dictionary and a method for dealing with grammatical roles between languages, based on the grammatical system of Esperanto. This system was separated into three stages: stage one consisted of a native-speaking editor in the source language to organize the words into their logical forms and to exercise the syntactic functions; stage two required the machine to "translate" these forms into the target language; and stage three required a native-speaking editor in the target language to normalize this output. Troyanskii's proposal remained unknown until the late 1950s, by which time computers were well-known and utilized. == The early years == The first set of proposals for computer based machine translation was presented in 1949 by Warren Weaver, a researcher at the Rockefeller Foundation, "Translation memorandum". These proposals were based on information theory, successes in code breaking during the Second World War, and theories about the universal principles underlying natural language. A few years after Weaver submitted his proposals, research began in earnest at many universities in the United States. On 7 January 1954 the Georgetown–IBM experiment was held in New York at the head office of IBM. This was the first public demonstration of a machine translation system. The demonstration was widely reported in the newspapers and garnered public interest. The system itself, however, was no more than a "toy" system. It had only 250 words and translated 49 carefully selected Russian sentences into English – mainly in the field of chemistry. Nevertheless, it encouraged the idea that machine translation was imminent and stimulated the financing of the research, not only in the US but worldwide. Early systems used large bilingual dictionaries and hand-coded rules for fixing the word order in the final output which was eventually considered too restrictive in linguistic developments at the time. For example, generative linguistics and transformational grammar were exploited to improve the quality of translations. During this period operational systems were installed. The United States Air Force used a system produced by IBM and Washington University in St. Louis, while the Atomic Energy Commission and Euratom, in Italy, used a system developed at Georgetown University. While the quality of the output was poor it met many of the customers' needs, particularly in terms of speed. At the end of the 1950s, Yehoshua Bar-Hillel was asked by the US government to look into machine translation, to assess the possibility of fully automatic high-quality translation by machines. Bar-Hillel described the problem of semantic ambiguity or double-meaning, as illustrated in the following sentence: Little John was looking for his toy box. Finally he found it. The box was in the pen. The word pen may have two meanings: the first meaning, something used to write in ink with; the second meaning, a container of some kind. To a human, the meaning is obvious, but Bar-Hillel claimed that without a "universal encyclopedia" a machine would never be able to deal with this problem. At the time, this type of semantic ambiguity could only be solved by writing source texts for machine translation in a controlled language that uses a vocabulary in which each word has exactly one meaning. == The 1960s, the ALPAC report and the seventies == Research in the 1960s in both the Soviet Union and the United States concentrated mainly on the Russian–English language pair. The objects of translation were chiefly scientific and technical documents, such as articles from scientific journals. The rough translations produced were sufficient to get a basic understanding of the articles. If an article discussed a subject deemed to be confidential, it was sent to a human translator for a complete translation; if not, it was discarded. A great blow came to machine-translation research in 1966 with the publication of the ALPAC report. The report was commissioned by the US government and delivered by ALPAC, the Automatic Language Processing Advisory Committee, a group of seven scientists convened by the US government in 1964. The US government was concerned that there was a lack of progress being made despite significant expenditure. The report concluded that machine translation was more expensive, less accurate and slower than human translation, and that despite the expenditures, machine translation was not likely to reach the quality of a human translator in the near future. The report recommended, however, that tools be developed to aid translators – automatic dictionaries, for example – and that some research in computational linguistics should continue to be supported. The publication of the report had a profound impact on research into machine translation in the United States, and to a lesser extent the Soviet Union and United Kingdom. Research, at least in the US, was almost completely abandoned for over a decade. In Canada, France and Germany, however, research continued. In the US the main exceptions were the founders of SYSTRAN (Peter Toma) and Logos (Bernard Scott), who established their companies in 1968 and 1970 respectively and served the US Department of Defense. In 1970, the SYSTRAN system was installed for the United States Air Force, and subsequently by the Commission of the European Communities in 1976. The METEO System, developed at the Université de Montréal, was installed in Canada in 1977 to translate weather forecasts from English to French, and was translating close to 80,000 words per day or 30 million words per year until it was replaced by a competitor's system on 30 September 2001. While research in the 1960s concentrated on limited language pairs and input, demand in the 1970s was for low-cost systems that could translate a range of technical and commercial documents. This demand was spurred by the increase of globalisation and the demand for translation in Canada, Europe, and Japan. == The 1980s and early 1990s == By the 1980s, both the diversity and the number of installed systems for machine translation had increased. A number of systems relying on mainframe technology were in use, such as SYSTRAN, Logos, Ariane-G5, and Metal. As a result of the improved availability of microcomputers, there was a market for lower-end machine translation systems. Many companies took advantage of this in Europe, Japan, and the USA. Systems were also brought onto the market in China, Eastern Europe, Korea, and the Soviet Union. During the 1980s there was a lot of activity in MT in Japan especially. With the fifth-generation co

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  Ni1000

  The Ni1000 is an artificial neural network chip developed by Nestor Corporation and Intel, developed in the 1990s. It is Intel's second-generation neural network chip, but the first all-digital chip. The chip is aimed at image analysis applications– containing more than 3 million transistors – and can analyze 40,000 patterns per second. Prototypes running Nestor's OCR software in 1994 were capable of recognizing around 100 handwritten characters per second. The development was funded with money from DARPA and Office of Naval Research.

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  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.

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  Generalized blockmodeling of binary networks

  Generalized blockmodeling of binary networks (also relational blockmodeling) is an approach of generalized blockmodeling, analysing the binary network(s). As most network analyses deal with binary networks, this approach is also considered as the fundamental approach of blockmodeling. This is especially noted, as the set of ideal blocks, when used for interpretation of blockmodels, have binary link patterns, which precludes them to be compared with valued empirical blocks. When analysing the binary networks, the criterion function is measuring block inconsistencies, while also reporting the possible errors. The ideal block in binary blockmodeling has only three types of conditions: "a certain cell must be (at least) 1, a certain cell must be 0 and the f {\displaystyle f} over each row (or column) must be at least 1". It is also used as a basis for developing the generalized blockmodeling of valued networks.

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  ZeroPC

  ZeroPC was a commercial webtop developed by ZeroDesktop, Inc. located in San Mateo, California. ZeroPC has been called a personal cloud OS. It mimicked the look, feel and functionality of the desktop environment of a real operating system. The software was launched in September 2011 through Disrupt SF 2011 event and recently selected to the finalist of SXSW 2012 in Innovative Web Technology category. ZeroPC is web-based and required a Java applet to operate bundled productivity tool Thinkfree. The web applications found on ZeroPC are built on Java in the back end. Features included drag-and-drop functionality, cloud dashboard and personal cloud storage meta services. ZeroPC belonged to a category of services that intended to turn the Web into a full-fledged platform by using Web services as a foundation along with presentation technologies that replicated the experience of desktop applications for users. ZeroPC aggregates content so users can easily access, transfer and share whatever content they want, using a web browser from any device. Its meta-cloud layer supports Dropbox, Box, SugarSync, OneDrive, 4Shared, Google Drive, Evernote, Picasa, Flickr, Instagram, Facebook, Twitter, and Photobucket. ZeroPC Cloud OS platform also provides extensive APIs for iOS and Android App developers. Some of the features found on ZeroPC are: File sharing, Webmail, Cloud Content Navigator, Instant messenger, Sticky Note, Audio/Video Player and Office productivity applications. ZeroPC 2.0 platform ran on AWS for free and paid users. Its platform is licensable to Telco and ISV for commercial purpose. Their clients are SFR, SK Telecom, Hancom and others. As of June 1, 2017, ZeroPC's servers were switched off completely, and ZeroPC is no longer in service since its parent company, NComputing, had launched Virtual Desktop Service in the cloud (AWS) to public. == Browser and Platform Compatibility == The ZeroPC web desktop was compatible with Mac OS X and Microsoft Windows platforms. It is certified to operate on Safari 6.0, Firefox 15.0.1, Google Chrome 22.0.1229.79 m and Internet Explorer 8 and 9. The ZeroPC front end user interface executes entirely within a web browser (see above) and uses HTML, some features of HTML5, JavaScript, AJAX and an optional Java plug-in. == Security == All communication between the ZeroPC front end user interface and the ZeroPC back end servers is encrypted using SSL (HTTPS) protocol. Furthermore, any content stored in the ZeroPC server-side repository is also encrypted using 256-bit Advanced Encryption Standard (AES-256) by Amazon S3 on AWS. ZeroPC users could connect their ZeroPC profile to other storage services such as Dropbox and Box. This connection allows the ZeroPC user to fully manage their content stored in these other storage services. To establish the connection ZeroPC rigorously adhered to the Oauth implementation provided by the target storage service. Upon completion of the Oauth process, ZeroPC stores the relevant access token in the user's profile. This token, along with all other sensitive password related data was encrypted using AES 256-bit key size. == Implementations == As noted above, the ZeroPC platform was hosted on Amazon Web Services infrastructure and is available to the general consumer. A user was allowed to sign up by selecting one of three account plans including a no-cost option. The ZeroPC could also be white-labeled for organizations wishing to provide this functionality to their own users. The white-label options include managed hosting on Amazon Web Services infrastructure and also installation within the organization's IT infrastructure. == User Access Points == The ZeroPC infrastructure provided user access to content and features in several different ways. As described in this article the user can access their information by signing into the ZeroPC web desktop. Additionally, ZeroPC offers native applications designed to run on popular mobile devices including smartphones and tablets. == Leadership == ZeroPC was founded by Chief Executive Officer, Young Song, an entrepreneur who previously founded NComputing, a $60 million venture-backed company. He also co-founded eMachines, Inc., a low-cost computer brand (later acquired by Gateway).

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  Softplus

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

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  Count sketch

  Count sketch is a type of dimensionality reduction that is particularly efficient in statistics, machine learning and algorithms. It was invented by Moses Charikar, Kevin Chen and Martin Farach-Colton in an effort to speed up the AMS Sketch by Alon, Matias and Szegedy for approximating the frequency moments of streams (these calculations require counting of the number of occurrences for the distinct elements of the stream). The sketch is nearly identical to the Feature hashing algorithm by John Moody, but differs in its use of hash functions with low dependence, which makes it more practical. In order to still have a high probability of success, the median trick is used to aggregate multiple count sketches, rather than the mean. These properties allow use for explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms. == Intuitive explanation == The inventors of this data structure offer the following iterative explanation of its operation: at the simplest level, the output of a single hash function s mapping stream elements q into {+1, -1} is feeding a single up/down counter C. After a single pass over the data, the frequency n ( q ) {\displaystyle n(q)} of a stream element q can be approximated, although extremely poorly, by the expected value E [ C ⋅ s ( q ) ] {\displaystyle {\mathbf {E}}[C\cdot s(q)]} ; a straightforward way to improve the variance of the previous estimate is to use an array of different hash functions s i {\displaystyle s_{i}} , each connected to its own counter C i {\displaystyle C_{i}} . For each i, the E [ C i ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i}\cdot s_{i}(q)]=n(q)} still holds, so averaging across the i range will tighten the approximation; the previous construct still has a major deficiency: if a lower-frequency-but-still-important output element a exhibits a hash collision with a high-frequency element even for one of the s i {\displaystyle s_{i}} hashes, n ( a ) {\displaystyle n(a)} estimate can be significantly affected. Avoiding this requires reducing the frequency of collision counter updates between any two distinct elements. This is achieved by replacing each C i {\displaystyle C_{i}} in the previous construct with an array of m counters (making the counter set into a two-dimensional matrix C i , j {\displaystyle C_{i,j}} ), with index j of a particular counter to be incremented/decremented selected via another set of hash functions h i {\displaystyle h_{i}} that map element q into the range {1..m}. Since E [ C i , h i ( q ) ⋅ s i ( q ) ] = n ( q ) {\displaystyle {\mathbf {E}}[C_{i,h_{i}(q)}\cdot s_{i}(q)]=n(q)} , averaging across all values of i will work. == Mathematical definition == 1. For constants w {\displaystyle w} and t {\displaystyle t} (to be defined later) independently choose d = 2 t + 1 {\displaystyle d=2t+1} random hash functions h 1 , … , h d {\displaystyle h_{1},\dots ,h_{d}} and s 1 , … , s d {\displaystyle s_{1},\dots ,s_{d}} such that h i : [ n ] → [ w ] {\displaystyle h_{i}:[n]\to [w]} and s i : [ n ] → { ± 1 } {\displaystyle s_{i}:[n]\to \{\pm 1\}} . It is necessary that the hash families from which h i {\displaystyle h_{i}} and s i {\displaystyle s_{i}} are chosen be pairwise independent. 2. For each item q i {\displaystyle q_{i}} in the stream, add s j ( q i ) {\displaystyle s_{j}(q_{i})} to the h j ( q i ) {\displaystyle h_{j}(q_{i})} th bucket of the j {\displaystyle j} th hash. At the end of this process, one has w d {\displaystyle wd} sums ( C i j ) {\displaystyle (C_{ij})} where C i , j = ∑ h i ( k ) = j s i ( k ) . {\displaystyle C_{i,j}=\sum _{h_{i}(k)=j}s_{i}(k).} To estimate the count of q {\displaystyle q} s one computes the following value: r q = median i = 1 d s i ( q ) ⋅ C i , h i ( q ) . {\displaystyle r_{q}={\text{median}}_{i=1}^{d}\,s_{i}(q)\cdot C_{i,h_{i}(q)}.} The values s i ( q ) ⋅ C i , h i ( q ) {\displaystyle s_{i}(q)\cdot C_{i,h_{i}(q)}} are unbiased estimates of how many times q {\displaystyle q} has appeared in the stream. The estimate r q {\displaystyle r_{q}} has variance O ( m i n { m 1 2 / w 2 , m 2 2 / w } ) {\displaystyle O(\mathrm {min} \{m_{1}^{2}/w^{2},m_{2}^{2}/w\})} , where m 1 {\displaystyle m_{1}} is the length of the stream and m 2 2 {\displaystyle m_{2}^{2}} is ∑ q ( ∑ i [ q i = q ] ) 2 {\displaystyle \sum _{q}(\sum _{i}[q_{i}=q])^{2}} . Furthermore, r q {\displaystyle r_{q}} is guaranteed to never be more than 2 m 2 / w {\displaystyle 2m_{2}/{\sqrt {w}}} off from the true value, with probability 1 − e − O ( t ) {\displaystyle 1-e^{-O(t)}} . === Vector formulation === Alternatively Count-Sketch can be seen as a linear mapping with a non-linear reconstruction function. Let M ( i ∈ [ d ] ) ∈ { − 1 , 0 , 1 } w × n {\displaystyle M^{(i\in [d])}\in \{-1,0,1\}^{w\times n}} , be a collection of d = 2 t + 1 {\displaystyle d=2t+1} matrices, defined by M h i ( j ) , j ( i ) = s i ( j ) {\displaystyle M_{h_{i}(j),j}^{(i)}=s_{i}(j)} for j ∈ [ w ] {\displaystyle j\in [w]} and 0 everywhere else. Then a vector v ∈ R n {\displaystyle v\in \mathbb {R} ^{n}} is sketched by C ( i ) = M ( i ) v ∈ R w {\displaystyle C^{(i)}=M^{(i)}v\in \mathbb {R} ^{w}} . To reconstruct v {\displaystyle v} we take v j ∗ = median i C j ( i ) s i ( j ) {\displaystyle v_{j}^{}={\text{median}}_{i}C_{j}^{(i)}s_{i}(j)} . This gives the same guarantees as stated above, if we take m 1 = ‖ v ‖ 1 {\displaystyle m_{1}=\|v\|_{1}} and m 2 = ‖ v ‖ 2 {\displaystyle m_{2}=\|v\|_{2}} . == Relation to Tensor sketch == The count sketch projection of the outer product of two vectors is equivalent to the convolution of two component count sketches. The count sketch computes a vector convolution C ( 1 ) x ∗ C ( 2 ) x T {\displaystyle C^{(1)}x\ast C^{(2)}x^{T}} , where C ( 1 ) {\displaystyle C^{(1)}} and C ( 2 ) {\displaystyle C^{(2)}} are independent count sketch matrices. Pham and Pagh show that this equals C ( x ⊗ x T ) {\displaystyle C(x\otimes x^{T})} – a count sketch C {\displaystyle C} of the outer product of vectors, where ⊗ {\displaystyle \otimes } denotes Kronecker product. The fast Fourier transform can be used to do fast convolution of count sketches. By using the face-splitting product such structures can be computed much faster than normal matrices.

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  Sigmoid function

  A sigmoid function is any mathematical function whose graph has a characteristic S-shaped or sigmoid curve. A common example of a sigmoid function is the logistic function. Other sigmoid functions are given in the Examples section. In some fields, most notably in the context of artificial neural networks, the term "sigmoid function" is used as a synonym for "logistic function". Special cases of sigmoid functions include the Gompertz curve (used in modeling systems that saturate at large values of x) and the ogee curve (used in the spillway of some dams). Sigmoid functions have domain of all real numbers, with return (response) value commonly monotonically increasing but could be decreasing. Sigmoid functions most often show a return value (y axis) in the range 0 to 1. Another commonly used range is from −1 to 1. There is also the Heaviside step function, which instantaneously transitions between 0 and 1. A wide variety of sigmoid functions including the logistic and hyperbolic tangent functions have been used as the activation function of artificial neurons. Sigmoid curves are also common in statistics as cumulative distribution functions (which go from 0 to 1), such as the integrals of the logistic density, the normal density, and Student's t probability density functions. The logistic sigmoid function is invertible, and its inverse is the logit function. == Theory == In mathematics, a unitary sigmoid function is a bounded sigmoid-type function normalized to the unit range, typically with lower and upper asymptotes at 0 and 1. The theory proposed by Grebenc distinguishes three kinds of unitary sigmoid functions according to their asymptotic behavior and the presence or absence of oscillation near the asymptotes. A general form of a unitary sigmoid function is y = A S ( f ( x ) ) + B , {\displaystyle y=A\,S(f(x))+B,} where S {\displaystyle S} is an increasing sigmoid function, f ( x ) {\displaystyle f(x)} is a transformation of the independent variable, and A {\displaystyle A} and B {\displaystyle B} are constants controlling scaling and translation. === Classification === ==== 1st kind ==== A unitary sigmoid function of the first kind is a bounded increasing function that approaches its lower and upper asymptotes monotonically, without oscillation. This class includes many of the standard sigmoid functions used in statistics, biomathematics, and engineering, such as the logistic function and related generalizations. ==== 2nd kind ==== A unitary sigmoid function of the second kind is a bounded increasing function that oscillates near the upper asymptote while preserving an overall sigmoid transition. ==== 3rd kind ==== A unitary sigmoid function of the third kind is a bounded increasing function that oscillates near both the lower and upper asymptotes. These functions retain the global shape of a sigmoid curve but exhibit oscillatory behavior in the vicinity of both limiting states. === Taxonomy === The tables below show the taxonomy of unitary sigmoid functions of all three kinds. Table 1. Taxonomy matrix with examples of sigmoid functions of the 1st kind Table 2. Taxonomy matrix with examples of sigmoid functions of the 2nd kind on the unbounded interval Table 3. Taxonomy matrix with examples of sigmoid functions of the 3rd kind === Construction methods === The same theory presents a list of 30 methods for constructing sigmoid functions.. These include algebraic transformations, integration and convolution methods, constructions from bell-shaped functions, solutions of ordinary and partial differential equations, recursive schemes, stochastic differential equations, feedback systems, and chaotic systems. M0: Construction method for sigmoid functions not evident or intuitive M1: Inverse of singularity functions M2: Sigmoid functions of embedded positive functions M3: Rising a sigmoid function to the power M4: Exponentiating a sigmoid function M5: Symmetric sigmoid functions derived from asymmetric ones M6: Sigmoid functions of the reciprocal independent variable M7: Embedding a sigmoid function into other function M8: Sum of sigmoid functions M9: Multiplication of sigmoid functions M10: Integral of the product of an increasing and a decreasing function M11: Derivation from lambda (bell-shaped) functions M12: Integration of lambda (bell-shaped) function M13: Integration of the sum of lambda (bell-shaped) functions M14: Integration of the product of two lambda (bell-shaped) functions M15: Integration of the difference of two shifted sigmoid functions M16: Integration of the product of two shifted sigmoid functions M17: Convolution of sigmoid functions M18: Integration of the product of lambda and sigmoid function M19: Solutions of ordinary differential equations M20: Solutions of partial differential equation (PDE) M21: Solutions of functional differential equation (FDE) M22: Sum of a sigmoid function and some derivatives M23: Combination of sigmoid functions, its derivative and integral M24: Filtering sigmoid functions M25: Special cases of Gauss hypergeometric functions M26: Feedback closed-loop systems M27: Recursive functions M28: Recursive time-delayed feed-forward loops M29: Solutions of stochastic differential equation M30: Chaotic sigmoid functions Consult reference for more details. == Definition == A sigmoid function is a bounded, differentiable, real function that is defined for all real input values and has a positive derivative at each point. == Properties == In general, a sigmoid function is monotonic, and has a first derivative which is bell shaped. Conversely, the integral of any continuous, non-negative, bell-shaped function (with one local maximum and no local minimum, unless degenerate) will be sigmoidal. Thus the cumulative distribution functions for many common probability distributions are sigmoidal. One such example is the error function, which is related to the cumulative distribution function of a normal distribution; another is the arctan function, which is related to the cumulative distribution function of a Cauchy distribution. A sigmoid function is constrained by a pair of horizontal asymptotes as x → ± ∞ {\displaystyle x\rightarrow \pm \infty } . A sigmoid function is convex for values less than a particular point, and it is concave for values greater than that point: in many of the examples here, that point is 0. == Examples == Logistic function f ( x ) = 1 1 + e − x {\displaystyle f(x)={\frac {1}{1+e^{-x}}}} Hyperbolic tangent (shifted and scaled version of the logistic function, above) f ( x ) = tanh ⁡ x = e x − e − x e x + e − x {\displaystyle f(x)=\tanh x={\frac {e^{x}-e^{-x}}{e^{x}+e^{-x}}}} Arctangent function f ( x ) = arctan ⁡ x {\displaystyle f(x)=\arctan x} Gudermannian function f ( x ) = gd ⁡ ( x ) = ∫ 0 x d t cosh ⁡ t = 2 arctan ⁡ ( tanh ⁡ ( x 2 ) ) {\displaystyle f(x)=\operatorname {gd} (x)=\int _{0}^{x}{\frac {dt}{\cosh t}}=2\arctan \left(\tanh \left({\frac {x}{2}}\right)\right)} Error function f ( x ) = erf ⁡ ( x ) = 2 π ∫ 0 x e − t 2 d t {\displaystyle f(x)=\operatorname {erf} (x)={\frac {2}{\sqrt {\pi }}}\int _{0}^{x}e^{-t^{2}}\,dt} Generalised logistic function f ( x ) = ( 1 + e − x ) − α , α > 0 {\displaystyle f(x)=\left(1+e^{-x}\right)^{-\alpha },\quad \alpha >0} Smoothstep function f ( x ) = { ( ∫ 0 1 ( 1 − u 2 ) N d u ) − 1 ∫ 0 x ( 1 − u 2 ) N d u , | x | ≤ 1 sgn ⁡ ( x ) | x | ≥ 1 N ∈ Z ≥ 1 {\displaystyle f(x)={\begin{cases}{\displaystyle \left(\int _{0}^{1}\left(1-u^{2}\right)^{N}du\right)^{-1}\int _{0}^{x}\left(1-u^{2}\right)^{N}\ du},&|x|\leq 1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\quad N\in \mathbb {Z} \geq 1} Some algebraic functions, for example f ( x ) = x 1 + x 2 {\displaystyle f(x)={\frac {x}{\sqrt {1+x^{2}}}}} and in a more general form f ( x ) = x ( 1 + | x | k ) 1 / k {\displaystyle f(x)={\frac {x}{\left(1+|x|^{k}\right)^{1/k}}}} Up to shifts and scaling, many sigmoids are special cases of f ( x ) = φ ( φ ( x , β ) , α ) , {\displaystyle f(x)=\varphi (\varphi (x,\beta ),\alpha ),} where φ ( x , λ ) = { ( 1 − λ x ) 1 / λ λ ≠ 0 e − x λ = 0 {\displaystyle \varphi (x,\lambda )={\begin{cases}(1-\lambda x)^{1/\lambda }&\lambda \neq 0\\e^{-x}&\lambda =0\\\end{cases}}} is the inverse of the negative Box–Cox transformation, and α < 1 {\displaystyle \alpha <1} and β < 1 {\displaystyle \beta <1} are shape parameters. Smooth transition function normalized to (−1,1): f ( x ) = { 2 1 + e − 2 m x 1 − x 2 − 1 , | x | < 1 sgn ⁡ ( x ) | x | ≥ 1 = { tanh ⁡ ( m x 1 − x 2 ) , | x | < 1 sgn ⁡ ( x ) | x | ≥ 1 {\displaystyle {\begin{aligned}f(x)&={\begin{cases}{\displaystyle {\frac {2}{1+e^{-2m{\frac {x}{1-x^{2}}}}}}-1},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\\&={\begin{cases}{\displaystyle \tanh \left(m{\frac {x}{1-x^{2}}}\right)},&|x|<1\\\\\operatorname {sgn}(x)&|x|\geq 1\\\end{cases}}\end{aligned}}} using the hyperbolic tangent mentioned above. Here, m {\displaystyle m} is a free parameter encoding the slope at x = 0 {\displaystyle x=0} , which must be great

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  Lingua Libre

  Lingua Libre is an online collaborative project and tool by the Wikimédia France association, which aims to build a collaborative, multilingual, audiovisual speech corpus under a free license. It mostly consists of a rapid recording online service which allows the user to chain hundreds of recordings. Contributors have produced content in 310+ languages. == Description == Lingua Libre enables the recording of words, phrases or sentences of any language, oral (audio recording) or signed (video recording). Words are presented to the speaker in the form of a list, created on the spot, in advance, or by reusing an existing Wikimedia category. The speaker simply reads the word displayed on the screen, and the software moves on to the next word when it detects a silence after the read word. This principle, borrowed from the open source software Shtooka recorder with the help of its creator, Nicolas Vion, makes it possible to record several hundreds of words per hour. The recordings are then uploaded automatically from the web client to the Wikimedia Commons media library. In spring 2021, Lingua Libre was offline due to a fire in Strasbourg, but no audio recordings were lost. === Use of the recordings === The recordings can be consulted either on Lingua Libre or on Commons. They are mainly used on other Wikimedia projects, for example to illustrate entries on Wiktionaries or proper nouns in Wikipedia articles. The re-use of the recordings in a language teaching context is envisaged. Language learners can freely download pronunciations and use them on GoldenDict, a popular dictionary software. Thus, audio recordings can be used as “Pronunciation Dictionaries” on GoldenDict without needing internet connection. The recordings are also reused in Natural Language Processing projects, for example to drive Mozilla's DeepSpeech speech recognition engines. == Versions == Lingua Libre was initiated on January 23, 2015 and has had three successive versions: === Lingua Libre v.1 (2016) === As part of the Languages of France project, which aims to document and promote the regional languages of France on Wikimedia and Internet projects in general, the conception of Lingua Libre started in November 2015, partly funded by the DGLFLF (General Delegation for the French language and the languages of France). The first version of the project was launched in August 2016. Only suitable for audio recording, Lingua Libre was shown during a workshop on Occitan language in December 2016, and then presented to the online Wikimedia community and at international events in 2017. === Lingua Libre v.2 (2018) === A complete rebuilding was launched at the end of 2017. The new version of Lingua Libre is based on MediaWiki, uses Wikibase and OAuth to better integrate into the Wikimedia environment. The interface is translated via Translatewiki.net so that the project can be used by a large number of communities. The new version of the site was ready in June 2018 and opened to the public in August 2018. === Lingua Libre v.2.2 (2020) === In 2020, important changes were made to the platform; a new look was developed especially for the site, the .org domain replaced the .fr domain used until then, and added support for sign languages through video recording. == Statistics == In the first two years of the project's launch, approximately 10,000 recordings were made. The transition to v.2 was accompanied by a sharp increase in the contributions. The number of recordings multiplied by 10 in less than a year, exceeding the 100,000 threshold in May 2019. These recordings were made by 127 speakers in almost 50 languages. By September 2020, the platform had more than 300,000 recordings in 90 languages with more than 350 speakers. The 500,000 recordings milestone was reached in June 2021, thanks to 540 speakers of 120 languages.

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  Santa Fe Trail problem

  The Santa Fe Trail problem is a genetic programming exercise in which artificial ants search for food pellets according to a programmed set of instructions. The layout of food pellets in the Santa Fe Trail problem has become a standard for comparing different genetic programming algorithms and solutions. One method for programming and testing algorithms on the Santa Fe Trail problem is by using the NetLogo application. There is at least one case of a student creating a Lego robotic ant to solve the problem.

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  Multinomial logistic regression

  In statistics, multinomial logistic regression is a classification method that generalizes logistic regression to multiclass problems, i.e. with more than two possible discrete outcomes. That is, it is a model that is used to predict the probabilities of the different possible outcomes of a categorically distributed dependent variable, given a set of independent variables (which may be real-valued, binary-valued, categorical-valued, etc.). Multinomial logistic regression is known by a variety of other names, including polytomous LR, multiclass LR, softmax regression, multinomial logit (mlogit), the maximum entropy (MaxEnt) classifier, and the conditional maximum entropy model. == Background == Multinomial logistic regression is used when the dependent variable in question is nominal (equivalently categorical, meaning that it falls into any one of a set of categories that cannot be ordered in any meaningful way) and for which there are more than two categories. Some examples would be: Which major will a college student choose, given their grades, stated likes and dislikes, etc.? Which blood type does a person have, given the results of various diagnostic tests? In a hands-free mobile phone dialing application, which person's name was spoken, given various properties of the speech signal? Which candidate will a person vote for, given particular demographic characteristics? Which country will a firm locate an office in, given the characteristics of the firm and of the various candidate countries? These are all statistical classification problems. They all have in common a dependent variable to be predicted that comes from one of a limited set of items that cannot be meaningfully ordered, as well as a set of independent variables (also known as features, explanators, etc.), which are used to predict the dependent variable. Multinomial logistic regression is a particular solution to classification problems that use a linear combination of the observed features and some problem-specific parameters to estimate the probability of each particular value of the dependent variable. The best values of the parameters for a given problem are usually determined from some training data (e.g. some people for whom both the diagnostic test results and blood types are known, or some examples of known words being spoken). == Assumptions == The multinomial logistic model assumes that data are case-specific; that is, each independent variable has a single value for each case. As with other types of regression, there is no need for the independent variables to be statistically independent from each other (unlike, for example, in a naive Bayes classifier); however, collinearity is assumed to be relatively low, as it becomes difficult to differentiate between the impact of several variables if this is not the case. If the multinomial logit is used to model choices, it relies on the assumption of independence of irrelevant alternatives (IIA), which is not always desirable. This assumption states that the odds of preferring one class over another do not depend on the presence or absence of other "irrelevant" alternatives. For example, the relative probabilities of taking a car or bus to work do not change if a bicycle is added as an additional possibility. This allows the choice of K alternatives to be modeled as a set of K − 1 independent binary choices, in which one alternative is chosen as a "pivot" and the other K − 1 compared against it, one at a time. The IIA hypothesis is a core hypothesis in rational choice theory; however numerous studies in psychology show that individuals often violate this assumption when making choices. An example of a problem case arises if choices include a car and a blue bus. Suppose the odds ratio between the two is 1 : 1. Now if the option of a red bus is introduced, a person may be indifferent between a red and a blue bus, and hence may exhibit a car : blue bus : red bus odds ratio of 1 : 0.5 : 0.5, thus maintaining a 1 : 1 ratio of car : any bus while adopting a changed car : blue bus ratio of 1 : 0.5. Here the red bus option was not in fact irrelevant, because a red bus was a perfect substitute for a blue bus. If the multinomial logit is used to model choices, it may in some situations impose too much constraint on the relative preferences between the different alternatives. It is especially important to take into account if the analysis aims to predict how choices would change if one alternative were to disappear (for instance if one political candidate withdraws from a three candidate race). Other models like the nested logit or the multinomial probit may be used in such cases as they allow for violation of the IIA. == Model == === Introduction === There are multiple equivalent ways to describe the mathematical model underlying multinomial logistic regression. This can make it difficult to compare different treatments of the subject in different texts. The article on logistic regression presents a number of equivalent formulations of simple logistic regression, and many of these have analogues in the multinomial logit model. The idea behind all of them, as in many other statistical classification techniques, is to construct a linear predictor function that constructs a score from a set of weights that are linearly combined with the explanatory variables (features) of a given observation using a dot product: score ⁡ ( X i , k ) = β k ⋅ X i , {\displaystyle \operatorname {score} (\mathbf {X} _{i},k)={\boldsymbol {\beta }}_{k}\cdot \mathbf {X} _{i},} where Xi is the vector of explanatory variables describing observation i, βk is a vector of weights (or regression coefficients) corresponding to outcome k, and score(Xi, k) is the score associated with assigning observation i to category k. In discrete choice theory, where observations represent people and outcomes represent choices, the score is considered the utility associated with person i choosing outcome k. The predicted outcome is the one with the highest score. The difference between the multinomial logit model and numerous other methods, models, algorithms, etc. with the same basic setup (the perceptron algorithm, support vector machines, linear discriminant analysis, etc.) is the procedure for determining (training) the optimal weights/coefficients and the way that the score is interpreted. In particular, in the multinomial logit model, the score can directly be converted to a probability value, indicating the probability of observation i choosing outcome k given the measured characteristics of the observation. This provides a principled way of incorporating the prediction of a particular multinomial logit model into a larger procedure that may involve multiple such predictions, each with a possibility of error. Without such means of combining predictions, errors tend to multiply. For example, imagine a large predictive model that is broken down into a series of submodels where the prediction of a given submodel is used as the input of another submodel, and that prediction is in turn used as the input into a third submodel, etc. If each submodel has 90% accuracy in its predictions, and there are five submodels in series, then the overall model has only 0.95 = 59% accuracy. If each submodel has 80% accuracy, then overall accuracy drops to 0.85 = 33% accuracy. This issue is known as error propagation and is a serious problem in real-world predictive models, which are usually composed of numerous parts. Predicting probabilities of each possible outcome, rather than simply making a single optimal prediction, is one means of alleviating this issue. === Setup === The basic setup is the same as in logistic regression, the only difference being that the dependent variables are categorical rather than binary, i.e. there are K possible outcomes rather than just two. The following description is somewhat shortened; for more details, consult the logistic regression article. ==== Data points ==== Specifically, it is assumed that we have a series of N observed data points. Each data point i (ranging from 1 to N) consists of a set of M explanatory variables x1,i ... xM,i (also known as independent variables, predictor variables, features, etc.), and an associated categorical outcome Yi (also known as dependent variable, response variable), which can take on one of K possible values. These possible values represent logically separate categories (e.g. different political parties, blood types, etc.), and are often described mathematically by arbitrarily assigning each a number from 1 to K. The explanatory variables and outcome represent observed properties of the data points, and are often thought of as originating in the observations of N "experiments" — although an "experiment" may consist of nothing more than gathering data. The goal of multinomial logistic regression is to construct a model that explains the relationship between the explanatory variables and the outcome, so tha

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  Neocognitron

  The neocognitron is a hierarchical, multilayered artificial neural network proposed by Kunihiko Fukushima in 1979. It has been used for Japanese handwritten character recognition and other pattern recognition tasks, and served as the inspiration for convolutional neural networks. Previously in 1969, he published a similar architecture, but with hand-designed kernels inspired by convolutions in mammalian vision. In 1975 he improved it to the Cognitron, and in 1979 he improved it to the neocognitron, which learns all convolutional kernels by unsupervised learning (in his terminology, "self-organized by 'learning without a teacher'"). The neocognitron was inspired by the model proposed by Hubel & Wiesel in 1959. They found two types of cells in the visual primary cortex called simple cell and complex cell, and also proposed a cascading model of these two types of cells for use in pattern recognition tasks. The neocognitron is a natural extension of these cascading models. The neocognitron consists of multiple types of cells, the most important of which are called S-cells and C-cells. The local features are extracted by S-cells, and these features' deformation, such as local shifts, are tolerated by C-cells. Local features in the input are integrated gradually and classified in the higher layers. The idea of local feature integration is found in several other models, such as the Convolutional Neural Network model, the SIFT method, and the HoG method. There are various kinds of neocognitron. For example, some types of neocognitron can detect multiple patterns in the same input by using backward signals to achieve selective attention.

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  Security type system

  In computer science, a type system can be described as a syntactic framework which contains a set of rules that are used to assign a type property (int, boolean, char etc.) to various components of a computer program, such as variables or functions. A security type system works in a similar way, only with a main focus on the security of the computer program, through information flow control. Thus, the various components of the program are assigned security types, or labels. The aim of a such system is to ultimately be able to verify that a given program conforms to the type system rules and satisfies non-interference. Security type systems is one of many security techniques used in the field of language-based security, and is tightly connected to information flow and information flow policies. In simple terms, a security type system can be used to detect if there exists any kind of violation of confidentiality or integrity in a program, i.e. the programmer wants to detect if the program is in line with the information flow policy or not. == A simple information flow policy == Suppose there are two users, A and B. In a program, the following security classes (SC) are introduced: SC = {∅, {A}, {B}, {A,B}}, where ∅ is the empty set. The information flow policy should define the direction that information is allowed to flow, which is dependent on whether the policy allows read or write operations. This example considers read operations (confidentiality). The following flows are allowed: → = {({A}, {A}), ({B}, {B}), ({A,B}, {A,B}), ({A,B}, {A}), ({A,B}, {B}), ({A}, ∅), ({B}, ∅), ({A,B}, ∅)} This can also be described as a superset (⊇). In words: information is allowed to flow towards stricter levels of confidentiality. The combination operator (⊕) can express how security classes can perform read operations with respect to other security classes. For example: {A} ⊕ {A,B} = {A} — the only security class that can read from both {A} and {A,B} is {A}. {A} ⊕ {B} = ∅ — neither {A} nor {B} are allowed to read from both {A} and {B}. This can also be described as an intersection (∩) between security classes. An information flow policy can be illustrated as a Hasse diagram. The policy should also be a lattice, that is, it has a greatest lower-bound and least upper-bound (there always exists a combination between security classes). In the case of integrity, information will flow in the opposite direction, thus the policy will be inverted. == Information flow policy in security type systems == Once the policy is in place, the software developer can apply the security classes to the program components. Use of a security type system is usually combined with a compiler that can perform the verification of the information flow according to the type system rules. For the sake of simplicity, a very simple computer program, together with the information flow policy as described in the previous section, can be used as a demonstration. The simple program is given in the following pseudocode: if y{A} = 1 then x{A,B} := 0 else x{A,B} := 1 Here, an equality check is made on a variable y that is assigned the security class {A}. A variable x with a lower security class ({A,B}) is influenced by this check. This means that information is leaking from class {A} to class {A,B}, which is a violation of the confidentiality policy. This leak should be detected by the security type system. === Example === Designing a security type system requires a function (also known as a security environment) that creates a mapping from variables to security types, or classes. This function can be called Γ, such that Γ(x) = τ, where x is a variable and τ is the security class, or type. Security classes are assigned (also called "judgement") to program components, using the following notation: Types are assigned to read operations by: Γ ⊢ e : τ. Types are assigned to write operations by: Γ ⊢ S : τ cmd. Constants can be assigned any type. The following bottom-up notation can be used to decompose the program: ⁠assumption1 ... assumptionn/conclusion⁠. Once the program is decomposed into trivial judgements, by which the type can easily be determined, the types for the less trivial parts of the program can be derived. Each "numerator" is considered in isolation, looking at the type of each statement to see if an allowed type can be derived for the "denominator", based on the defined type system "rules". ==== Rules ==== The main part of the security type system is the rules. They say how the program should be decomposed and how type verification should be performed. This toy program consists of a conditional test and two possible variable assignments. Rules for these two events are defined as follows: Applying this to the simple program introduced above yields: The type system detects the policy violation in line 2, where a read operation of security class {A} is performed, followed by two write operations of a less strict security class {A,B}. In more formalized terms, {A} ⋢ {A,B}, {A,B} (from the rule of the conditional test). Thus, the program is classified as "not typeable". === Soundness === The soundness of a security type system can be informally defined as: If program P is well typed, P satisfies non-interference. Volpano, Smith and Irvine were the first to prove soundness of a security type system for a deterministic imperative programming language with a standard (non-instrumented) semantics using the notion of non-interference.

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  Multidimensional analysis

  In statistics, econometrics and related fields, multidimensional analysis (MDA) is a data analysis process that groups data into two categories: data dimensions and measurements. For example, a data set consisting of the number of wins for a single football team at each of several years is a single-dimensional (in this case, longitudinal) data set. A data set consisting of the number of wins for several football teams in a single year is also a single-dimensional (in this case, cross-sectional) data set. A data set consisting of the number of wins for several football teams over several years is a two-dimensional data set. == Higher dimensions == In many disciplines, two-dimensional data sets are also called panel data. While, strictly speaking, two- and higher-dimensional data sets are "multi-dimensional", the term "multidimensional" tends to be applied only to data sets with three or more dimensions. For example, some forecast data sets provide forecasts for multiple target periods, conducted by multiple forecasters, and made at multiple horizons. The three dimensions provide more information than can be gleaned from two-dimensional panel data sets. == Software == Computer software for MDA include Online analytical processing (OLAP) for data in relational databases, pivot tables for data in spreadsheets, and Array DBMSs for general multi-dimensional data (such as raster data) in science, engineering, and business.

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  Hinge loss

  In machine learning, the hinge loss is a loss function used for training classifiers. The hinge loss is used for "maximum-margin" classification, most notably for support vector machines (SVMs). For an intended output t = ±1 and a classifier score y, the hinge loss of the prediction y is defined as ℓ ( y ) = max ( 0 , 1 − t ⋅ y ) {\displaystyle \ell (y)=\max(0,1-t\cdot y)} Note that y {\displaystyle y} should be the "raw" output of the classifier's decision function, not the predicted class label. For instance, in linear SVMs, y = w ⋅ x + b {\displaystyle y=\mathbf {w} \cdot \mathbf {x} +b} , where ( w , b ) {\displaystyle (\mathbf {w} ,b)} are the parameters of the hyperplane and x {\displaystyle \mathbf {x} } is the input variable(s). When t and y have the same sign (meaning y predicts the right class) and | y | ≥ 1 {\displaystyle |y|\geq 1} , the hinge loss ℓ ( y ) = 0 {\displaystyle \ell (y)=0} . When they have opposite signs, ℓ ( y ) {\displaystyle \ell (y)} increases linearly with y, and similarly if | y | < 1 {\displaystyle |y|<1} , even if it has the same sign (correct prediction, but not by enough margin). The Hinge loss is not a proper scoring rule. == Extensions == While binary SVMs are commonly extended to multiclass classification in a one-vs.-all or one-vs.-one fashion, it is also possible to extend the hinge loss itself for such an end. Several different variations of multiclass hinge loss have been proposed. For example, Crammer and Singer defined it for a linear classifier as ℓ ( y ) = max ( 0 , 1 + max y ≠ t w y x − w t x ) {\displaystyle \ell (y)=\max(0,1+\max _{y\neq t}\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} , where t {\displaystyle t} is the target label, w t {\displaystyle \mathbf {w} _{t}} and w y {\displaystyle \mathbf {w} _{y}} are the model parameters. Weston and Watkins provided a similar definition, but with a sum rather than a max: ℓ ( y ) = ∑ y ≠ t max ( 0 , 1 + w y x − w t x ) {\displaystyle \ell (y)=\sum _{y\neq t}\max(0,1+\mathbf {w} _{y}\mathbf {x} -\mathbf {w} _{t}\mathbf {x} )} . In structured prediction, the hinge loss can be further extended to structured output spaces. Structured SVMs with margin rescaling use the following variant, where w denotes the SVM's parameters, y the SVM's predictions, φ the joint feature function, and Δ the Hamming loss: ℓ ( y ) = max ( 0 , Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ − ⟨ w , ϕ ( x , t ) ⟩ ) = max ( 0 , max y ∈ Y ( Δ ( y , t ) + ⟨ w , ϕ ( x , y ) ⟩ ) − ⟨ w , ϕ ( x , t ) ⟩ ) {\displaystyle {\begin{aligned}\ell (\mathbf {y} )&=\max(0,\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle -\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\\&=\max(0,\max _{y\in {\mathcal {Y}}}\left(\Delta (\mathbf {y} ,\mathbf {t} )+\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {y} )\rangle \right)-\langle \mathbf {w} ,\phi (\mathbf {x} ,\mathbf {t} )\rangle )\end{aligned}}} . == Optimization == The hinge loss is a convex function, so many of the usual convex optimizers used in machine learning can work with it. It is not differentiable, but has a subgradient with respect to model parameters w of a linear SVM with score function y = w ⋅ x {\displaystyle y=\mathbf {w} \cdot \mathbf {x} } that is given by ∂ ℓ ∂ w i = { − t ⋅ x i if t ⋅ y < 1 , 0 otherwise . {\displaystyle {\frac {\partial \ell }{\partial w_{i}}}={\begin{cases}-t\cdot x_{i}&{\text{if }}t\cdot y<1,\\0&{\text{otherwise}}.\end{cases}}} However, since the derivative of the hinge loss at t y = 1 {\displaystyle ty=1} is undefined, smoothed versions may be preferred for optimization, such as Rennie and Srebro's ℓ ( y ) = { 1 2 − t y if t y ≤ 0 , 1 2 ( 1 − t y ) 2 if 0 < t y < 1 , 0 if 1 ≤ t y {\displaystyle \ell (y)={\begin{cases}{\frac {1}{2}}-ty&{\text{if}}~~ty\leq 0,\\{\frac {1}{2}}(1-ty)^{2}&{\text{if}}~~0 Read more →