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  Open-source software security

  Open-source software security is the measure of assurance or guarantee in the freedom from danger and risk inherent to an open-source software system. == Implementation debate == === Benefits === Proprietary software forces the user to accept the level of security that the software vendor is willing to deliver and to accept the rate that patches and updates are released. It is assumed that any compiler that is used creates code that can be trusted, but it has been demonstrated by Ken Thompson that a compiler can be subverted using a compiler backdoor to create faulty executables that are unwittingly produced by a well-intentioned developer. With access to the source code for the compiler, the developer has at least the ability to discover if there is any mal-intention. Kerckhoffs' principle is based on the idea that an enemy can steal a secure military system and not be able to compromise the information. His ideas were the basis for many modern security practices, and followed that security through obscurity is a bad practice. === Drawbacks === Simply making source code available does not guarantee review. An example of this occurring is when Marcus Ranum, an expert on security system design and implementation, released his first public firewall toolkit. At one time, there were over 2,000 sites using his toolkit, but only 10 people gave him any feedback or patches. Having a large amount of eyes reviewing code can "lull a user into a false sense of security". Having many users look at source code does not guarantee that security flaws will be found and fixed. == Metrics and models == There are a variety of models and metrics to measure the security of a system. These are a few methods that can be used to measure the security of software systems. === Number of days between vulnerabilities === It is argued that a system is most vulnerable after a potential vulnerability is discovered, but before a patch is created. By measuring the number of days between the vulnerability and when the vulnerability is fixed, a basis can be determined on the security of the system. There are a few caveats to such an approach: not every vulnerability is equally bad, and fixing a lot of bugs quickly might not be better than only finding a few and taking a little bit longer to fix them, taking into account the operating system, or the effectiveness of the fix. === Poisson process === The Poisson process can be used to measure the rates at which different people find security flaws between open and closed source software. The process can be broken down by the number of volunteers Nv and paid reviewers Np. The rates at which volunteers find a flaw is measured by λv and the rate that paid reviewers find a flaw is measured by λp. The expected time that a volunteer group is expected to find a flaw is 1/(Nv λv) and the expected time that a paid group is expected to find a flaw is 1/(Np λp). === Morningstar model === By comparing a large variety of open source and closed source projects a star system could be used to analyze the security of the project similar to how Morningstar, Inc. rates mutual funds. With a large enough data set, statistics could be used to measure the overall effectiveness of one group over the other. An example of such as system is as follows: 1 Star: Many security vulnerabilities. 2 Stars: Reliability issues. 3 Stars: Follows best security practices. 4 Stars: Documented secure development process. 5 Stars: Passed independent security review. === Coverity scan === Coverity in collaboration with Stanford University has established a new baseline for open-source quality and security. The development is being completed through a contract with the Department of Homeland Security. They are utilizing innovations in automated defect detection to identify critical types of bugs found in software. The level of quality and security is measured in rungs. Rungs do not have a definitive meaning, and can change as Coverity releases new tools. Rungs are based on the progress of fixing issues found by the Coverity Analysis results and the degree of collaboration with Coverity. They start with Rung 0 and currently go up to Rung 2. Rung 0 The project has been analyzed by Coverity's Scan infrastructure, but no representatives from the open-source software have come forward for the results. Rung 1 At rung 1, there is collaboration between Coverity and the development team. The software is analyzed with a subset of the scanning features to prevent the development team from being overwhelmed. Rung 2 There are 11 projects that have been analyzed and upgraded to the status of Rung 2 by reaching zero defects in the first year of the scan. These projects include: AMANDA, ntp, OpenPAM, OpenVPN, Overdose, Perl, PHP, Postfix, Python, Samba, and Tcl.

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  Liquid state machine

  A liquid state machine (LSM) is a type of reservoir computer that uses a spiking neural network. An LSM consists of a large collection of units (called nodes, or neurons). Each node receives time varying input from external sources (the inputs) as well as from other nodes. Nodes are randomly connected to each other. The recurrent nature of the connections turns the time varying input into a spatio-temporal pattern of activations in the network nodes. The spatio-temporal patterns of activation are read out by linear discriminant units. The soup of recurrently connected nodes will end up computing a large variety of nonlinear functions on the input. Given a large enough variety of such nonlinear functions, it is theoretically possible to obtain linear combinations (using the read out units) to perform whatever mathematical operation is needed to perform a certain task, such as speech recognition or computer vision. The word liquid in the name comes from the analogy drawn to dropping a stone into a still body of water or other liquid. The falling stone will generate ripples in the liquid. The input (motion of the falling stone) has been converted into a spatio-temporal pattern of liquid displacement (ripples). LSMs have been put forward as a way to explain the operation of brains. LSMs are argued to be an improvement over the theory of artificial neural networks because: Circuits are not hard coded to perform a specific task. Continuous time inputs are handled "naturally". Computations on various time scales can be done using the same network. The same network can perform multiple computations. Criticisms of LSMs as used in computational neuroscience are that LSMs don't actually explain how the brain functions. At best they can replicate some parts of brain functionality. There is no guaranteed way to dissect a working network and figure out how or what computations are being performed. There is very little control over the process. == Universal function approximation == If a reservoir has fading memory and input separability, with help of a readout, it can be proven the liquid state machine is a universal function approximator using Stone–Weierstrass theorem.

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  Weka (software)

  Waikato Environment for Knowledge Analysis (Weka) is a collection of machine learning and data analysis free software licensed under the GNU General Public License. It was developed at the University of Waikato, New Zealand, and is the companion software to the book "Data Mining: Practical Machine Learning Tools and Techniques". == Description == Weka contains a collection of visualization tools and algorithms for data analysis and predictive modeling, together with graphical user interfaces for easy access to these functions. The original non-Java version of Weka was a Tcl/Tk front-end to (mostly third-party) modeling algorithms implemented in other programming languages, plus data preprocessing utilities in C, and a makefile-based system for running machine learning experiments. This original version was primarily designed as a tool for analyzing data from agricultural domains, but the more recent fully Java-based version (Weka 3), for which development started in 1997, is now used in many different application areas, in particular for educational purposes and research. Advantages of Weka include: Free availability under the GNU General Public License. Portability, since it is fully implemented in the Java programming language and thus runs on almost any modern computing platform. A comprehensive collection of data preprocessing and modeling techniques. Ease of use due to its graphical user interfaces. Weka supports several standard data mining tasks, more specifically, data preprocessing, clustering, classification, regression, visualization, and feature selection. Input to Weka is expected to be formatted according the Attribute-Relational File Format and with the filename bearing the .arff extension. All of Weka's techniques are predicated on the assumption that the data is available as one flat file or relation, where each data point is described by a fixed number of attributes (normally, numeric or nominal attributes, but some other attribute types are also supported). Weka provides access to SQL databases using Java Database Connectivity and can process the result returned by a database query. Weka provides access to deep learning with Deeplearning4j. It is not capable of multi-relational data mining, but there is separate software for converting a collection of linked database tables into a single table that is suitable for processing using Weka. Another important area that is currently not covered by the algorithms included in the Weka distribution is sequence modeling. == Extension packages == In version 3.7.2, a package manager was added to allow the easier installation of extension packages. Some functionality that used to be included with Weka prior to this version has since been moved into such extension packages, but this change also makes it easier for others to contribute extensions to Weka and to maintain the software, as this modular architecture allows independent updates of the Weka core and individual extensions. == History == In 1993, the University of Waikato in New Zealand began development of the original version of Weka, which became a mix of Tcl/Tk, C, and makefiles. In 1997, the decision was made to redevelop Weka from scratch in Java, including implementations of modeling algorithms. In 2005, Weka received the SIGKDD Data Mining and Knowledge Discovery Service Award. In 2006, Pentaho Corporation acquired an exclusive licence to use Weka for business intelligence. It forms the data mining and predictive analytics component of the Pentaho business intelligence suite. Pentaho has since been acquired by Hitachi Vantara, and Weka now underpins the PMI (Plugin for Machine Intelligence) open source component. == Related tools == Auto-WEKA is an automated machine learning system for Weka. Environment for DeveLoping KDD-Applications Supported by Index-Structures (ELKI) is a similar project to Weka with a focus on cluster analysis, i.e., unsupervised methods. H2O.ai is an open-source data science and machine learning platform KNIME is a machine learning and data mining software implemented in Java. Massive Online Analysis (MOA) is an open-source project for large scale mining of data streams, also developed at the University of Waikato in New Zealand. Neural Designer is a data mining software based on deep learning techniques written in C++. Orange is a similar open-source project for data mining, machine learning and visualization based on scikit-learn. RapidMiner is a commercial machine learning framework implemented in Java which integrates Weka. scikit-learn is a popular machine learning library in Python.

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  Discrete diffusion model

  In machine learning, discrete diffusion models are a class of diffusion models, which themselves are a class of latent variable generative models. Each discrete diffusion model consists of two major components: the forward jump diffusion process, and the reverse jump diffusion process. The goal of diffusion modeling is, given a given dataset and a forward process, to learn a model for the reverse process, such that the reverse process can generate new elements that are distributed similarly as the original dataset. A trained discrete diffusion model can be sampled in many ways, which trades off computational efficiency and sample quality. In general, higher quality data can be obtained, but at the price of higher computational cost. In standard diffusion modeling, the diffusion process takes place over a state space that is continuous space of R n {\displaystyle \mathbb {R} ^{n}} , but over a discrete set S {\displaystyle S} . A discrete set is simply a set where one cannot speak of "infinitesimally close" points. Points can be more or less separated from each other, but the separation is always a finite number. This in particular means the standard framework of continuous diffusion does not apply, since it uses gaussian noise, which is continuous. Nevertheless, an analogous theory can be produced. Discrete diffusion is usually used for language modeling. In practice, the state space S {\displaystyle S} is not only discrete, but finite, so this is what we will assume from now on. == Continuous time Markov process == In the case of continuous state space, during the forward discrete diffusion process, at each step t → t + d t {\displaystyle t\to t+dt} , we mix in an infinitesimal amount of gaussian noise d x t = − 1 2 β ( t ) x t d t + β ( t ) d W t {\displaystyle dx_{t}=-{\frac {1}{2}}\beta (t)x_{t}dt+{\sqrt {\beta (t)}}dW_{t}} . This changes the probability density function, by first a convolution with the density of a gaussian, followed by a scaling. In the case of discrete state space, the gaussian noise must be replaced by a noise that takes values over a finite set. For example, if the noise is the uniform distribution over S {\displaystyle S} , then the probability distribution at time t + d t {\displaystyle t+dt} satisfies q t + d t ( x ) = ( 1 − d t ) q t ( x ) + d t ( 1 | S | ∑ y ∈ S q t ( y ) ) {\displaystyle q_{t+dt}(x)=(1-dt)q_{t}(x)+dt\left({\frac {1}{|S|}}\sum _{y\in S}q_{t}(y)\right)} More succinctly, ∂ t q t ( x ) = − ( 1 − 1 | S | ) q t ( x ) + ∑ y ∈ S , y ≠ x 1 | S | q t ( y ) {\displaystyle \partial _{t}q_{t}(x)=-\left(1-{\frac {1}{|S|}}\right)q_{t}(x)+\sum _{y\in S,y\neq x}{\frac {1}{|S|}}q_{t}(y)} In general, we do not need to convolve with a uniformly distributed noise, but with an arbitrary noise process. That is, we use an arbitrary matrix Q t {\displaystyle Q_{t}} such that ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} where Q t {\displaystyle Q_{t}} is called the rate matrix. Any matrix may be used as a rate matrix if it has non-negative off-diagonals, and each column sums to 0: Q t ( y , x ) ≥ 0 ∀ y ≠ x , ∑ y ∈ S Q t ( y , x ) = 0 ∀ x {\displaystyle Q_{t}(y,x)\geq 0\quad \forall y\neq x,\quad \sum _{y\in S}Q_{t}(y,x)=0\quad \forall x} A continuous time Markov chain (CTMC) is defined by a continuous function Q {\displaystyle Q} that maps any time t ∈ [ 0 , T ) {\displaystyle t\in [0,T)} to a rate matrix Q t {\displaystyle Q_{t}} . Given the function Q {\displaystyle Q} , time-evolution under the CTMC is done as follows: Given state x t {\displaystyle x_{t}} at time t {\displaystyle t} , and given an infinitesimal d t {\displaystyle dt} , the state at t + d t {\displaystyle t+dt} is x t + d t {\displaystyle x_{t+dt}} , such that Pr ( x t + d t | x t ) = { 1 + Q t ( x t + d t , x t ) d t if x t + d t = x t Q t ( x t + d t , x t ) d t else {\displaystyle \Pr(x_{t+dt}|x_{t})={\begin{cases}1+Q_{t}(x_{t+dt},x_{t})dt&{\text{if }}x_{t+dt}=x_{t}\\Q_{t}(x_{t+dt},x_{t})dt&{\text{else}}\end{cases}}} This implies that the probability distribution function evolves according to ∂ t q t ( y ) = ∑ x ∈ S Q t ( y , x ) q t ( x ) {\displaystyle \partial _{t}q_{t}(y)=\sum _{x\in S}Q_{t}(y,x)q_{t}(x)} which is what we previously specified. === Backward process === Similarly to the case of continuous diffusion, in discrete diffusion, there exists a backward diffusion process Q ¯ t {\displaystyle {\bar {Q}}_{t}} : s ( x , t ) y := q t ( y ) q t ( x ) , Q ¯ t ( y , x ) := { s ( x , t ) y Q t ( x , y ) if y ≠ x − ∑ y : y ≠ x Q ¯ t ( y , x ) if y = x {\displaystyle s(x,t)_{y}:={\frac {q_{t}(y)}{q_{t}(x)}},\quad {\bar {Q}}_{t}(y,x):={\begin{cases}s(x,t)_{y}Q_{t}(x,y)&{\text{if }}y\neq x\\-\sum _{y:y\neq x}{\bar {Q}}_{t}(y,x)&{\text{if }}y=x\end{cases}}} where s ( x , t ) y {\displaystyle s(x,t)_{y}} should be interpreted as the discrete score or concrete score, since, abusing notation a bit, the score function is ∇ ln ⁡ ρ t ( x ) = 1 d x ( ρ t ( x + d x ) ρ t ( x ) − 1 ) {\displaystyle \nabla \ln \rho _{t}(x)={\frac {1}{dx}}\left({\frac {\rho _{t}(x+dx)}{\rho _{t}(x)}}-1\right)} . If we picture the distribution q t {\displaystyle q_{t}} as a bunch of point-masses, one per state x ∈ S {\displaystyle x\in S} , then the forward diffusion from time t {\displaystyle t} to t + d t {\displaystyle t+dt} is performed by removing Q t ( x , y ) q t ( y ) d t {\displaystyle Q_{t}(x,y)q_{t}(y)dt} from the mass at y {\displaystyle y} and moving it to the mass at x {\displaystyle x} , for each pair x ≠ y {\displaystyle x\neq y} . Thus, the process is reversed in detail by the CTMC defined by Q ¯ {\displaystyle {\bar {Q}}} , since Q ¯ t ( y , x ) q t ( x ) = Q t ( x , y ) q t ( y ) {\displaystyle {\bar {Q}}_{t}(y,x)q_{t}(x)=Q_{t}(x,y)q_{t}(y)} . Given Q ¯ t {\displaystyle {\bar {Q}}_{t}} , if we have a way to sample from q t {\displaystyle q_{t}} , then we can sample from q t − d t {\displaystyle q_{t-dt}} by first sampling x t ∼ q t {\displaystyle x_{t}\sim q_{t}} , then sampling x t − d t {\displaystyle x_{t-dt}} according to Pr ( x t − d t | x t ) = { 1 + Q ¯ t ( x t − d t , x t ) d t if x t − d t = x t Q ¯ t ( x t − d t , x t ) d t else {\displaystyle \Pr(x_{t-dt}|x_{t})={\begin{cases}1+{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{if }}x_{t-dt}=x_{t}\\{\bar {Q}}_{t}(x_{t-dt},x_{t})dt&{\text{else}}\end{cases}}} === Overall plan of score-matching discrete diffusion modeling === Similar to score-matching continuous diffusion, score-matching discrete diffusion is a method to sample an initial distribution. If we have a certain function s θ {\displaystyle s_{\theta }} that approximates the true score function s θ ( x , t ) y ≈ s ( x , t ) y {\displaystyle s_{\theta }(x,t)_{y}\approx s(x,t)_{y}} , then it allows a corresponding Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} to be defined in the same way. If we also have a base distribution q base {\displaystyle q_{\text{base}}} such that it is easy to sample from, and approximately equal to the true terminal distribution q base ≈ q T {\displaystyle q_{\text{base}}\approx q_{T}} , then we can perform the backward CTMC with Q ¯ θ {\displaystyle {\bar {Q}}^{\theta }} and q T θ := q terminal {\displaystyle q_{T}^{\theta }:=q_{\text{terminal}}} . When both approximations are good, the backward CTMC would give q 0 θ ≈ q 0 {\displaystyle q_{0}^{\theta }\approx q_{0}} . This is the idea of score-matching discrete diffusion modeling. If q data {\displaystyle q_{\text{data}}} is sharp, in the sense that for some x , x ′ {\displaystyle x,x'} , we have q data ( x ) ≫ q data ( x ′ ) {\displaystyle q_{\text{data}}(x)\gg q_{\text{data}}(x')} , then the score function would diverge as 1 / t {\displaystyle 1/t} at the t → 0 {\displaystyle t\to 0} limit. To avoid this in practice, it is common to use early stopping, which is to stop the backward process at some time δ > 0 {\displaystyle \delta >0} , and sample from q δ θ {\displaystyle q_{\delta }^{\theta }} instead of q 0 θ {\displaystyle q_{0}^{\theta }} . === Tractable forward processes === The theory of CTMC works for any continuous choice of rate matrices Q {\displaystyle Q} . However, most choices are computationally expensive and cannot be used in practice. In the case of continuous diffusion, the gaussian noise is used for the simple reason that the sum of any number of gaussians is still a gaussian. This allows one to sample any x t ∼ ρ t {\displaystyle x_{t}\sim \rho _{t}} by sampling a single x 0 ∼ ρ 0 {\displaystyle x_{0}\sim \rho _{0}} , followed by a single gaussian noise z ∼ N ( 0 , I ) {\displaystyle z\sim {\mathcal {N}}(0,I)} , and let x t = α ¯ t x 0 + σ t z {\displaystyle x_{t}={\sqrt {{\bar {\alpha }}_{t}}}x_{0}+\sigma _{t}z} , without needing any x s {\displaystyle x_{s}} for any 0 < s < t {\displaystyle 0 Read more →

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  Progressive Graphics File

  PGF (Progressive Graphics File) is a wavelet-based bitmapped image format that employs lossless and lossy data compression. PGF was created to improve upon and replace the JPEG format. It was developed at the same time as JPEG 2000 but with a focus on speed over compression ratio. PGF can operate at higher compression ratios without taking more encoding/decoding time and without generating the characteristic "blocky and blurry" artifacts of the original DCT-based JPEG standard. It also allows more sophisticated progressive downloads. == Color models == PGF supports a wide variety of color models: Grayscale with 1, 8, 16, or 31 bits per pixel Indexed color with palette size of 256 RGB color image with 12, 16 (red: 5 bits, green: 6 bits, blue: 5 bits), 24, or 48 bits per pixel ARGB color image with 32 bits per pixel Lab color image with 24 or 48 bits per pixel CMYK color image with 32 or 64 bits per pixel == Technical discussion == PGF claims to achieve an improved compression quality over JPEG adding or improving features such as scalability. Its compression performance is similar to the original JPEG standard. Very low and very high compression rates (including lossless compression) are also supported in PGF. The ability of the design to handle a very large range of effective bit rates is one of the strengths of PGF. For example, to reduce the number of bits for a picture below a certain amount, the advisable thing to do with the first JPEG standard is to reduce the resolution of the input image before encoding it — something that is ordinarily not necessary for that purpose when using PGF because of its wavelet scalability properties. The PGF process chain contains the following four steps: Color space transform (in case of color images) Discrete Wavelet Transform Quantization (in case of lossy data compression) Hierarchical bit-plane run-length encoding === Color components transformation === Initially, images have to be transformed from the RGB color space to another color space, leading to three components that are handled separately. PGF uses a fully reversible modified YUV color transform. The transformation matrices are: [ Y r U r V r ] = [ 1 4 1 2 1 4 1 − 1 0 0 − 1 1 ] [ R G B ] ; [ R G B ] = [ 1 3 4 − 1 4 1 − 1 4 − 1 4 1 − 1 4 3 4 ] [ Y r U r V r ] {\displaystyle {\begin{bmatrix}Y_{r}\\U_{r}\\V_{r}\end{bmatrix}}={\begin{bmatrix}{\frac {1}{4}}&{\frac {1}{2}}&{\frac {1}{4}}\\1&-1&0\\0&-1&1\end{bmatrix}}{\begin{bmatrix}R\\G\\B\end{bmatrix}};\qquad \qquad {\begin{bmatrix}R\\G\\B\end{bmatrix}}={\begin{bmatrix}1&{\frac {3}{4}}&-{\frac {1}{4}}\\1&-{\frac {1}{4}}&-{\frac {1}{4}}\\1&-{\frac {1}{4}}&{\frac {3}{4}}\end{bmatrix}}{\begin{bmatrix}Y_{r}\\U_{r}\\V_{r}\end{bmatrix}}} The chrominance components can be, but do not necessarily have to be, down-scaled in resolution. === Wavelet transform === The color components are then wavelet transformed to an arbitrary depth. In contrast to JPEG 1992 which uses an 8x8 block-size discrete cosine transform, PGF uses one reversible wavelet transform: a rounded version of the biorthogonal CDF 5/3 wavelet transform. This wavelet filter bank is exactly the same as the reversible wavelet used in JPEG 2000. It uses only integer coefficients, so the output does not require rounding (quantization) and so it does not introduce any quantization noise. === Quantization === After the wavelet transform, the coefficients are scalar-quantized to reduce the amount of bits to represent them, at the expense of a loss of quality. The output is a set of integer numbers which have to be encoded bit-by-bit. The parameter that can be changed to set the final quality is the quantization step: the greater the step, the greater is the compression and the loss of quality. With a quantization step that equals 1, no quantization is performed (it is used in lossless compression). In contrast to JPEG 2000, PGF uses only powers of two, therefore the parameter value i represents a quantization step of 2i. Just using powers of two makes no need of integer multiplication and division operations. === Coding === The result of the previous process is a collection of sub-bands which represent several approximation scales. A sub-band is a set of coefficients — integer numbers which represent aspects of the image associated with a certain frequency range as well as a spatial area of the image. The quantized sub-bands are split further into blocks, rectangular regions in the wavelet domain. They are typically selected in a way that the coefficients within them across the sub-bands form approximately spatial blocks in the (reconstructed) image domain and collected in a fixed size macroblock. The encoder has to encode the bits of all quantized coefficients of a macroblock, starting with the most significant bits and progressing to less significant bits. In this encoding process, each bit-plane of the macroblock gets encoded in two so-called coding passes, first encoding bits of significant coefficients, then refinement bits of significant coefficients. Clearly, in lossless mode all bit-planes have to be encoded, and no bit-planes can be dropped. Only significant coefficients are compressed with an adaptive run-length/Rice (RLR) coder, because they contain long runs of zeros. The RLR coder with parameter k (logarithmic length of a run of zeros) is also known as the elementary Golomb code of order 2k. === Comparison with other file formats === JPEG 2000 is slightly more space-efficient in handling natural images. The PSNR for the same compression ratio is on average 3% better than the PSNR of PGF. It has a small advantage in compression ratio but longer encoding and decoding times. PNG (Portable Network Graphics) is more space-efficient in handling images with many pixels of the same color. There are several self-proclaimed advantages of PGF over the ordinary JPEG standard: Superior compression performance: The image quality (measured in PSNR) for the same compression ratio is on average 3% better than the PSNR of JPEG. At lower bit rates (e.g. less than 0.25 bits/pixel for gray-scale images), PGF has a much more significant advantage over certain modes of JPEG: artifacts are less visible and there is almost no blocking. The compression gains over JPEG are attributed to the use of DWT. Multiple resolution representation: PGF provides seamless compression of multiple image components, with each component carrying from 1 to 31 bits per component sample. With this feature there is no need for separately stored preview images (thumbnails). Progressive transmission by resolution accuracy, commonly referred to as progressive decoding: PGF provides efficient code-stream organizations which are progressive by resolution. This way, after a smaller part of the whole file has been received, it is possible to see a lower quality of the final picture, the quality can be improved monotonically getting more data from the source. Lossless and lossy compression: PGF provides both lossless and lossy compression in a single compression architecture. Both lossy and lossless compression are provided by the use of a reversible (integer) wavelet transform. Side channel spatial information: Transparency and alpha planes are fully supported ROI extraction: Since version 5, PGF supports extraction of regions of interest (ROI) without decoding the whole image. == Available software == The author published libPGF via a SourceForge, under the GNU Lesser General Public License version 2.0. Xeraina offers a free Windows console encoder and decoder, and PGF viewers based on WIC for 32bit and 64bit Windows platforms. Other WIC applications including File Explorer are able to display PGF images after installing this viewer. Digikam is a popular open-source image editing and cataloging software that uses libPGF for its thumbnails. It makes use of the progressive decoding feature of PGF images to store a single version of each thumbnail, which can then be decoded to different resolutions without loss, thus allowing users to dynamically change the size of the thumbnails without having to recalculate them again.

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  ID3 algorithm

  In decision tree learning, ID3 (Iterative Dichotomiser 3) is a greedy algorithm invented by Ross Quinlan used to generate a decision tree from a dataset. ID3 is the precursor to the C4.5 algorithm. The 3 in the name is meant to signify that this was Quinlan's third attempt at a model based on entropy-based splitting, and the term dichotimser is a misnomer as it implies a binary split, but the ID3 algorithm can split on multi-valued attributes. == Algorithm == The ID3 algorithm begins with the original set S {\displaystyle S} as the root node. On each iteration of the algorithm, it iterates through every unused attribute of the set S {\displaystyle S} and calculates the entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} or the information gain I G ( S ) {\displaystyle IG(S)} of that attribute. It then selects the attribute which has the smallest entropy (or largest information gain) value. The set S {\displaystyle S} is then split or partitioned by the selected attribute to produce subsets of the data. (For example, a node can be split into child nodes based upon the subsets of the population whose ages are less than 50, between 50 and 100, and greater than 100.) The algorithm continues to recurse on each subset, considering only attributes never selected before. Recursion on a subset may stop in one of these cases: every element in the subset belongs to the same class; in which case the node is turned into a leaf node and labelled with the class of the examples. there are no more attributes to be selected, but the examples still do not belong to the same class. In this case, the node is made a leaf node and labelled with the most common class of the examples in the subset. there are no examples in the subset, which happens when no example in the parent set was found to match a specific value of the selected attribute. An example could be the absence of a person among the population with age over 100 years. Then a leaf node is created and labelled with the most common class of the examples in the parent node's set. Throughout the algorithm, the decision tree is constructed with each non-terminal node (internal node) representing the selected attribute on which the data was split, and terminal nodes (leaf nodes) representing the class label of the final subset of this branch. === Summary === Calculate the entropy of every attribute a {\displaystyle a} of the data set S {\displaystyle S} . Partition ("split") the set S {\displaystyle S} into subsets using the attribute for which the resulting entropy after splitting is minimized; or, equivalently, information gain is maximum. Make a decision tree node containing that attribute. Recurse on subsets using the remaining attributes. === Properties === ID3 does not guarantee an optimal solution. It can converge upon local optima. It uses a greedy strategy by selecting the locally best attribute to split the dataset on each iteration. The algorithm's optimality can be improved by using backtracking during the search for the optimal decision tree at the cost of possibly taking longer. ID3 can overfit the training data. To avoid overfitting, smaller decision trees should be preferred over larger ones. This algorithm usually produces small trees, but it does not always produce the smallest possible decision tree. ID3 is harder to use on continuous data than on factored data (factored data has a discrete number of possible values, thus reducing the possible branch points). If the values of any given attribute are continuous, then there are many more places to split the data on this attribute, and searching for the best value to split by can be time-consuming. === Usage === The ID3 algorithm is used by training on a data set S {\displaystyle S} to produce a decision tree which is stored in memory. At runtime, this decision tree is used to classify new test cases (feature vectors) by traversing the decision tree using the features of the datum to arrive at a leaf node. == The ID3 metrics == === Entropy === Entropy H ( S ) {\displaystyle \mathrm {H} {(S)}} is a measure of the amount of uncertainty in the (data) set S {\displaystyle S} (i.e. entropy characterizes the (data) set S {\displaystyle S} ). H ( S ) = ∑ x ∈ X − p ( x ) log 2 ⁡ p ( x ) {\displaystyle \mathrm {H} {(S)}=\sum _{x\in X}{-p(x)\log _{2}p(x)}} Where, S {\displaystyle S} – The current dataset for which entropy is being calculated This changes at each step of the ID3 algorithm, either to a subset of the previous set in the case of splitting on an attribute or to a "sibling" partition of the parent in case the recursion terminated previously. X {\displaystyle X} – The set of classes in S {\displaystyle S} p ( x ) {\displaystyle p(x)} – The proportion of the number of elements in class x {\displaystyle x} to the number of elements in set S {\displaystyle S} When H ( S ) = 0 {\displaystyle \mathrm {H} {(S)}=0} , the set S {\displaystyle S} is perfectly classified (i.e. all elements in S {\displaystyle S} are of the same class). In ID3, entropy is calculated for each remaining attribute. The attribute with the smallest entropy is used to split the set S {\displaystyle S} on this iteration. Entropy in information theory measures how much information is expected to be gained upon measuring a random variable; as such, it can also be used to quantify the amount to which the distribution of the quantity's values is unknown. A constant quantity has zero entropy, as its distribution is perfectly known. In contrast, a uniformly distributed random variable (discretely or continuously uniform) maximizes entropy. Therefore, the greater the entropy at a node, the less information is known about the classification of data at this stage of the tree; and therefore, the greater the potential to improve the classification here. As such, ID3 is a greedy heuristic performing a best-first search for locally optimal entropy values. Its accuracy can be improved by preprocessing the data. === Information gain === Information gain I G ( A ) {\displaystyle IG(A)} is the measure of the difference in entropy from before to after the set S {\displaystyle S} is split on an attribute A {\displaystyle A} . In other words, how much uncertainty in S {\displaystyle S} was reduced after splitting set S {\displaystyle S} on attribute A {\displaystyle A} . I G ( S , A ) = H ( S ) − ∑ t ∈ T p ( t ) H ( t ) = H ( S ) − H ( S | A ) . {\displaystyle IG(S,A)=\mathrm {H} {(S)}-\sum _{t\in T}p(t)\mathrm {H} {(t)}=\mathrm {H} {(S)}-\mathrm {H} {(S|A)}.} Where, H ( S ) {\displaystyle \mathrm {H} (S)} – Entropy of set S {\displaystyle S} T {\displaystyle T} – The subsets created from splitting set S {\displaystyle S} by attribute A {\displaystyle A} such that S = ⋃ t ∈ T t {\displaystyle S=\bigcup _{t\in T}t} p ( t ) {\displaystyle p(t)} – The proportion of the number of elements in t {\displaystyle t} to the number of elements in set S {\displaystyle S} H ( t ) {\displaystyle \mathrm {H} (t)} – Entropy of subset t {\displaystyle t} In ID3, information gain can be calculated (instead of entropy) for each remaining attribute. The attribute with the largest information gain is used to split the set S {\displaystyle S} on this iteration.

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  Charge based boundary element fast multipole method

  The charge-based formulation of the boundary element method (BEM) is a dimensionality reduction numerical technique that is used to model quasistatic electromagnetic phenomena in highly complex conducting media (targeting, e.g., the human brain) with a very large (up to approximately 1 billion) number of unknowns. The charge-based BEM solves an integral equation of the potential theory written in terms of the induced surface charge density. This formulation is naturally combined with fast multipole method (FMM) acceleration, and the entire method is known as charge-based BEM-FMM. The combination of BEM and FMM is a common technique in different areas of computational electromagnetics and, in the context of bioelectromagnetism, it provides improvements over the finite element method. == Historical development == Along with more common electric potential-based BEM, the quasistatic charge-based BEM, derived in terms of the single-layer (charge) density, for a single-compartment medium has been known in the potential theory since the beginning of the 20th century. For multi-compartment conducting media, the surface charge density formulation first appeared in discretized form (for faceted interfaces) in the 1964 paper by Gelernter and Swihart. A subsequent continuous form, including time-dependent and dielectric effects, appeared in the 1967 paper by Barnard, Duck, and Lynn. The charge-based BEM has also been formulated for conducting, dielectric, and magnetic media, and used in different applications. In 2009, Greengard et al. successfully applied the charge-based BEM with fast multipole acceleration to molecular electrostatics of dielectrics. A similar approach to realistic modeling of the human brain with multiple conducting compartments was first described by Makarov et al. in 2018. Along with this, the BEM-based multilevel fast multipole method has been widely used in radar and antenna studies at microwave frequencies as well as in acoustics. == Physical background - surface charges in biological media == The charge-based BEM is based on the concept of an impressed (or primary) electric field E i {\displaystyle \mathbf {E} ^{i}} and a secondary electric field E s {\displaystyle \mathbf {E} ^{s}} . The impressed field is usually known a priori or is trivial to find. For the human brain, the impressed electric field can be classified as one of the following: A conservative field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed density of EEG or MEG current sources in a homogeneous infinite medium with the conductivity σ {\displaystyle \sigma } at the source location; An instantaneous solenoidal field E i {\displaystyle \mathbf {E} ^{i}} of an induction coil obtained from Faraday's law of induction in a homogeneous infinite medium (air), when transcranial magnetic stimulation (TMS) problems are concerned; A surface field E i {\displaystyle \mathbf {E} ^{i}} derived from an impressed surface current density J i = σ E i {\displaystyle \mathbf {J} ^{i}=\sigma \mathbf {E} ^{i}} of current electrodes injecting electric current at a boundary of a compartment with conductivity σ {\displaystyle \sigma } when transcranial direct-current stimulation (tDCS) or deep brain stimulation (DBS) are concerned; A conservative field E i {\displaystyle \mathbf {E} ^{i}} of charges deposited on voltage electrodes for tDCS or DBS. This specific problem requires a coupled treatment since these charges will depend on the environment; In application to multiscale modeling, a field E i {\displaystyle \mathbf {E} ^{i}} obtained from any other macroscopic numerical solution in a small (mesoscale or microscale) spatial domain within the brain. For example, a constant field can be used. When the impressed field is "turned on", free charges located within a conducting volume D immediately begin to redistribute and accumulate at the boundaries (interfaces) of regions of different conductivity in D. A surface charge density ρ ( r ) {\displaystyle \rho (\mathbf {r} )} appears on the conductivity interfaces. This charge density induces a secondary conservative electric field E s {\displaystyle \mathbf {E} ^{s}} following Coulomb's law. One example is a human under a direct current powerline with the known field E i {\displaystyle \mathbf {E} ^{i}} directed down. The superior surface of the human's conducting body will be charged negatively while its inferior portion is charged positively. These surface charges create a secondary electric field that effectively cancels or blocks the primary field everywhere in the body so that no current will flow within the body under DC steady state conditions. Another example is a human head with electrodes attached. At any conductivity interface with a normal vector n {\displaystyle \mathbf {n} } pointing from an "inside" (-) compartment of conductivity σ − {\displaystyle \sigma ^{-}} to an "outside" (+) compartment of conductivity σ + {\displaystyle \sigma ^{+}} , Kirchhoff's current law requires continuity of the normal component of the electric current density. This leads to the interfacial boundary condition in the form for every facet at a triangulated interface. As long as σ ± {\displaystyle \sigma ^{\pm }} are different from each other, the two normal components of the electric field, E ± ⋅ n {\displaystyle \mathbf {E} ^{\pm }\cdot \mathbf {n} } , must also be different. Such a jump across the interface is only possible when a sheet of surface charge exists at that interface. Thus, if an electric current or voltage is applied, the surface charge density follows. The goal of the numerical analysis is to find the unknown surface charge distribution and thus the total electric field E = E i + E s {\displaystyle \mathbf {E} =\mathbf {E} ^{i}+\mathbf {E} ^{s}} (and the total electric potential if required) anywhere in space. == System of equations for surface charges == Below, a derivation is given based on Gauss's law and Coulomb's law. All conductivity interfaces, denoted by S, are discretized into planar triangular facets t m {\displaystyle t_{m}} with centers r m {\displaystyle \mathbf {r} _{m}} . Assume that an m-th facet with the normal vector n m {\displaystyle \mathbf {n} _{m}} and area A m {\displaystyle A_{m}} carries a uniform surface charge density ρ m {\displaystyle \rho _{m}} . If a volumetric tetrahedral mesh were present, the charged facets would belong to tetrahedra with different conductivity values. We first compute the electric field E m + {\displaystyle \mathbf {E} _{m}^{+}} at the point r m + δ n m {\displaystyle \mathbf {r} _{m}+\delta \mathbf {n} _{m}} , for δ → 0 + {\displaystyle \delta \rightarrow 0^{+}} i.e., just outside facet 𝑚 at its center. This field contains three contributions: The continuous impressed electric field E i {\displaystyle \mathbf {E} ^{i}} itself; An electric field of the m-th charged facet itself. Very close to the facet, it can be approximated as the electric field of an infinite sheet of uniform surface charge ρ m {\displaystyle \rho _{m}} . By Gauss's law, it is given by + ρ m / 2 ε 0 ⋅ n m {\displaystyle +\rho _{m}/2\varepsilon _{0}\cdot \mathbf {n} _{m}} where ε 0 {\displaystyle \varepsilon _{0}} is a background electrical permittivity; An electric field generated by all other facets t n {\displaystyle t_{n}} , which we approximate as point charges of charge A n ρ n {\displaystyle A_{n}\rho _{n}} at each center r n {\displaystyle \mathbf {r} _{n}} . A similar treatment holds for the electric field E m − {\displaystyle \mathbf {E} _{m}^{-}} just inside facet 𝑚, but the electric field of the flat sheet of charge changes its sign. Using Coulomb's law to calculate the contribution of facets different from t m {\displaystyle t_{m}} , we find From this equation, we see that the normal component of the electric field indeed undergoes a jump through the charged interface. This is equivalent to a jump relation of the potential theory. As a second step, the two expressions for E m ± {\displaystyle \mathbf {E} _{m}^{\pm }} are substituted into the interfacial boundary condition σ − E m − ⋅ n m = σ + E m + ⋅ n m {\displaystyle \sigma ^{-}\mathbf {E} _{m}^{-}\cdot \mathbf {n} _{m}=\sigma ^{+}\mathbf {E} _{m}^{+}\cdot \mathbf {n} _{m}} , applied to every facet 𝑚. This operation leads to a system of linear equations for unknown charge densities ρ m {\displaystyle \rho _{m}} which solves the problem: where K m = σ − − σ + σ − + σ + {\displaystyle K_{m}={\frac {\sigma ^{-}-\sigma ^{+}}{\sigma ^{-}+\sigma ^{+}}}} is the electric conductivity contrast at the m-th facet. The normalization constant ε 0 {\displaystyle \varepsilon _{0}} will cancel out after the solution is substituted in the expression for E s {\displaystyle \mathbf {E} ^{s}} and becomes redundant. == Application of fast multipole method == For modern characterizations of brain topologies with ever-increasing levels of complexity, the above system of equations for ρ m {\displaystyle \rho _{m}} is very large; it is t

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  Radial basis function network

  In the field of mathematical modeling, a radial basis function network is an artificial neural network that uses radial basis functions as activation functions. The output of the network is a linear combination of radial basis functions of the inputs and neuron parameters. Radial basis function networks have many uses, including function approximation, time series prediction, classification, and system control. They were first formulated in a 1988 paper by Broomhead and Lowe, both researchers at the Royal Signals and Radar Establishment. == Network architecture == Radial basis function (RBF) networks typically have three layers: an input layer, a hidden layer with a non-linear RBF activation function and a linear output layer. The input can be modeled as a vector of real numbers x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} . The output of the network is then a scalar function of the input vector, φ : R n → R {\displaystyle \varphi :\mathbb {R} ^{n}\to \mathbb {R} } , and is given by φ ( x ) = ∑ i = 1 N a i ρ ( | | x − c i | | ) {\displaystyle \varphi (\mathbf {x} )=\sum _{i=1}^{N}a_{i}\rho (||\mathbf {x} -\mathbf {c} _{i}||)} where N {\displaystyle N} is the number of neurons in the hidden layer, c i {\displaystyle \mathbf {c} _{i}} is the center vector for neuron i {\displaystyle i} , and a i {\displaystyle a_{i}} is the weight of neuron i {\displaystyle i} in the linear output neuron. Functions that depend only on the distance from a center vector are radially symmetric about that vector, hence the name radial basis function. In the basic form, all inputs are connected to each hidden neuron. The norm is typically taken to be the Euclidean distance (although the Mahalanobis distance appears to perform better with pattern recognition) and the radial basis function is commonly taken to be Gaussian ρ ( ‖ x − c i ‖ ) = exp ⁡ [ − β i ‖ x − c i ‖ 2 ] {\displaystyle \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}=\exp \left[-\beta _{i}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert ^{2}\right]} . The Gaussian basis functions are local to the center vector in the sense that lim | | x | | → ∞ ρ ( ‖ x − c i ‖ ) = 0 {\displaystyle \lim _{||x||\to \infty }\rho (\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert )=0} i.e. changing parameters of one neuron has only a small effect for input values that are far away from the center of that neuron. Given certain mild conditions on the shape of the activation function, RBF networks are universal approximators on a compact subset of R n {\displaystyle \mathbb {R} ^{n}} . This means that an RBF network with enough hidden neurons can approximate any continuous function on a closed, bounded set with arbitrary precision. The parameters a i {\displaystyle a_{i}} , c i {\displaystyle \mathbf {c} _{i}} , and β i {\displaystyle \beta _{i}} are determined in a manner that optimizes the fit between φ {\displaystyle \varphi } and the data. === Normalization === ==== Normalized architecture ==== In addition to the above unnormalized architecture, RBF networks can be normalized. In this case the mapping is φ ( x ) = d e f ∑ i = 1 N a i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N a i u ( ‖ x − c i ‖ ) {\displaystyle \varphi (\mathbf {x} )\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\sum _{i=1}^{N}a_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}a_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} where u ( ‖ x − c i ‖ ) = d e f ρ ( ‖ x − c i ‖ ) ∑ j = 1 N ρ ( ‖ x − c j ‖ ) {\displaystyle u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{j=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{j}\right\Vert {\big )}}}} is known as a normalized radial basis function. ==== Theoretical motivation for normalization ==== There is theoretical justification for this architecture in the case of stochastic data flow. Assume a stochastic kernel approximation for the joint probability density P ( x ∧ y ) = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) σ ( | y − e i | ) {\displaystyle P\left(\mathbf {x} \land y\right)={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,\sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}} where the weights c i {\displaystyle \mathbf {c} _{i}} and e i {\displaystyle e_{i}} are exemplars from the data and we require the kernels to be normalized ∫ ρ ( ‖ x − c i ‖ ) d n x = 1 {\displaystyle \int \rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}\,d^{n}\mathbf {x} =1} and ∫ σ ( | y − e i | ) d y = 1 {\displaystyle \int \sigma {\big (}\left\vert y-e_{i}\right\vert {\big )}\,dy=1} . The probability densities in the input and output spaces are P ( x ) = ∫ P ( x ∧ y ) d y = 1 N ∑ i = 1 N ρ ( ‖ x − c i ‖ ) {\displaystyle P\left(\mathbf {x} \right)=\int P\left(\mathbf {x} \land y\right)\,dy={1 \over N}\sum _{i=1}^{N}\,\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and The expectation of y given an input x {\displaystyle \mathbf {x} } is φ ( x ) = d e f E ( y ∣ x ) = ∫ y P ( y ∣ x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)\ {\stackrel {\mathrm {def} }{=}}\ E\left(y\mid \mathbf {x} \right)=\int y\,P\left(y\mid \mathbf {x} \right)dy} where P ( y ∣ x ) {\displaystyle P\left(y\mid \mathbf {x} \right)} is the conditional probability of y given x {\displaystyle \mathbf {x} } . The conditional probability is related to the joint probability through Bayes' theorem P ( y ∣ x ) = P ( x ∧ y ) P ( x ) {\displaystyle P\left(y\mid \mathbf {x} \right)={\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}} which yields φ ( x ) = ∫ y P ( x ∧ y ) P ( x ) d y {\displaystyle \varphi \left(\mathbf {x} \right)=\int y\,{\frac {P\left(\mathbf {x} \land y\right)}{P\left(\mathbf {x} \right)}}\,dy} . This becomes φ ( x ) = ∑ i = 1 N e i ρ ( ‖ x − c i ‖ ) ∑ i = 1 N ρ ( ‖ x − c i ‖ ) = ∑ i = 1 N e i u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)={\frac {\sum _{i=1}^{N}e_{i}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}{\sum _{i=1}^{N}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}}}=\sum _{i=1}^{N}e_{i}u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} when the integrations are performed. === Local linear models === It is sometimes convenient to expand the architecture to include local linear models. In that case the architectures become, to first order, φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) ρ ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} and φ ( x ) = ∑ i = 1 N ( a i + b i ⋅ ( x − c i ) ) u ( ‖ x − c i ‖ ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{N}\left(a_{i}+\mathbf {b} _{i}\cdot \left(\mathbf {x} -\mathbf {c} _{i}\right)\right)u{\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )}} in the unnormalized and normalized cases, respectively. Here b i {\displaystyle \mathbf {b} _{i}} are weights to be determined. Higher order linear terms are also possible. This result can be written φ ( x ) = ∑ i = 1 2 N ∑ j = 1 n e i j v i j ( x − c i ) {\displaystyle \varphi \left(\mathbf {x} \right)=\sum _{i=1}^{2N}\sum _{j=1}^{n}e_{ij}v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}} where e i j = { a i , if i ∈ [ 1 , N ] b i j , if i ∈ [ N + 1 , 2 N ] {\displaystyle e_{ij}={\begin{cases}a_{i},&{\mbox{if }}i\in [1,N]\\b_{ij},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} and v i j ( x − c i ) = d e f { δ i j ρ ( ‖ x − c i ‖ ) , if i ∈ [ 1 , N ] ( x i j − c i j ) ρ ( ‖ x − c i ‖ ) , if i ∈ [ N + 1 , 2 N ] {\displaystyle v_{ij}{\big (}\mathbf {x} -\mathbf {c} _{i}{\big )}\ {\stackrel {\mathrm {def} }{=}}\ {\begin{cases}\delta _{ij}\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [1,N]\\\left(x_{ij}-c_{ij}\right)\rho {\big (}\left\Vert \mathbf {x} -\mathbf {c} _{i}\right\Vert {\big )},&{\mbox{if }}i\in [N+1,2N]\end{cases}}} in the unnormalized case and in the normalized case. Here δ i j {\displaystyle \delta _{ij}} is a Kronecker delta function defined as δ i j = { 1 , if i = j 0 , if i ≠ j {\displaystyle \delta _{ij}={\begin{cases}1,&{\mbox{if }}i=j\\0,&{\mbox{if }}i\neq j\end{cases}}} . == Training == RBF networks are typically trained from pairs of input and target values x ( t ) , y ( t ) {\displaystyle \mathbf {x} (t),y(t)} , t = 1 , … , T {\displaystyle t=1,\dots ,T} by a two-step algorithm. In the first step, the center vectors c i {\displaystyle \mathbf {c} _{i}} of the RBF functions in the hidden layer

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  Web Intents

  Web Intents was an experimental framework for web-based inter-application communication and service discovery. Web Intents consists of a discovery mechanism and a very light-weight RPC system between web applications, modelled after the Intents system in Android. In the context of the framework an Intent equals an action to be performed by a provider. Web Intents allow two web applications to communicate with each other, without either of them having to actually know what the other one is. == Support == === Client === Google Chrome versions 18 to 23 natively supported Web Intents. This support was disabled in version 24, citing the existence of a "number of areas for development in both the API and specific user experience in Chrome". There is a JavaScript shim with support for IE 8, IE 9, Opera, Safari, Firefox 3+ and Chrome 3+. === Server === There are some Web Intents proxy pages that make available some real services that don't yet support intents. AddThis supports Web Intents by their sharing tools regardless of browser support. == History == Paul Kinlan of Google announced the Web Intents project in December 2010. He soon released a prototype API to GitHub. In August 2011 Google announced that Chrome would support Web Intents. Google and Mozilla have started co-operating to unify Web Intents and Mozilla's Web Activities (which tries to solve the same problem) into one proposal. In November 2012, Greg Billock of Google announced that experimental support of Web Intents had been removed from Chrome.

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  GNU Octave

  GNU Octave is a scientific programming language for scientific computing and numerical computation. Among other things, Octave can be used to solve linear and nonlinear problems numerically and to perform other numerical experiments using a language that is mostly compatible with MATLAB. It may also be used as a batch-oriented language. As part of the GNU Project, it is free software under the terms of the GNU General Public License. == History == The project was conceived around 1988. At first it was intended to be a companion to a chemical reactor design course. Full development was started by John W. Eaton in 1992. The first alpha release dates back to 4 January 1993 and on 17 February 1994 version 1.0 was released. Version 9.2.0 was released on 7 June 2024. The program is named after Octave Levenspiel, a former professor of the principal author. Levenspiel was known for his ability to perform quick back-of-the-envelope calculations. == Development history == == Developments == In addition to use on desktops for personal scientific computing, Octave is used in academia and industry. For example, Octave was used on a massive parallel computer at Pittsburgh Supercomputing Center to find vulnerabilities related to guessing social security numbers. Acceleration with OpenCL or CUDA is also possible with use of GPUs. == Technical details == Octave is written in C++ using the C++ standard library. Octave uses an interpreter to execute the Octave scripting language. Octave is extensible using dynamically loadable modules. Octave interpreter has an OpenGL-based graphics engine to create plots, graphs and charts and to save or print them. Alternatively, gnuplot can be used for the same purpose. Octave includes a graphical user interface (GUI) in addition to the traditional command-line interface (CLI); see #User interfaces for details. == Octave, the language == The Octave language is an interpreted programming language. It is a structured programming language (similar to C) and supports many common C standard library functions, and also certain UNIX system calls and functions. However, it does not support passing arguments by reference although function arguments are copy-on-write to avoid unnecessary duplication. Octave programs consist of a list of function calls or a script. The syntax is matrix-based and provides various functions for matrix operations. It supports various data structures and allows object-oriented programming. Its syntax is very similar to MATLAB, and careful programming of a script will allow it to run on both Octave and MATLAB. Because Octave is made available under the GNU General Public License, it may be freely changed, copied and used. The program runs on Microsoft Windows and most Unix and Unix-like operating systems, including Linux, Android, and macOS. == Notable features == === Command and variable name completion === Typing a TAB character on the command line causes Octave to attempt to complete variable, function, and file names (similar to Bash's tab completion). Octave uses the text before the cursor as the initial portion of the name to complete. === Command history === When running interactively, Octave saves the commands typed in an internal buffer so that they can be recalled and edited. === Data structures === Octave includes a limited amount of support for organizing data in structures. In this example, we see a structure x with elements a, b, and c, (an integer, an array, and a string, respectively): === Short-circuit Boolean operators === Octave's && and || logical operators are evaluated in a short-circuit fashion (like the corresponding operators in the C language), in contrast to the element-by-element operators & and |. === Increment and decrement operators === Octave includes the C-like increment and decrement operators ++ and -- in both their prefix and postfix forms. Octave also does augmented assignment, e.g. x += 5. === Unwind-protect === Octave supports a limited form of exception handling modelled after the unwind_protect of Lisp. The general form of an unwind_protect block looks like this: As a general rule, GNU Octave recognizes as termination of a given block either the keyword end (which is compatible with the MATLAB language) or a more specific keyword endblock or, in some cases, end_block. As a consequence, an unwind_protect block can be terminated either with the keyword end_unwind_protect as in the example, or with the more portable keyword end. The cleanup part of the block is always executed. In case an exception is raised by the body part, cleanup is executed immediately before propagating the exception outside the block unwind_protect. GNU Octave also supports another form of exception handling (compatible with the MATLAB language): This latter form differs from an unwind_protect block in two ways. First, exception_handling is only executed when an exception is raised by body. Second, after the execution of exception_handling the exception is not propagated outside the block (unless a rethrow( lasterror ) statement is explicitly inserted within the exception_handling code). === Variable-length argument lists === Octave has a mechanism for handling functions that take an unspecified number of arguments without explicit upper limit. To specify a list of zero or more arguments, use the special argument varargin as the last (or only) argument in the list. varargin is a cell array containing all the input arguments. === Variable-length return lists === A function can be set up to return any number of values by using the special return value varargout. For example: === C++ integration === It is also possible to execute Octave code directly in a C++ program. For example, here is a code snippet for calling rand([10,1]): C and C++ code can be integrated into GNU Octave by creating oct files, or using the MATLAB compatible MEX files. == MATLAB compatibility == Octave has been built with MATLAB compatibility in mind, and shares many features with MATLAB: % Script: myscript.m a = 5; b = a 2 % Function: myfunc.m function result = myfunc(x) result = x^2 + 3; end Matrices as fundamental data type. Built-in support for complex numbers. Powerful built-in math functions and extensive function libraries. Extensibility in the form of user-defined functions. Octave treats incompatibility with MATLAB as a bug; therefore, it could be considered a software clone, which does not infringe software copyright as per Lotus v. Borland court case. MATLAB scripts from the MathWorks' FileExchange repository in principle are compatible with Octave. However, while they are often provided and uploaded by users under an Octave compatible and proper open source BSD license, the FileExchange Terms of use prohibit any usage beside MathWorks' proprietary MATLAB. === Syntax compatibility === There are a few purposeful, albeit minor, syntax additions Archived 2012-04-26 at the Wayback Machine: Comment lines can be prefixed with the # character as well as the % character; Various C-based operators ++, --, +=, =, /= are supported; Elements can be referenced without creating a new variable by cascaded indexing, e.g. [1:10](3); Strings can be defined with the double-quote " character as well as the single-quote ' character; When the variable type is single (a single-precision floating-point number), Octave calculates the "mean" in the single-domain (MATLAB in double-domain) which is faster but gives less accurate results; Blocks can also be terminated with more specific Control structure keywords, i.e., endif, endfor, endwhile, etc.; Functions can be defined within scripts and at the Octave prompt; Presence of a do-until loop (similar to do-while in C). === Function compatibility === Many, but not all, of the numerous MATLAB functions are available in GNU Octave, some of them accessible through packages in Octave Forge. The functions available as part of either core Octave or Forge packages are listed online Archived 2024-03-14 at the Wayback Machine. A list of unavailable functions is included in the Octave function __unimplemented.m__. Unimplemented functions are also listed under many Octave Forge packages in the Octave Wiki. When an unimplemented function is called the following error message is shown: == User interfaces == Octave comes with an official graphical user interface (GUI) and an integrated development environment (IDE) based on Qt. It has been available since Octave 3.8, and has become the default interface (over the command-line interface) with the release of Octave 4.0. It was well-received by an EDN contributor, who wrote "[Octave] now has a very workable GUI" in reviewing the then-new GUI in 2014. Several 3rd-party graphical front-ends have also been developed, like ToolboX for coding education. == GUI applications == With Octave code, the user can create GUI applications. See GUI Development (GNU Octave (version 7.1.0)). Below are some examples: Button, edit control, checkboxTextboxListbox wit

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  Influence diagram

  An influence diagram (ID) (also called a relevance diagram, decision diagram or a decision network) is a compact graphical and mathematical representation of a decision situation. It is a generalization of a Bayesian network, in which not only probabilistic inference problems but also decision making problems (following the maximum expected utility criterion) can be modeled and solved. ID was first developed in the mid-1970s by decision analysts with an intuitive semantic that is easy to understand. It is now adopted widely and becoming an alternative to the decision tree which typically suffers from exponential growth in number of branches with each variable modeled. ID is directly applicable in team decision analysis, since it allows incomplete sharing of information among team members to be modeled and solved explicitly. Extensions of ID also find their use in game theory as an alternative representation of the game tree. == Semantics == An ID is a directed acyclic graph with three types (plus one subtype) of node and three types of arc (or arrow) between nodes. Nodes: Decision node (corresponding to each decision to be made) is drawn as a rectangle. Uncertainty node (corresponding to each uncertainty to be modeled) is drawn as an oval. Deterministic node (corresponding to special kind of uncertainty that its outcome is deterministically known whenever the outcome of some other uncertainties are also known) is drawn as a double oval. Value node (corresponding to each component of additively separable Von Neumann-Morgenstern utility function) is drawn as an octagon (or diamond). Arcs: Functional arcs (ending in value node) indicate that one of the components of additively separable utility function is a function of all the nodes at their tails. Conditional arcs (ending in uncertainty node) indicate that the uncertainty at their heads is probabilistically conditioned on all the nodes at their tails. Conditional arcs (ending in deterministic node) indicate that the uncertainty at their heads is deterministically conditioned on all the nodes at their tails. Informational arcs (ending in decision node) indicate that the decision at their heads is made with the outcome of all the nodes at their tails known beforehand. Given a properly structured ID: Decision nodes and incoming information arcs collectively state the alternatives (what can be done when the outcome of certain decisions and/or uncertainties are known beforehand) Uncertainty/deterministic nodes and incoming conditional arcs collectively model the information (what are known and their probabilistic/deterministic relationships) Value nodes and incoming functional arcs collectively quantify the preference (how things are preferred over one another). Alternative, information, and preference are termed decision basis in decision analysis, they represent three required components of any valid decision situation. Formally, the semantic of influence diagram is based on sequential construction of nodes and arcs, which implies a specification of all conditional independencies in the diagram. The specification is defined by the d {\displaystyle d} -separation criterion of Bayesian network. According to this semantic, every node is probabilistically independent on its non-successor nodes given the outcome of its immediate predecessor nodes. Likewise, a missing arc between non-value node X {\displaystyle X} and non-value node Y {\displaystyle Y} implies that there exists a set of non-value nodes Z {\displaystyle Z} , e.g., the parents of Y {\displaystyle Y} , that renders Y {\displaystyle Y} independent of X {\displaystyle X} given the outcome of the nodes in Z {\displaystyle Z} . == Example == Consider the simple influence diagram representing a situation where a decision-maker is planning their vacation. There is 1 decision node (Vacation Activity), 2 uncertainty nodes (Weather Condition, Weather Forecast), and 1 value node (Satisfaction). There are 2 functional arcs (ending in Satisfaction), 1 conditional arc (ending in Weather Forecast), and 1 informational arc (ending in Vacation Activity). Functional arcs ending in Satisfaction indicate that Satisfaction is a utility function of Weather Condition and Vacation Activity. In other words, their satisfaction can be quantified if they know what the weather is like and what their choice of activity is. (Note that they do not value Weather Forecast directly) Conditional arc ending in Weather Forecast indicates their belief that Weather Forecast and Weather Condition can be dependent. Informational arc ending in Vacation Activity indicates that they will only know Weather Forecast, not Weather Condition, when making their choice. In other words, actual weather will be known after they make their choice, and only forecast is what they can count on at this stage. It also follows semantically, for example, that Vacation Activity is independent on (irrelevant to) Weather Condition given Weather Forecast is known. == Applicability to value of information == The above example highlights the power of the influence diagram in representing an extremely important concept in decision analysis known as the value of information. Consider the following three scenarios; Scenario 1: The decision-maker could make their Vacation Activity decision while knowing what Weather Condition will be like. This corresponds to adding extra informational arc from Weather Condition to Vacation Activity in the above influence diagram. Scenario 2: The original influence diagram as shown above. Scenario 3: The decision-maker makes their decision without even knowing the Weather Forecast. This corresponds to removing informational arc from Weather Forecast to Vacation Activity in the above influence diagram. Scenario 1 is the best possible scenario for this decision situation since there is no longer any uncertainty on what they care about (Weather Condition) when making their decision. Scenario 3, however, is the worst possible scenario for this decision situation since they need to make their decision without any hint (Weather Forecast) on what they care about (Weather Condition) will turn out to be. The decision-maker is usually better off (definitely no worse off, on average) to move from scenario 3 to scenario 2 through the acquisition of new information. The most they should be willing to pay for such move is called the value of information on Weather Forecast, which is essentially the value of imperfect information on Weather Condition. The applicability of this simple ID and the value of information concept is tremendous, especially in medical decision making when most decisions have to be made with imperfect information about their patients, diseases, etc. == Related concepts == Influence diagrams are hierarchical and can be defined either in terms of their structure or in greater detail in terms of the functional and numerical relation between diagram elements. An ID that is consistently defined at all levels—structure, function, and number—is a well-defined mathematical representation and is referred to as a well-formed influence diagram (WFID). WFIDs can be evaluated using reversal and removal operations to yield answers to a large class of probabilistic, inferential, and decision questions. More recent techniques have been developed by artificial intelligence researchers concerning Bayesian network inference (belief propagation). An influence diagram having only uncertainty nodes (i.e., a Bayesian network) is also called a relevance diagram. An arc connecting node A to B implies not only that "A is relevant to B", but also that "B is relevant to A" (i.e., relevance is a symmetric relationship).

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  Shattered set

  A class of sets is said to shatter another set if it is possible to "pick out" any element of that set using intersection. The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory. == Definition == Suppose A is a set and C is a class of sets. The class C shatters the set A if for each subset a of A, there is some element c of C such that a = c ∩ A . {\displaystyle a=c\cap A.} Equivalently, C shatters A when their intersection is equal to A's power set: P(A) = { c ∩ A | c ∈ C }. We employ the letter C to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set A is often assumed to be finite because, in empirical processes, we are interested in the shattering of finite sets of data points. == Example == We will show that the class of all discs in the plane (two-dimensional space) does not shatter every set of four points on the unit circle, yet the class of all convex sets in the plane does shatter every finite set of points on the unit circle. Let A be a set of four points on the unit circle and let C be the class of all discs. To test where C shatters A, we attempt to draw a disc around every subset of points in A. First, we draw a disc around the subsets of each isolated point. Next, we try to draw a disc around every subset of point pairs. This turns out to be doable for adjacent points, but impossible for points on opposite sides of the circle. Any attempt to include those points on the opposite side will necessarily include other points not in that pair. Hence, any pair of opposite points cannot be isolated out of A using intersections with class C and so C does not shatter A. As visualized below: Because there is some subset which can not be isolated by any disc in C, we conclude then that A is not shattered by C. And, with a bit of thought, we can prove that no set of four points is shattered by this C. However, if we redefine C to be the class of all elliptical discs, we find that we can still isolate all the subsets from above, as well as the points that were formerly problematic. Thus, this specific set of 4 points is shattered by the class of elliptical discs. Visualized below: With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all convex sets (visualize connecting the dots). == Shatter coefficient == To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function). For a collection C of sets s ⊂ Ω {\displaystyle s\subset \Omega } , Ω {\displaystyle \Omega } being any space, often a sample space, we define the nth shattering coefficient of C as S C ( n ) = max ∀ x 1 , x 2 , … , x n ∈ Ω card ⁡ { { x 1 , x 2 , … , x n } ∩ s , s ∈ C } {\displaystyle S_{C}(n)=\max _{\forall x_{1},x_{2},\dots ,x_{n}\in \Omega }\operatorname {card} \{\,\{\,x_{1},x_{2},\dots ,x_{n}\}\cap s,s\in C\}} where card {\displaystyle \operatorname {card} } denotes the cardinality of the set and x 1 , x 2 , … , x n ∈ Ω {\displaystyle x_{1},x_{2},\dots ,x_{n}\in \Omega } is any set of n points,. S C ( n ) {\displaystyle S_{C}(n)} is the largest number of subsets of any set A of n points that can be formed by intersecting A with the sets in collection C. For example, if set A contains 3 points, its power set, P ( A ) {\displaystyle P(A)} , contains 2 3 = 8 {\displaystyle 2^{3}=8} elements. If C shatters A, its shattering coefficient(3) would be 8 and S C ( 2 ) {\displaystyle S_{C}(2)} would be 2 2 = 4 {\displaystyle 2^{2}=4} . However, if one of those sets in P ( A ) {\displaystyle P(A)} cannot be obtained through intersections in c, then S C ( 3 ) {\displaystyle S_{C}(3)} would only be 7. If none of those sets can be obtained, S C ( 3 ) {\displaystyle S_{C}(3)} would be 0. Additionally, if S C ( 2 ) = 3 {\displaystyle S_{C}(2)=3} , for example, then there is an element in the set of all 2-point sets from A that cannot be obtained from intersections with C. It follows from this that S C ( 3 ) {\displaystyle S_{C}(3)} would also be less than 8 (i.e. C would not shatter A) because we have already located a "missing" set in the smaller power set of 2-point sets. This example illustrates some properties of S C ( n ) {\displaystyle S_{C}(n)} : S C ( n ) ≤ 2 n {\displaystyle S_{C}(n)\leq 2^{n}} for all n because { s ∩ A | s ∈ C } ⊆ P ( A ) {\displaystyle \{s\cap A|s\in C\}\subseteq P(A)} for any A ⊆ Ω {\displaystyle A\subseteq \Omega } . If S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} , that means there is a set of cardinality n, which can be shattered by C. If S C ( N ) < 2 N {\displaystyle S_{C}(N)<2^{N}} for some N > 1 {\displaystyle N>1} then S C ( n ) < 2 n {\displaystyle S_{C}(n)<2^{n}} for all n ≥ N {\displaystyle n\geq N} . The third property means that if C cannot shatter any set of cardinality N then it can not shatter sets of larger cardinalities. == Vapnik–Chervonenkis class == If A cannot be shattered by C, there will be a smallest value of n that makes the shatter coefficient(n) less than 2 n {\displaystyle 2^{n}} because as n gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of n for which the S C ( n ) {\displaystyle S_{C}(n)} is still 2 n {\displaystyle 2^{n}} , because as n gets smaller, there are fewer sets that could be omitted. The extreme of this is S C ( 0 ) {\displaystyle S_{C}(0)} (the shattering coefficient of the empty set), which must always be 2 0 = 1 {\displaystyle 2^{0}=1} . These statements lends themselves to defining the VC dimension of a class C as: V C ( C ) = min n { n : S C ( n ) < 2 n } {\displaystyle VC(C)={\underset {n}{\min }}\{n:S_{C}(n)<2^{n}\}\,} or, alternatively, as V C 0 ( C ) = max n { n : S C ( n ) = 2 n } . {\displaystyle VC_{0}(C)={\underset {n}{\max }}\{n:S_{C}(n)=2^{n}\}.\,} Note that V C ( C ) = V C 0 ( C ) + 1. {\displaystyle VC(C)=VC_{0}(C)+1.} . The VC dimension is usually defined as V C 0 {\displaystyle VC_{0}} , the largest cardinality of points chosen that will still shatter A (i.e. n such that S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} ). Altneratively, if for any n there is a set of cardinality n which can be shattered by C, then S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} for all n and the VC dimension of this class C is infinite. A class with finite VC dimension is called a Vapnik–Chervonenkis class or VC class. A class C is uniformly Glivenko–Cantelli if and only if it is a VC class.

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  Lesk algorithm

  The Lesk algorithm is a classical algorithm for word sense disambiguation introduced by Michael E. Lesk in 1986. It operates on the premise that words within a given context are likely to share a common meaning. This algorithm compares the dictionary definitions of an ambiguous word with the words in its surrounding context to determine the most appropriate sense. Variations, such as the Simplified Lesk algorithm, have demonstrated improved precision and efficiency. However, the Lesk algorithm has faced criticism for its sensitivity to definition wording and its reliance on brief glosses. Researchers have sought to enhance its accuracy by incorporating additional resources like thesauruses and syntactic models. == Overview == The Lesk algorithm is based on the assumption that words in a given "neighborhood" (section of text) will tend to share a common topic. A simplified version of the Lesk algorithm is to compare the dictionary definition of an ambiguous word with the terms contained in its neighborhood. Versions have been adapted to use WordNet. An implementation might look like this: for every sense of the word being disambiguated one should count the number of words that are in both the neighborhood of that word and in the dictionary definition of that sense the sense that is to be chosen is the sense that has the largest number of this count. A frequently used example illustrating this algorithm is for the context "pine cone". The following dictionary definitions are used: PINE 1. kinds of evergreen tree with needle-shaped leaves 2. waste away through sorrow or illness CONE 1. solid body which narrows to a point 2. something of this shape whether solid or hollow 3. fruit of certain evergreen trees As can be seen, the best intersection is Pine #1 ⋂ Cone #3 = 2. == Simplified Lesk algorithm == In Simplified Lesk algorithm, the correct meaning of each word in a given context is determined individually by locating the sense that overlaps the most between its dictionary definition and the given context. Rather than simultaneously determining the meanings of all words in a given context, this approach tackles each word individually, independent of the meaning of the other words occurring in the same context. "A comparative evaluation performed by Vasilescu et al. (2004) has shown that the simplified Lesk algorithm can significantly outperform the original definition of the algorithm, both in terms of precision and efficiency. By evaluating the disambiguation algorithms on the Senseval-2 English all words data, they measure a 58% precision using the simplified Lesk algorithm compared to the only 42% under the original algorithm. Note: Vasilescu et al. implementation considers a back-off strategy for words not covered by the algorithm, consisting of the most frequent sense defined in WordNet. This means that words for which all their possible meanings lead to zero overlap with current context or with other word definitions are by default assigned sense number one in WordNet." Simplified LESK Algorithm with smart default word sense (Vasilescu et al., 2004) The COMPUTEOVERLAP function returns the number of words in common between two sets, ignoring function words or other words on a stop list. The original Lesk algorithm defines the context in a more complex way. == Criticisms == Unfortunately, Lesk’s approach is very sensitive to the exact wording of definitions, so the absence of a certain word can radically change the results. Further, the algorithm determines overlaps only among the glosses of the senses being considered. This is a significant limitation in that dictionary glosses tend to be fairly short and do not provide sufficient vocabulary to relate fine-grained sense distinctions. A lot of work has appeared offering different modifications of this algorithm. These works use other resources for analysis (thesauruses, synonyms dictionaries or morphological and syntactic models): for instance, it may use such information as synonyms, different derivatives, or words from definitions of words from definitions. == Lesk variants == Original Lesk (Lesk, 1986) Adapted/Extended Lesk (Banerjee and Pederson, 2002/2003): In the adaptive lesk algorithm, a word vector is created corresponds to every content word in the wordnet gloss. Concatenating glosses of related concepts in WordNet can be used to augment this vector. The vector contains the co-occurrence counts of words co-occurring with w in a large corpus. Adding all the word vectors for all the content words in its gloss creates the Gloss vector g for a concept. Relatedness is determined by comparing the gloss vector using the Cosine similarity measure. There are a lot of studies concerning Lesk and its extensions: Wilks and Stevenson, 1998, 1999; Mahesh et al., 1997; Cowie et al., 1992; Yarowsky, 1992; Pook and Catlett, 1988; Kilgarriff and Rosensweig, 2000; Kwong, 2001; Nastase and Szpakowicz, 2001; Gelbukh and Sidorov, 2004.

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  Universal approximation theorem

  In the field of machine learning, the universal approximation theorems (UATs) state that neural networks with a certain structure can, in principle, approximate any continuous function to any desired degree of accuracy. These theorems provide a mathematical justification for using neural networks, assuring researchers that a sufficiently large or deep network can model the complex, non-linear relationships often found in real-world data. The best-known version of the theorem applies to feedforward networks with a single hidden layer. It states that if the layer's activation function is non-polynomial (which is true for common choices like the sigmoid function or ReLU), then the network can act as a "universal approximator." Universality is achieved by increasing the number of neurons in the hidden layer, making the network "wider." Other versions of the theorem show that universality can also be achieved by keeping the network's width fixed but increasing its number of layers, making it "deeper." These are existence theorems. They guarantee that a network with the right structure exists, but they do not provide a method for finding the network's parameters (training it), nor do they specify exactly how large the network must be for a given function. Finding a suitable network remains a practical challenge that is typically addressed with optimization algorithms like backpropagation. == Setup == Artificial neural networks are combinations of multiple simple mathematical functions that implement more complicated functions from (typically) real-valued vectors to real-valued vectors. The spaces of multivariate functions that can be implemented by a network are determined by the structure of the network, the set of simple functions, and its multiplicative parameters. A great deal of theoretical work has gone into characterizing these function spaces. Most universal approximation theorems are in one of two classes. The first quantifies the approximation capabilities of neural networks with an arbitrary number of artificial neurons ("arbitrary width" case) and the second focuses on the case with an arbitrary number of hidden layers, each containing a limited number of artificial neurons ("arbitrary depth" case). In addition to these two classes, there are also universal approximation theorems for neural networks with bounded number of hidden layers and a limited number of neurons in each layer ("bounded depth and bounded width" case). == History == === Arbitrary width === The first results concerned the arbitrary width case. Ken-ichi Funahashi (May 1989) showed that Rumelhart–Hinton–Williams type backpropagation networks possess universal approximation capability with a class of sigmoidal activation functions, extending the result to multi-output mappings as well. Kurt Hornik, Maxwell Stinchcombe, and Halbert White (July 1989) showed that multilayer feed-forward networks with as few as one hidden layer are universal approximators, provided that the activation function satisfies certain conditions. George Cybenko (December 1989) independently established a related result for sigmoid activation functions using functional-analytic methods. Hornik also showed in 1991 that it is not the specific choice of the activation function but rather the multilayer feed-forward architecture itself that gives neural networks the potential of being universal approximators. Moshe Leshno et al in 1993 and later Allan Pinkus in 1999 showed that the universal approximation property is equivalent to having a nonpolynomial activation function. === Arbitrary depth === The arbitrary depth case was also studied by a number of authors such as Gustaf Gripenberg in 2003, Dmitry Yarotsky, Zhou Lu et al in 2017, Boris Hanin and Mark Sellke in 2018 who focused on neural networks with ReLU activation function. In 2020, Patrick Kidger and Terry Lyons extended those results to neural networks with general activation functions such, e.g. tanh or GeLU. One special case of arbitrary depth is that each composition component comes from a finite set of mappings. In 2024, Cai constructed a finite set of mappings, named a vocabulary, such that any continuous function can be approximated by compositing a sequence from the vocabulary. This is similar to the concept of compositionality in linguistics, which is the idea that a finite vocabulary of basic elements can be combined via grammar to express an infinite range of meanings. === Bounded depth and bounded width === The bounded depth and bounded width case was first studied by Maiorov and Pinkus in 1999. They showed that there exists an analytic sigmoidal activation function such that two hidden layer neural networks with bounded number of units in hidden layers are universal approximators. In 2018, Guliyev and Ismailov constructed a smooth sigmoidal activation function providing universal approximation property for two hidden layer feedforward neural networks with fewer units in hidden layers. In 2018, they also constructed single hidden layer networks with bounded width that are still universal approximators for univariate functions. However, this does not apply for multivariable functions. In 2022, Shen et al. obtained precise quantitative information on the depth and width required to approximate a target function by deep and wide ReLU neural networks. === Quantitative bounds === The question of minimal possible width for universality was first studied in 2021, Park et al obtained the minimum width required for the universal approximation of Lp functions using feed-forward neural networks with ReLU as activation functions. Similar results that can be directly applied to residual neural networks were also obtained in the same year by Paulo Tabuada and Bahman Gharesifard using control-theoretic arguments. In 2023, Cai obtained the optimal minimum width bound for the universal approximation. For the arbitrary depth case, Leonie Papon and Anastasis Kratsios derived explicit depth estimates depending on the regularity of the target function and of the activation function. === Kolmogorov network === The Kolmogorov–Arnold representation theorem is similar in spirit. Indeed, certain neural network families can directly apply the Kolmogorov–Arnold theorem to yield a universal approximation theorem. Robert Hecht-Nielsen showed that a three-layer neural network can approximate any continuous multivariate function. This was extended to the discontinuous case by Vugar Ismailov. In 2024, Ziming Liu and co-authors showed a practical application. === Reservoir computing and quantum reservoir computing === In reservoir computing a sparse recurrent neural network with fixed weights equipped of fading memory and echo state property is followed by a trainable output layer. Its universality has been demonstrated separately for what concerns networks of rate neurons and spiking neurons, respectively. In 2024, the framework has been generalized and extended to quantum reservoirs where the reservoir is based on qubits defined over Hilbert spaces. === Variants === Variants include discontinuous activation functions, noncompact domains, certifiable networks, random neural networks, and alternative network architectures and topologies. The universal approximation property of width-bounded networks has been studied as a dual of classical universal approximation results on depth-bounded networks. For input dimension d x {\displaystyle d_{x}} and output dimension d y {\displaystyle d_{y}} the minimum width required for the universal approximation of the Lp functions is exactly m a x { d x + 1 , d y } {\displaystyle max\{d_{x}+1,d_{y}\}} (for a ReLU network). More generally this also holds if both ReLU and a threshold activation function are used. Universal function approximation on graphs (or rather on graph isomorphism classes) by popular graph convolutional neural networks (GCNs or GNNs) can be made as discriminative as the Weisfeiler–Leman graph isomorphism test. In 2020, a universal approximation theorem result was established by Brüel-Gabrielsson, showing that graph representation with certain injective properties is sufficient for universal function approximation on bounded graphs and restricted universal function approximation on unbounded graphs, with an accompanying O ( | V | ⋅ | E | ) {\displaystyle {\mathcal {O}}(\left|V\right|\cdot \left|E\right|)} -runtime method that performed at state of the art on a collection of benchmarks (where V {\displaystyle V} and E {\displaystyle E} are the sets of nodes and edges of the graph respectively). There are also a variety of results between non-Euclidean spaces and other commonly used architectures and, more generally, algorithmically generated sets of functions, such as the convolutional neural network (CNN) architecture, radial basis functions, or neural networks with specific properties. == Arbitrary-width case == A universal approximation theorem formally states that a family of neural network funct

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  Moral graph

  In graph theory, a moral graph is used to find the equivalent undirected form of a directed acyclic graph. It is a key step of the junction tree algorithm, used in belief propagation on graphical models. The moralized counterpart of a directed acyclic graph is formed by adding edges between all pairs of non-adjacent nodes that have a common child, and then making all edges in the graph undirected. Equivalently, a moral graph of a directed acyclic graph G is an undirected graph in which each node of the original G is now connected to its Markov blanket. The name stems from the fact that, in a moral graph, two nodes that have a common child are required to be married by sharing an edge. Moralization may also be applied to mixed graphs, called in this context "chain graphs". In a chain graph, a connected component of the undirected subgraph is called a chain. Moralization adds an undirected edge between any two vertices that both have outgoing edges to the same chain, and then forgets the orientation of the directed edges of the graph. == Weakly recursively simplicial == A graph is weakly recursively simplicial if it has a simplicial vertex and the subgraph after removing a simplicial vertex and some edges (possibly none) between its neighbours is weakly recursively simplicial. A graph is moral if and only if it is weakly recursively simplicial. A chordal graph (a.k.a., recursive simplicial) is a special case of weakly recursively simplicial when no edge is removed during the elimination process. Therefore, a chordal graph is also moral. But a moral graph is not necessarily chordal. == Recognising moral graphs == Unlike chordal graphs that can be recognised in polynomial time, Verma & Pearl (1993) proved that deciding whether or not a graph is moral is NP-complete.

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