AI Image Generators

Explore the best AI Image Generators — independent reviews, comparisons, pricing and step-by-step how-to guides, curated by Aizhi.

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  Cyclodisparity

  In vision science, cyclodisparity is the difference in the rotation angle of an object or scene viewed by the left and right eyes. Cyclodisparity can result from the eyes' torsional rotation (cyclorotation) or can be created artificially by presenting to the eyes two images that need to be rotated relative to each other for binocular fusion to take place. == Human and animal vision == The eyes and visual system can compensate for cyclodisparity up to a certain point; if the cyclodisparity is larger than a threshold, the images cannot be fused, resulting stereoblindness, and in double vision in subjects who otherwise have full stereo vision. When a human subject is presented with images that have artificial cyclodisparity, cyclovergence is evoked, that is, a motor response of the eye muscles that rotates the two eyes in opposite directions, thereby reducing cyclodisparity. Visually-induced cyclovergence of up to 8 degrees has been observed in normal subjects. Furthermore, up to about 8 degrees can usually be compensated by purely sensory means, that is, without physical eye rotation. This means that the normal human observer can achieve binocular image fusion in presence of cyclodisparity of up to approximately 16 degrees. Cyclodisparity due to images having been rotated inward can be compensated better when the gaze is directed downwards, and cyclodisparity due to an outward rotation can be compensated better when the gaze is directed upwards. A proposed explanation for this phenomenon is that the motor system is coordinated in such a way that the eyes perform a torsional movement to reduce the size of the search zones and thus the computational load required for solving the correspondence problem. The resulting cyclovergence at near gaze is smaller than the cyclovergence predicted by Listing's law. == Video processing and computer vision == Active camera torsion can be used in machine and computer vision for several purposes. For instance, camera torsion can be used to make improved use of the search range over which matching detectors or stereo matching algorithms operate, or to make a 3D slanted surface appear frontoparallel for further stereo processing. For image compression purposes, images with cyclodisparity are advantageously encoded using global motion compensation using a rotational motion model.

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  Neural cryptography

  Neural cryptography is a branch of cryptography dedicated to analyzing the application of stochastic algorithms, especially artificial neural network algorithms, for use in encryption and cryptanalysis. == Definition == Artificial neural networks are well known for their ability to selectively explore the solution space of a given problem. This feature finds a natural niche of application in the field of cryptanalysis. At the same time, neural networks offer a new approach to attack ciphering algorithms based on the principle that any function could be reproduced by a neural network, which is a powerful proven computational tool that can be used to find the inverse-function of any cryptographic algorithm. The ideas of mutual learning, self learning, and stochastic behavior of neural networks and similar algorithms can be used for different aspects of cryptography, like public-key cryptography, solving the key distribution problem using neural network mutual synchronization, hashing or generation of pseudo-random numbers. Another idea is the ability of a neural network to separate space in non-linear pieces using "bias". It gives different probabilities of activating the neural network or not. This is very useful in the case of Cryptanalysis. Two names are used to design the same domain of research: Neuro-Cryptography and Neural Cryptography. The first work that it is known on this topic can be traced back to 1995 in an IT Master Thesis. == Applications == In 1995, Sebastien Dourlens applied neural networks to cryptanalyze DES by allowing the networks to learn how to invert the S-tables of the DES. The bias in DES studied through Differential Cryptanalysis by Adi Shamir is highlighted. The experiment shows about 50% of the key bits can be found, allowing the complete key to be found in a short time. Hardware application with multi micro-controllers have been proposed due to the easy implementation of multilayer neural networks in hardware. One example of a public-key protocol is given by Khalil Shihab . He describes the decryption scheme and the public key creation that are based on a backpropagation neural network. The encryption scheme and the private key creation process are based on Boolean algebra. This technique has the advantage of small time and memory complexities. A disadvantage is the property of backpropagation algorithms: because of huge training sets, the learning phase of a neural network is very long. Therefore, the use of this protocol is only theoretical so far. == Neural key exchange protocol == The most used protocol for key exchange between two parties A and B in the practice is Diffie–Hellman key exchange protocol. Neural key exchange, which is based on the synchronization of two tree parity machines, should be a secure replacement for this method. Synchronizing these two machines is similar to synchronizing two chaotic oscillators in chaos communications. === Tree parity machine === The tree parity machine is a special type of multi-layer feedforward neural network. It consists of one output neuron, K hidden neurons and K×N input neurons. Inputs to the network take three values: x i j ∈ { − 1 , 0 , + 1 } {\displaystyle x_{ij}\in \left\{-1,0,+1\right\}} The weights between input and hidden neurons take the values: w i j ∈ { − L , . . . , 0 , . . . , + L } {\displaystyle w_{ij}\in \left\{-L,...,0,...,+L\right\}} Output value of each hidden neuron is calculated as a sum of all multiplications of input neurons and these weights: σ i = sgn ⁡ ( ∑ j = 1 N w i j x i j ) {\displaystyle \sigma _{i}=\operatorname {sgn}(\sum _{j=1}^{N}w_{ij}x_{ij})} Signum is a simple function, which returns −1,0 or 1: sgn ⁡ ( x ) = { − 1 if x < 0 , 0 if x = 0 , 1 if x > 0. {\displaystyle \operatorname {sgn}(x)={\begin{cases}-1&{\text{if }}x<0,\\0&{\text{if }}x=0,\\1&{\text{if }}x>0.\end{cases}}} If the scalar product is 0, the output of the hidden neuron is mapped to −1 in order to ensure a binary output value. The output of neural network is then computed as the multiplication of all values produced by hidden elements: τ = ∏ i = 1 K σ i {\displaystyle \tau =\prod _{i=1}^{K}\sigma _{i}} Output of the tree parity machine is binary. === Protocol === Each party (A and B) uses its own tree parity machine. Synchronization of the tree parity machines is achieved in these steps Initialize random weight values Execute these steps until the full synchronization is achieved Generate random input vector X Compute the values of the hidden neurons Compute the value of the output neuron Compare the values of both tree parity machines Outputs are the same: one of the suitable learning rules is applied to the weights Outputs are different: go to 2.1 After the full synchronization is achieved (the weights wij of both tree parity machines are same), A and B can use their weights as keys. This method is known as a bidirectional learning. One of the following learning rules can be used for the synchronization: Hebbian learning rule: w i + = g ( w i + σ i x i Θ ( σ i τ ) Θ ( τ A τ B ) ) {\displaystyle w_{i}^{+}=g(w_{i}+\sigma _{i}x_{i}\Theta (\sigma _{i}\tau )\Theta (\tau ^{A}\tau ^{B}))} Anti-Hebbian learning rule: w i + = g ( w i − σ i x i Θ ( σ i τ ) Θ ( τ A τ B ) ) {\displaystyle w_{i}^{+}=g(w_{i}-\sigma _{i}x_{i}\Theta (\sigma _{i}\tau )\Theta (\tau ^{A}\tau ^{B}))} Random walk: w i + = g ( w i + x i Θ ( σ i τ ) Θ ( τ A τ B ) ) {\displaystyle w_{i}^{+}=g(w_{i}+x_{i}\Theta (\sigma _{i}\tau )\Theta (\tau ^{A}\tau ^{B}))} Where: Θ ( a , b ) = 0 {\displaystyle \Theta (a,b)=0} if a ≠ b {\displaystyle a\neq b} otherwise Θ ( a , b ) = 1 {\displaystyle \Theta (a,b)=1} And: g ( x ) {\displaystyle g(x)} is a function that keeps the w i {\displaystyle w_{i}} in the range { − L , − L + 1 , . . . , 0 , . . . , L − 1 , L } {\displaystyle \{-L,-L+1,...,0,...,L-1,L\}} === Attacks and security of this protocol === In every attack it is considered, that the attacker E can eavesdrop messages between the parties A and B, but does not have an opportunity to change them. ==== Brute force ==== To provide a brute force attack, an attacker has to test all possible keys (all possible values of weights wij). By K hidden neurons, K×N input neurons and boundary of weights L, this gives (2L+1)KN possibilities. For example, the configuration K = 3, L = 3 and N = 100 gives us 310253 key possibilities, making the attack impossible with today's computer power. ==== Learning with own tree parity machine ==== One of the basic attacks can be provided by an attacker, who owns the same tree parity machine as the parties A and B. He wants to synchronize his tree parity machine with these two parties. In each step there are three situations possible: Output(A) ≠ Output(B): None of the parties updates its weights. Output(A) = Output(B) = Output(E): All the three parties update weights in their tree parity machines. Output(A) = Output(B) ≠ Output(E): Parties A and B update their tree parity machines, but the attacker can not do that. Because of this situation his learning is slower than the synchronization of parties A and B. It has been proven, that the synchronization of two parties is faster than learning of an attacker. It can be improved by increasing of the synaptic depth L of the neural network. That gives this protocol enough security and an attacker can find out the key only with small probability. ==== Other attacks ==== For conventional cryptographic systems, we can improve the security of the protocol by increasing of the key length. In the case of neural cryptography, we improve it by increasing of the synaptic depth L of the neural networks. Changing this parameter increases the cost of a successful attack exponentially, while the effort for the users grows polynomially. Therefore, breaking the security of neural key exchange belongs to the complexity class NP. Alexander Klimov, Anton Mityaguine, and Adi Shamir say that the original neural synchronization scheme can be broken by at least three different attacks—geometric, probabilistic analysis, and using genetic algorithms. Even though this particular implementation is insecure, the ideas behind chaotic synchronization could potentially lead to a secure implementation. === Permutation parity machine === The permutation parity machine is a binary variant of the tree parity machine. It consists of one input layer, one hidden layer and one output layer. The number of neurons in the output layer depends on the number of hidden units K. Each hidden neuron has N binary input neurons: x i j ∈ { 0 , 1 } {\displaystyle x_{ij}\in \left\{0,1\right\}} The weights between input and hidden neurons are also binary: w i j ∈ { 0 , 1 } {\displaystyle w_{ij}\in \left\{0,1\right\}} Output value of each hidden neuron is calculated as a sum of all exclusive disjunctions (exclusive or) of input neurons and these weights: σ i = θ N ( ∑ j = 1 N w i j ⊕ x i j ) {\displaystyle \sigma _{i}=\theta _{N}(\sum _{j=1}^{N}w_{ij}\oplus x_{ij})} (⊕ means XOR). Th

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  Independent component analysis

  In signal processing, independent component analysis (ICA) is a computational method for separating a multivariate signal into additive subcomponents. This is done by assuming that at most one subcomponent is Gaussian and that the subcomponents are statistically independent from each other. ICA was invented by Jeanny Hérault and Christian Jutten in 1985. ICA is a special case of blind source separation. A common example application of ICA is the "cocktail party problem" of listening in on one person's speech in a noisy room. == Introduction == Independent component analysis attempts to decompose a multivariate signal into independent non-Gaussian signals. As an example, sound is usually a signal that is composed of the numerical addition, at each time t, of signals from several sources. The question then is whether it is possible to separate these contributing sources from the observed total signal. When the statistical independence assumption is correct, blind ICA separation of a mixed signal gives very good results. It is also used for signals that are not supposed to be generated by mixing for analysis purposes. A simple application of ICA is the "cocktail party problem", where the underlying speech signals are separated from a sample data consisting of people talking simultaneously in a room. Usually the problem is simplified by assuming no time delays or echoes. Note that a filtered and delayed signal is a copy of a dependent component, and thus the statistical independence assumption is not violated. Mixing weights for constructing the M {\textstyle M} observed signals from the N {\textstyle N} components can be placed in an M × N {\textstyle M\times N} matrix. An important thing to consider is that if N {\textstyle N} sources are present, at least N {\textstyle N} observations (e.g. microphones if the observed signal is audio) are needed to recover the original signals. When there are an equal number of observations and source signals, the mixing matrix is square ( M = N {\textstyle M=N} ). Other cases of underdetermined ( M < N {\textstyle M N {\textstyle M>N} ) have been investigated. The success of ICA separation of mixed signals relies on two assumptions and three effects of mixing source signals. Two assumptions: The source signals are independent of each other. The values in each source signal have non-Gaussian distributions. Three effects of mixing source signals: Independence: As per assumption 1, the source signals are independent; however, their signal mixtures are not. This is because the signal mixtures share the same source signals. Normality: According to the Central Limit Theorem, the distribution of a sum of independent random variables with finite variance tends towards a Gaussian distribution.Loosely speaking, a sum of two independent random variables usually has a distribution that is closer to Gaussian than any of the two original variables. Here we consider the value of each signal as the random variable. Complexity: The temporal complexity of any signal mixture is greater than that of its simplest constituent source signal. Those principles contribute to the basic establishment of ICA. If the signals extracted from a set of mixtures are independent and have non-Gaussian distributions or have low complexity, then they must be source signals. Another common example is image steganography, where ICA is used to embed one image within another. For instance, two grayscale images can be linearly combined to create mixed images in which the hidden content is visually imperceptible. ICA can then be used to recover the original source images from the mixtures. This technique underlies digital watermarking, which allows the embedding of ownership information into images, as well as more covert applications such as undetected information transmission. The method has even been linked to real-world cyberespionage cases. In such applications, ICA serves to unmix the data based on statistical independence, making it possible to extract hidden components that are not apparent in the observed data. Steganographic techniques, including those potentially involving ICA-based analysis, have been used in real-world cyberespionage cases. In 2010, the FBI uncovered a Russian spy network known as the "Illegals Program" (Operation Ghost Stories), where agents used custom-built steganography tools to conceal encrypted text messages within image files shared online. In another case, a former General Electric engineer, Xiaoqing Zheng, was convicted in 2022 for economic espionage. Zheng used steganography to exfiltrate sensitive turbine technology by embedding proprietary data within image files for transfer to entities in China. == Defining component independence == ICA finds the independent components (also called factors, latent variables or sources) by maximizing the statistical independence of the estimated components. We may choose one of many ways to define a proxy for independence, and this choice governs the form of the ICA algorithm. The two broadest definitions of independence for ICA are Minimization of mutual information Maximization of non-Gaussianity The Minimization-of-Mutual information (MMI) family of ICA algorithms uses measures like Kullback-Leibler Divergence and maximum entropy. The non-Gaussianity family of ICA algorithms, motivated by the central limit theorem, uses kurtosis and negentropy. Typical algorithms for ICA use centering (subtract the mean to create a zero mean signal), whitening (usually with the eigenvalue decomposition), and dimensionality reduction as preprocessing steps in order to simplify and reduce the complexity of the problem for the actual iterative algorithm. == Mathematical definitions == Linear independent component analysis can be divided into noiseless and noisy cases, where noiseless ICA is a special case of noisy ICA. Nonlinear ICA should be considered as a separate case. === General Derivation === In the classical ICA model, it is assumed that the observed data x i ∈ R m {\displaystyle \mathbf {x} _{i}\in \mathbb {R} ^{m}} at time t i {\displaystyle t_{i}} is generated from source signals s i ∈ R m {\displaystyle \mathbf {s} _{i}\in \mathbb {R} ^{m}} via a linear transformation x i = A s i {\displaystyle \mathbf {x} _{i}=A\mathbf {s} _{i}} , where A {\displaystyle A} is an unknown, invertible mixing matrix. To recover the source signals, the data is first centered (zero mean), and then whitened so that the transformed data has unit covariance. This whitening reduces the problem from estimating a general matrix A {\displaystyle A} to estimating an orthogonal matrix V {\displaystyle V} , significantly simplifying the search for independent components. If the covariance matrix of the centered data is Σ x = A A ⊤ {\displaystyle \Sigma _{x}=AA^{\top }} , then using the eigen-decomposition Σ x = Q D Q ⊤ {\displaystyle \Sigma _{x}=QDQ^{\top }} , the whitening transformation can be taken as D − 1 / 2 Q ⊤ {\displaystyle D^{-1/2}Q^{\top }} . This step ensures that the recovered sources are uncorrelated and of unit variance, leaving only the task of rotating the whitened data to maximize statistical independence. This general derivation underlies many ICA algorithms and is foundational in understanding the ICA model. ==== Reduced Mixing Problem ==== Independent component analysis (ICA) addresses the problem of recovering a set of unobserved source signals s i = ( s i 1 , s i 2 , … , s i m ) T {\displaystyle s_{i}=(s_{i1},s_{i2},\dots ,s_{im})^{T}} from observed mixed signals x i = ( x i 1 , x i 2 , … , x i m ) T {\displaystyle x_{i}=(x_{i1},x_{i2},\dots ,x_{im})^{T}} , based on the linear mixing model: x i = A s i , {\displaystyle x_{i}=A\,s_{i},} where the A {\displaystyle A} is an m × m {\displaystyle m\times m} invertible matrix called the mixing matrix, s i {\displaystyle s_{i}} represents the m‑dimensional vector containing the values of the sources at time t i {\displaystyle t_{i}} , and x i {\displaystyle x_{i}} is the corresponding vector of observed values at time t i {\displaystyle t_{i}} . The goal is to estimate both A {\displaystyle A} and the source signals { s i } {\displaystyle \{s_{i}\}} solely from the observed data { x i } {\displaystyle \{x_{i}\}} . After centering, the Gram matrix is computed as: ( X ∗ ) T X ∗ = Q D Q T , {\displaystyle (X^{})^{T}X^{}=Q\,D\,Q^{T},} where D is a diagonal matrix with positive entries (assuming X ∗ {\displaystyle X^{}} has maximum rank), and Q is an orthogonal matrix. Writing the SVD of the mixing matrix A = U Σ V T {\displaystyle A=U\Sigma V^{T}} and comparing with A A T = U Σ 2 U T {\displaystyle AA^{T}=U\Sigma ^{2}U^{T}} the mixing A has the form A = Q D 1 / 2 V T . {\displaystyle A=Q\,D^{1/2}\,V^{T}.} So, the normalized source values satisfy s i ∗ = V y i ∗ {\displaystyle s_{i}^{}=V\,y_{i}^{}} , where y i ∗ = D − 1 2 Q T x i ∗ . {\displaystyle y_{i}^{}=D^{-{\tfrac {1}{2}}}Q^{T}x_{i}^{}.} Thus, ICA reduces

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  Mean squared error

  In statistics, the mean squared error (MSE) or mean squared deviation (MSD) of an estimator (of a procedure for estimating an unobserved quantity) measures the average of the squares of the errors—that is, the average squared difference between the estimated values and the true value. MSE is a risk function, corresponding to the expected value of the squared error loss. The fact that MSE is almost always strictly positive (and not zero) is because of randomness or because the estimator does not account for information that could produce a more accurate estimate. In machine learning, specifically empirical risk minimization, MSE may refer to the empirical risk (the average loss on an observed data set), as an estimate of the true MSE (the true risk: the average loss on the actual population distribution). The MSE is a measure of the quality of an estimator. As it is derived from the square of Euclidean distance, it is always a positive value that decreases as the error approaches zero. The MSE is the second moment (about the origin) of the error, and thus incorporates both the variance of the estimator (how widely spread the estimates are from one data sample to another) and its bias (how far off the average estimated value is from the true value). For an unbiased estimator, the MSE is the variance of the estimator. Like the variance, MSE has the same units of measurement as the square of the quantity being estimated. In an analogy to standard deviation, taking the square root of MSE yields the root-mean-square error or root-mean-square deviation (RMSE or RMSD), which has the same units as the quantity being estimated; for an unbiased estimator, the RMSE is the square root of the variance, known as the standard error. == Definition and basic properties == The MSE either assesses the quality of a predictor (i.e., a function mapping arbitrary inputs to a sample of values of some random variable), or of an estimator (i.e., a mathematical function mapping a sample of data to an estimate of a parameter of the population from which the data is sampled). In the context of prediction, understanding the prediction interval can also be useful as it provides a range within which a future observation will fall, with a certain probability. The definition of an MSE differs according to whether one is describing a predictor or an estimator. === Predictor === If a vector of n {\displaystyle n} predictions is generated from a sample of n {\displaystyle n} data points on all variables, and Y {\displaystyle Y} is the vector of observed values of the variable being predicted, with Y ^ {\displaystyle {\hat {Y}}} being the predicted values (e.g. as from a least-squares fit), then the within-sample MSE of the predictor is computed as MSE = 1 n ∑ i = 1 n ( Y i − Y i ^ ) 2 {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} In other words, the MSE is the mean ( 1 n ∑ i = 1 n ) {\textstyle \left({\frac {1}{n}}\sum _{i=1}^{n}\right)} of the squares of the errors ( Y i − Y i ^ ) 2 {\textstyle \left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} . This is an easily computable quantity for a particular sample (and hence is sample-dependent). In matrix notation, MSE = 1 n ∑ i = 1 n ( e i ) 2 = 1 n e T e {\displaystyle \operatorname {MSE} ={\frac {1}{n}}\sum _{i=1}^{n}(e_{i})^{2}={\frac {1}{n}}\mathbf {e} ^{\mathsf {T}}\mathbf {e} } where e i {\displaystyle e_{i}} is Y i − Y i ^ {\displaystyle Y_{i}-{\hat {Y_{i}}}} and e {\displaystyle \mathbf {e} } is a n × 1 {\displaystyle n\times 1} column vector. The MSE can also be computed on q data points that were not used in estimating the model, either because they were held back for this purpose, or because these data have been newly obtained. Within this process, known as cross-validation, the MSE is often called the test MSE, and is computed as MSE = 1 q ∑ i = n + 1 n + q ( Y i − Y i ^ ) 2 {\displaystyle \operatorname {MSE} ={\frac {1}{q}}\sum _{i=n+1}^{n+q}\left(Y_{i}-{\hat {Y_{i}}}\right)^{2}} === Estimator === The MSE of an estimator θ ^ {\displaystyle {\hat {\theta }}} with respect to an unknown parameter θ {\displaystyle \theta } is defined as MSE ⁡ ( θ ^ ) = E θ ⁡ [ ( θ ^ − θ ) 2 ] . {\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {E} _{\theta }\left[({\hat {\theta }}-\theta )^{2}\right].} This definition depends on the unknown parameter, therefore the MSE is a priori property of an estimator. The MSE could be a function of unknown parameters, in which case any estimator of the MSE based on estimates of these parameters would be a function of the data (and thus a random variable). If the estimator θ ^ {\displaystyle {\hat {\theta }}} is derived as a sample statistic and is used to estimate some population parameter, then the expectation is with respect to the sampling distribution of the sample statistic. The MSE can be written as the sum of the variance of the estimator and the squared bias of the estimator, providing a useful way to calculate the MSE and implying that in the case of unbiased estimators, the MSE and variance are equivalent. MSE ⁡ ( θ ^ ) = Var θ ⁡ ( θ ^ ) + Bias ⁡ ( θ ^ , θ ) 2 . {\displaystyle \operatorname {MSE} ({\hat {\theta }})=\operatorname {Var} _{\theta }({\hat {\theta }})+\operatorname {Bias} ({\hat {\theta }},\theta )^{2}.} ==== Proof of variance and bias relationship ==== MSE ⁡ ( θ ^ ) = E θ ⁡ [ ( θ ^ − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] + E θ ⁡ [ θ ^ ] − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 + 2 ( θ ^ − E θ ⁡ [ θ ^ ] ) ( E θ ⁡ [ θ ^ ] − θ ) + ( E θ ⁡ [ θ ^ ] − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + E θ ⁡ [ 2 ( θ ^ − E θ ⁡ [ θ ^ ] ) ( E θ ⁡ [ θ ^ ] − θ ) ] + E θ ⁡ [ ( E θ ⁡ [ θ ^ ] − θ ) 2 ] = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + 2 ( E θ ⁡ [ θ ^ ] − θ ) E θ ⁡ [ θ ^ − E θ ⁡ [ θ ^ ] ] + ( E θ ⁡ [ θ ^ ] − θ ) 2 E θ ⁡ [ θ ^ ] − θ = constant = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + 2 ( E θ ⁡ [ θ ^ ] − θ ) ( E θ ⁡ [ θ ^ ] − E θ ⁡ [ θ ^ ] ) + ( E θ ⁡ [ θ ^ ] − θ ) 2 E θ ⁡ [ θ ^ ] = constant = E θ ⁡ [ ( θ ^ − E θ ⁡ [ θ ^ ] ) 2 ] + ( E θ ⁡ [ θ ^ ] − θ ) 2 = Var θ ⁡ ( θ ^ ) + Bias θ ⁡ ( θ ^ , θ ) 2 {\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})&=\operatorname {E} _{\theta }\left[({\hat {\theta }}-\theta )^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]+\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}+2\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+\operatorname {E} _{\theta }\left[2\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\right]+\operatorname {E} _{\theta }\left[\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\right]\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+2\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\operatorname {E} _{\theta }\left[{\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right]+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}&&\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta ={\text{constant}}\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+2\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}&&\operatorname {E} _{\theta }[{\hat {\theta }}]={\text{constant}}\\&=\operatorname {E} _{\theta }\left[\left({\hat {\theta }}-\operatorname {E} _{\theta }[{\hat {\theta }}]\right)^{2}\right]+\left(\operatorname {E} _{\theta }[{\hat {\theta }}]-\theta \right)^{2}\\&=\operatorname {Var} _{\theta }({\hat {\theta }})+\operatorname {Bias} _{\theta }({\hat {\theta }},\theta )^{2}\end{aligned}}} An even shorter proof can be achieved using the well-known formula that for a random variable X {\textstyle X} , E ( X 2 ) = Var ⁡ ( X ) + ( E ( X ) ) 2 {\textstyle \mathbb {E} (X^{2})=\operatorname {Var} (X)+(\mathbb {E} (X))^{2}} . By substituting X {\textstyle X} with, θ ^ − θ {\textstyle {\hat {\theta }}-\theta } , we have MSE ⁡ ( θ ^ ) = E [ ( θ ^ − θ ) 2 ] = Var ⁡ ( θ ^ − θ ) + ( E [ θ ^ − θ ] ) 2 = Var ⁡ ( θ ^ ) + Bias 2 ⁡ ( θ ^ , θ ) {\displaystyle {\begin{aligned}\operatorname {MSE} ({\hat {\theta }})&=\mathbb {E} [({\hat {\theta }}-\theta )^{2}]\\&=\operator

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  PCPaint

  PCPaint was one of the first IBM PC-based mouse-driven GUI paint programs, released in 1984. It followed after Microsoft Doodle, released in 1983 with the Microsoft Mouse version 1 drivers for DOS, and around the same time as Digital Research’s Draw program. It was developed and created by John Bridges and Doug Wolfgram. It was later developed into Pictor Paint. The hardware manufacturer Mouse Systems bundled PCPaint with millions of computer mice that they sold, making PCPaint one of the best-selling DOS-based paint programs of the mid 1980s. == History == In 1983, Doug Wolfgram bought a Microsoft Mouse and decided to write a drawing program for it. They named it “Mouse Draw”. The interface was primitive but the program functioned well. Wolfgram traveled to SoftCon in New Orleans where he demonstrated the program to Mouse Systems. Mouse Systems was developing an optical mouse and they wanted to bundle a painting program so they agreed to publish Mouse Draw. The original program was written entirely in assembly language with primitive graphics routines developed by Wolfgram. John Bridges worked for an educational software company, Classroom Consortia Media, Inc., developing and writing Apple and IBM graphics libraries for CCM's software. Bridges and Wolfgram were friends who had been connected through a bulletin board system developed and run by Wolfgram. The two collaborated cross country via the BBS, Wolfram in California and Bridges in New York. Mouse Systems wanted the paint program to capture the look and feel of MacPaint. John Bridges and Doug Wolfgram started reworking Mouse Draw into what became PCPaint. The program was completely re-written using Bridge's graphics library and the top-level elements were written in C rather than assembly language. Bridges developed the core graphics code for the first version of PCPaint while Wolfgram worked on the user interface and top-level code. Mouse Systems signed an exclusive agreement with Wolfgram's company, Microtex Industries, Inc., to bundle PCPaint with every mouse they sold. They began publishing PCPaint with their mice in 1984. Microsoft responded in 1985 by bundling a competing product, PC Paintbrush, with version 4 of its DOS drivers for the Microsoft Mouse, replacing its in-house Microsoft Doodle program which it published with version 1 of the DOS drivers in mid-1983. Microsoft’s mouse began to outsell Mouse Systems mouse. In November 1985 Microsoft bundled a cut-down version of PC Paintbrush with Windows 1.0 (called Microsoft Paint), later bundling an updated version of PC Paintbrush with Windows 3.0 (as Paintbrush), impacting PCPaint’s marketshare. In early 1987, Mouse Systems decided that PCPaint wasn't helping to sell mice any longer so they discontinued the bundle deal and returned rights to the code to MicroTex Industries, but retained rights to the name, PCPaint. Wolfgram then combined the paint program with a new animation system he was developing (called GRASP) and Paul Mace Software bought publishing rights to the animation system and PCPaint, which was to be renamed Pictor. Bridges again got involved and took over programming responsibilities for GRASP as well as PCPaint while Wolfgram focused on more of the business details. In creating the first version of PCPaint, Doug had a dual-floppy machine with a Computer Innovations compiler on one disk and source code on the other. John had the "luxury" of a 10MB hard disk in his XT. Data was exchanged daily via 1200, then 2400 baud modems. === Authorship and Ownership === John Bridges and Wolfgram continued to work on PCPaint and GRASP on behalf of Paul Mace Software until 1990. Also in that year, Doug Wolfgram sold his remaining rights to PCPaint (and its animation system, GRASP) to John Bridges. In 1994, GRASP development stopped and so did development of Pictor Paint. John Bridges terminated his GRASP publishing contract with Paul Mace Software, and went off to create GLPro (the next generation of GRASP) with GMEDIA. Along with GLPro, came GLPaint, the successor to PCPaint and Pictor Paint. == Versions == In June 1984, Mouse Systems shipped PCPaint 1.0, the first GUI based Paint program for the IBM PC family of computers. John Bridges and Doug Wolfgram, were the co-authors of PCPaint 1.0. PCPaint 1.0 saved its graphics in a modified BSaved image format with the extension of ".PIC". The release of PCPaint Version 1.5 followed in late 1984, with the additions of graphics image compression for the .PIC format and support for "larger-than-screen" images. PCjr support was also added in this version after overcoming severe memory shortage problems getting PCPaint to run on the 128k PCjr. October 1985 saw the release of PCPaint 2.0. EGA support and publishing features were added to this version. The .PIC format was further refined, offering support for the rapidly expanding graphics capabilities of the PC and efficient image compression. PCPaint 3.1 was released in 1989. Unlike previous versions, it was not bundled with mice but was sold as a stand-alone software product. PCPaint 3.1 offered improved text and image handling, provided 36 types of flood and fill, worked with VGA adapters in hi-res 16-color and 256-color modes, allowed the user to save and retrieve files in a variety of intercompatible formats (.PIC, .GIF, .PCX, .IMG), and printed selected portions of images on color or black-and-white dot matrix, ink jet, and laser printers such as PostScript and HP Laser Jet. PCPaint 3.1 is still in use today by some users of DOS emulation programs like DOSBox and available for free download. Pictor Paint was an improved version, written by John Bridges, and bundled with GRASP GRaphical System for Presentation also written by John Bridges. It was also called "The Painter's Easel". GLPaint, released in 1995, was the last in this series of paint programs written by John Bridges. By 1998 version 7.0 provided support for TrueColor images and the Pictor PIC format was expanded to handle these. == Pictor PIC Image Format == PCPaint 1.0 saved its graphics in a modified BSAVE image format (which was popular at the time) with the file type (extension) of ".PIC". By PCPaint 1.5 this format was extended further to accommodate image compression. With the release of version 2.0 the PICtor PIC image format was developed almost to its present state, with no similarity to the BSAVE format used by earlier versions. Pictor Paint saved its files in a compressed format with the file extension PIC, which was the same format used by PCPaint.

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  Learning classifier system

  Learning classifier systems, or LCS, are a paradigm of rule-based machine learning methods that combine a discovery component (e.g. typically a genetic algorithm in evolutionary computation) with a learning component (performing either supervised learning, reinforcement learning, or unsupervised learning). Learning classifier systems seek to identify a set of context-dependent rules that collectively store and apply knowledge in a piecewise manner in order to make predictions (e.g. behavior modeling, classification, data mining, regression, function approximation, or game strategy). This approach allows complex solution spaces to be broken up into smaller, simpler parts for the reinforcement learning that is inside artificial intelligence research. The founding concepts behind learning classifier systems came from attempts to model complex adaptive systems, using rule-based agents to form an artificial cognitive system (i.e. artificial intelligence). == Methodology == The architecture and components of a given learning classifier system can be quite variable. It is useful to think of an LCS as a machine consisting of several interacting components. Components may be added or removed, or existing components modified/exchanged to suit the demands of a given problem domain (like algorithmic building blocks) or to make the algorithm flexible enough to function in many different problem domains. As a result, the LCS paradigm can be flexibly applied to many problem domains that call for machine learning. The major divisions among LCS implementations are as follows: (1) Michigan-style architecture vs. Pittsburgh-style architecture, (2) reinforcement learning vs. supervised learning, (3) incremental learning vs. batch learning, (4) online learning vs. offline learning, (5) strength-based fitness vs. accuracy-based fitness, and (6) complete action mapping vs best action mapping. These divisions are not necessarily mutually exclusive. For example, XCS, the best known and best studied LCS algorithm, is Michigan-style, was designed for reinforcement learning but can also perform supervised learning, applies incremental learning that can be either online or offline, applies accuracy-based fitness, and seeks to generate a complete action mapping. === Elements of a generic LCS algorithm === Keeping in mind that LCS is a paradigm for genetic-based machine learning rather than a specific method, the following outlines key elements of a generic, modern (i.e. post-XCS) LCS algorithm. For simplicity let us focus on Michigan-style architecture with supervised learning. See the illustrations on the right laying out the sequential steps involved in this type of generic LCS. ==== Environment ==== The environment is the source of data upon which an LCS learns. It can be an offline, finite training dataset (characteristic of a data mining, classification, or regression problem), or an online sequential stream of live training instances. Each training instance is assumed to include some number of features (also referred to as attributes, or independent variables), and a single endpoint of interest (also referred to as the class, action, phenotype, prediction, or dependent variable). Part of LCS learning can involve feature selection, therefore not all of the features in the training data need to be informative. The set of feature values of an instance is commonly referred to as the state. For simplicity let's assume an example problem domain with Boolean/binary features and a Boolean/binary class. For Michigan-style systems, one instance from the environment is trained on each learning cycle (i.e. incremental learning). Pittsburgh-style systems perform batch learning, where rule sets are evaluated in each iteration over much or all of the training data. ==== Rule/classifier/population ==== A rule is a context dependent relationship between state values and some prediction. Rules typically take the form of an {IF:THEN} expression, (e.g. {IF 'condition' THEN 'action'}, or as a more specific example, {IF 'red' AND 'octagon' THEN 'stop-sign'}). A critical concept in LCS and rule-based machine learning alike, is that an individual rule is not in itself a model, since the rule is only applicable when its condition is satisfied. Think of a rule as a "local-model" of the solution space. Rules can be represented in many different ways to handle different data types (e.g. binary, discrete-valued, ordinal, continuous-valued). Given binary data LCS traditionally applies a ternary rule representation (i.e. rules can include either a 0, 1, or '#' for each feature in the data). The 'don't care' symbol (i.e. '#') serves as a wild card within a rule's condition allowing rules, and the system as a whole to generalize relationships between features and the target endpoint to be predicted. Consider the following rule (#1###0 ~ 1) (i.e. condition ~ action). This rule can be interpreted as: IF the second feature = 1 AND the sixth feature = 0 THEN the class prediction = 1. We would say that the second and sixth features were specified in this rule, while the others were generalized. This rule, and the corresponding prediction are only applicable to an instance when the condition of the rule is satisfied by the instance. This is more commonly referred to as matching. In Michigan-style LCS, each rule has its own fitness, as well as a number of other rule-parameters associated with it that can describe the number of copies of that rule that exist (i.e. the numerosity), the age of the rule, its accuracy, or the accuracy of its reward predictions, and other descriptive or experiential statistics. A rule along with its parameters is often referred to as a classifier. In Michigan-style systems, classifiers are contained within a population [P] that has a user defined maximum number of classifiers. Unlike most stochastic search algorithms (e.g. evolutionary algorithms), LCS populations start out empty (i.e. there is no need to randomly initialize a rule population). Classifiers will instead be initially introduced to the population with a covering mechanism. In any LCS, the trained model is a set of rules/classifiers, rather than any single rule/classifier. In Michigan-style LCS, the entire trained (and optionally, compacted) classifier population forms the prediction model. ==== Matching ==== One of the most critical and often time-consuming elements of an LCS is the matching process. The first step in an LCS learning cycle takes a single training instance from the environment and passes it to [P] where matching takes place. In step two, every rule in [P] is now compared to the training instance to see which rules match (i.e. are contextually relevant to the current instance). In step three, any matching rules are moved to a match set [M]. A rule matches a training instance if all feature values specified in the rule condition are equivalent to the corresponding feature value in the training instance. For example, assuming the training instance is (001001 ~ 0), these rules would match: (###0## ~ 0), (00###1 ~ 0), (#01001 ~ 1), but these rules would not (1##### ~ 0), (000##1 ~ 0), (#0#1#0 ~ 1). Notice that in matching, the endpoint/action specified by the rule is not taken into consideration. As a result, the match set may contain classifiers that propose conflicting actions. In the fourth step, since we are performing supervised learning, [M] is divided into a correct set [C] and an incorrect set [I]. A matching rule goes into the correct set if it proposes the correct action (based on the known action of the training instance), otherwise it goes into [I]. In reinforcement learning LCS, an action set [A] would be formed here instead, since the correct action is not known. ==== Covering ==== At this point in the learning cycle, if no classifiers made it into either [M] or [C] (as would be the case when the population starts off empty), the covering mechanism is applied (fifth step). Covering is a form of online smart population initialization. Covering randomly generates a rule that matches the current training instance (and in the case of supervised learning, that rule is also generated with the correct action. Assuming the training instance is (001001 ~ 0), covering might generate any of the following rules: (#0#0## ~ 0), (001001 ~ 0), (#010## ~ 0). Covering not only ensures that each learning cycle there is at least one correct, matching rule in [C], but that any rule initialized into the population will match at least one training instance. This prevents LCS from exploring the search space of rules that do not match any training instances. ==== Parameter updates/credit assignment/learning ==== In the sixth step, the rule parameters of any rule in [M] are updated to reflect the new experience gained from the current training instance. Depending on the LCS algorithm, a number of updates can take place at this step. For supervised learning, we can simply update the accuracy/error of a

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  Large margin nearest neighbor

  Large margin nearest neighbor (LMNN) classification is a statistical machine learning algorithm for metric learning. It learns a pseudometric designed for k-nearest neighbor classification. The algorithm is based on semidefinite programming, a sub-class of convex optimization. The goal of supervised learning (more specifically classification) is to learn a decision rule that can categorize data instances into pre-defined classes. The k-nearest neighbor rule assumes a training data set of labeled instances (i.e. the classes are known). It classifies a new data instance with the class obtained from the majority vote of the k closest (labeled) training instances. Closeness is measured with a pre-defined metric. Large margin nearest neighbors is an algorithm that learns this global (pseudo-)metric in a supervised fashion to improve the classification accuracy of the k-nearest neighbor rule. == Setup == The main intuition behind LMNN is to learn a pseudometric under which all data instances in the training set are surrounded by at least k instances that share the same class label. If this is achieved, the leave-one-out error (a special case of cross validation) is minimized. Let the training data consist of a data set D = { ( x → 1 , y 1 ) , … , ( x → n , y n ) } ⊂ R d × C {\displaystyle D=\{({\vec {x}}_{1},y_{1}),\dots ,({\vec {x}}_{n},y_{n})\}\subset R^{d}\times C} , where the set of possible class categories is C = { 1 , … , c } {\displaystyle C=\{1,\dots ,c\}} . The algorithm learns a pseudometric of the type d ( x → i , x → j ) = ( x → i − x → j ) ⊤ M ( x → i − x → j ) {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})=({\vec {x}}_{i}-{\vec {x}}_{j})^{\top }\mathbf {M} ({\vec {x}}_{i}-{\vec {x}}_{j})} . For d ( ⋅ , ⋅ ) {\displaystyle d(\cdot ,\cdot )} to be well defined, the matrix M {\displaystyle \mathbf {M} } needs to be positive semi-definite. The Euclidean metric is a special case, where M {\displaystyle \mathbf {M} } is the identity matrix. This generalization is often (falsely) referred to as Mahalanobis metric. Figure 1 illustrates the effect of the metric under varying M {\displaystyle \mathbf {M} } . The two circles show the set of points with equal distance to the center x → i {\displaystyle {\vec {x}}_{i}} . In the Euclidean case this set is a circle, whereas under the modified (Mahalanobis) metric it becomes an ellipsoid. The algorithm distinguishes between two types of special data points: target neighbors and impostors. === Target neighbors === Target neighbors are selected before learning. Each instance x → i {\displaystyle {\vec {x}}_{i}} has exactly k {\displaystyle k} different target neighbors within D {\displaystyle D} , which all share the same class label y i {\displaystyle y_{i}} . The target neighbors are the data points that should become nearest neighbors under the learned metric. Let us denote the set of target neighbors for a data point x → i {\displaystyle {\vec {x}}_{i}} as N i {\displaystyle N_{i}} . === Impostors === An impostor of a data point x → i {\displaystyle {\vec {x}}_{i}} is another data point x → j {\displaystyle {\vec {x}}_{j}} with a different class label (i.e. y i ≠ y j {\displaystyle y_{i}\neq y_{j}} ) which is one of the nearest neighbors of x → i {\displaystyle {\vec {x}}_{i}} . During learning the algorithm tries to minimize the number of impostors for all data instances in the training set. == Algorithm == Large margin nearest neighbors optimizes the matrix M {\displaystyle \mathbf {M} } with the help of semidefinite programming. The objective is twofold: For every data point x → i {\displaystyle {\vec {x}}_{i}} , the target neighbors should be close and the impostors should be far away. Figure 1 shows the effect of such an optimization on an illustrative example. The learned metric causes the input vector x → i {\displaystyle {\vec {x}}_{i}} to be surrounded by training instances of the same class. If it was a test point, it would be classified correctly under the k = 3 {\displaystyle k=3} nearest neighbor rule. The first optimization goal is achieved by minimizing the average distance between instances and their target neighbors ∑ i , j ∈ N i d ( x → i , x → j ) {\displaystyle \sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})} . The second goal is achieved by penalizing distances to impostors x → l {\displaystyle {\vec {x}}_{l}} that are less than one unit further away than target neighbors x → j {\displaystyle {\vec {x}}_{j}} (and therefore pushing them out of the local neighborhood of x → i {\displaystyle {\vec {x}}_{i}} ). The resulting value to be minimized can be stated as: ∑ i , j ∈ N i , l , y l ≠ y i [ d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ] + {\displaystyle \sum _{i,j\in N_{i},l,y_{l}\neq y_{i}}[d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})]_{+}} With a hinge loss function [ ⋅ ] + = max ( ⋅ , 0 ) {\textstyle [\cdot ]_{+}=\max(\cdot ,0)} , which ensures that impostor proximity is not penalized when outside the margin. The margin of exactly one unit fixes the scale of the matrix M {\displaystyle M} . Any alternative choice c > 0 {\displaystyle c>0} would result in a rescaling of M {\displaystyle M} by a factor of 1 / c {\displaystyle 1/c} . The final optimization problem becomes: min M ∑ i , j ∈ N i d ( x → i , x → j ) + λ ∑ i , j , l ξ i j l {\displaystyle \min _{\mathbf {M} }\sum _{i,j\in N_{i}}d({\vec {x}}_{i},{\vec {x}}_{j})+\lambda \sum _{i,j,l}\xi _{ijl}} ∀ i , j ∈ N i , l , y l ≠ y i {\displaystyle \forall _{i,j\in N_{i},l,y_{l}\neq y_{i}}} d ( x → i , x → j ) + 1 − d ( x → i , x → l ) ≤ ξ i j l {\displaystyle d({\vec {x}}_{i},{\vec {x}}_{j})+1-d({\vec {x}}_{i},{\vec {x}}_{l})\leq \xi _{ijl}} ξ i j l ≥ 0 {\displaystyle \xi _{ijl}\geq 0} M ⪰ 0 {\displaystyle \mathbf {M} \succeq 0} The hyperparameter λ > 0 {\textstyle \lambda >0} is some positive constant (typically set through cross-validation). Here the variables ξ i j l {\displaystyle \xi _{ijl}} (together with two types of constraints) replace the term in the cost function. They play a role similar to slack variables to absorb the extent of violations of the impostor constraints. The last constraint ensures that M {\displaystyle \mathbf {M} } is positive semi-definite. The optimization problem is an instance of semidefinite programming (SDP). Although SDPs tend to suffer from high computational complexity, this particular SDP instance can be solved very efficiently due to the underlying geometric properties of the problem. In particular, most impostor constraints are naturally satisfied and do not need to be enforced during runtime (i.e. the set of variables ξ i j l {\displaystyle \xi _{ijl}} is sparse). A particularly well suited solver technique is the working set method, which keeps a small set of constraints that are actively enforced and monitors the remaining (likely satisfied) constraints only occasionally to ensure correctness. == Extensions and efficient solvers == LMNN was extended to multiple local metrics in the 2008 paper. This extension significantly improves the classification error, but involves a more expensive optimization problem. In their 2009 publication in the Journal of Machine Learning Research, Weinberger and Saul derive an efficient solver for the semi-definite program. It can learn a metric for the MNIST handwritten digit data set in several hours, involving billions of pairwise constraints. An open source Matlab implementation is freely available at the authors web page. Kumal et al. extended the algorithm to incorporate local invariances to multivariate polynomial transformations and improved regularization.

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  Latent and observable variables

  In statistics, latent variables (from Latin: present participle of lateo 'lie hidden') are variables that can only be inferred indirectly through a mathematical model from other observable variables that can be directly observed or measured. Such latent variable models are used in many disciplines, including engineering, medicine, ecology, physics, machine learning/artificial intelligence, natural language processing, bioinformatics, chemometrics, demography, economics, management, political science, psychology and the social sciences. Latent variables may correspond to aspects of physical reality. These could in principle be measured, but may not be for practical reasons. Among the earliest expressions of this idea is Francis Bacon's polemic the Novum Organum, itself a challenge to the more traditional logic expressed in Aristotle's Organon: But the latent process of which we speak, is far from being obvious to men’s minds, beset as they now are. For we mean not the measures, symptoms, or degrees of any process which can be exhibited in the bodies themselves, but simply a continued process, which, for the most part, escapes the observation of the senses. In this situation, the term hidden variables is commonly used, reflecting the fact that the variables are meaningful, but not observable. Other latent variables correspond to abstract concepts, like categories, behavioral or mental states, or data structures. The terms hypothetical variables or hypothetical constructs may be used in these situations. The use of latent variables can serve to reduce the dimensionality of data. Many observable variables can be aggregated in a model to represent an underlying concept, making it easier to understand the data. In this sense, they serve a function similar to that of scientific theories. At the same time, latent variables link observable "sub-symbolic" data in the real world to symbolic data in the modeled world. == Examples == === Psychology === Latent variables, as created by factor analytic methods, generally represent "shared" variance, or the degree to which variables "move" together. Variables that have no correlation cannot result in a latent construct based on the common factor model. The "Big Five personality traits" have been inferred using factor analysis. extraversion spatial ability wisdom: “Two of the more predominant means of assessing wisdom include wisdom-related performance and latent variable measures.” Spearman's g, or the general intelligence factor in psychometrics === Economics === Examples of latent variables from the field of economics include quality of life, business confidence, morale, happiness and conservatism: these are all variables which cannot be measured directly. However, by linking these latent variables to other, observable variables, the values of the latent variables can be inferred from measurements of the observable variables. Quality of life is a latent variable which cannot be measured directly, so observable variables are used to infer quality of life. Observable variables to measure quality of life include wealth, employment, environment, physical and mental health, education, recreation and leisure time, and social belonging. === Medicine === Latent-variable methodology is used in many branches of medicine. A class of problems that naturally lend themselves to latent variables approaches are longitudinal studies where the time scale (e.g. age of participant or time since study baseline) is not synchronized with the trait being studied. For such studies, an unobserved time scale that is synchronized with the trait being studied can be modeled as a transformation of the observed time scale using latent variables. Examples of this include disease progression modeling and modeling of growth (see box). == Inferring latent variables == There exists a range of different model classes and methodology that make use of latent variables and allow inference in the presence of latent variables. Models include: linear mixed-effects models and nonlinear mixed-effects models Hidden Markov models Factor analysis Item response theory Analysis and inference methods include: Principal component analysis Instrumented principal component analysis Partial least squares regression Latent semantic analysis and probabilistic latent semantic analysis EM algorithms Metropolis–Hastings algorithm === Bayesian algorithms and methods === Bayesian statistics is often used for inferring latent variables. Latent Dirichlet allocation The Chinese restaurant process is often used to provide a prior distribution over assignments of objects to latent categories. The Indian buffet process is often used to provide a prior distribution over assignments of latent binary features to objects.

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  Scene statistics

  Scene statistics is a discipline within the field of perception. It is concerned with the statistical regularities related to scenes. It is based on the premise that a perceptual system is designed to interpret scenes. Biological perceptual systems have evolved in response to physical properties of natural environments. Therefore natural scenes receive a great deal of attention. Natural scene statistics are useful for defining the behavior of an ideal observer in a natural task, typically by incorporating signal detection theory, information theory or estimation theory. == Within-domain versus across-domain == Geisler (2008) distinguishes between four kinds of domains: (1) Physical environments (2) Images/Scenes (3) Neural responses and (4) Behavior. Within the domain of images/scenes one can study the characteristics of information related to redundancy and efficient coding. Across-domain statistics determine how an autonomous system should make inferences about its environment, process information and control its behavior. To study these statistics it is necessary to sample or register information in multiple domains simultaneously. == Applications == === Prediction of picture and video quality === One of the most successful applications of Natural Scenes Statistics Models has been perceptual picture and video quality prediction. For example, the Visual Information Fidelity (VIF) algorithm, which is used to measure the degree of distortion of pictures and videos, is used extensively by the image and video processing communities to assess perceptual quality. This is often after processing, such as compression, which can degrade the appearance of a visual signal. The premise is that the scene statistics are changed by distortion and that the visual system is sensitive to the changes in the scene statistics. VIF is heavily used in the streaming television industry. Other popular picture quality models that use natural scene statistics include BRISQUE and NIQE, both of which are no-reference since they do not require any reference picture to measure quality against.

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  Gremlin (query language)

  Gremlin is a graph traversal language and virtual machine developed by Apache TinkerPop of the Apache Software Foundation. Gremlin works for both OLTP-based graph databases as well as OLAP-based graph processors. Gremlin's automata and functional language foundation enable Gremlin to naturally support imperative and declarative querying, host language agnosticism, user-defined domain specific languages, an extensible compiler/optimizer, single- and multi-machine execution models, and hybrid depth- and breadth-first evaluation with Turing completeness. As an explanatory analogy, Apache TinkerPop and Gremlin are to graph databases what the JDBC and SQL are to relational databases. Likewise, the Gremlin traversal machine is to graph computing as what the Java virtual machine is to general purpose computing. == History == 2009-10-30 the project is born, and immediately named "TinkerPop" 2009-12-25 v0.1 is the first release 2011-05-21 v1.0 is released 2012-05-24 v2.0 is released 2015-01-16 TinkerPop becomes an Apache Incubator project 2015-07-09 v3.0.0-incubating is released 2016-05-23 Apache TinkerPop becomes a top-level project 2016-07-18 v3.1.3 and v3.2.1 are first releases as Apache TinkerPop 2017-12-17 v3.3.1 is released 2018-05-08 v3.3.3 is released 2019-08-05 v3.4.3 is released 2020-02-20 v3.4.6 is released 2021-05-01 v3.5.0 is released 2022-04-04 v3.6.0 is released 2023-07-31 v3.7.0 is released 2025-11-12 v3.8.0 is released == Vendor integration == Gremlin is an Apache2-licensed graph traversal language that can be used by graph system vendors. There are typically two types of graph system vendors: OLTP graph databases and OLAP graph processors. The table below outlines those graph vendors that support Gremlin. == Traversal examples == The following examples of Gremlin queries and responses in a Gremlin-Groovy environment are relative to a graph representation of the MovieLens dataset. The dataset includes users who rate movies. Users each have one occupation, and each movie has one or more categories associated with it. The MovieLens graph schema is detailed below. === Simple traversals === For each vertex in the graph, emit its label, then group and count each distinct label. What year was the oldest movie made? What is Die Hard's average rating? === Projection traversals === For each category, emit a map of its name and the number of movies it represents. For each movie with at least 11 ratings, emit a map of its name and average rating. Sort the maps in decreasing order by their average rating. Emit the first 10 maps (i.e. top 10). === Declarative pattern matching traversals === Gremlin supports declarative graph pattern matching similar to SPARQL. For instance, the following query below uses Gremlin's match()-step. What 80's action movies do 30-something programmers like? Group count the movies by their name and sort the group count map in decreasing order by value. Clip the map to the top 10 and emit the map entries. === OLAP traversal === Which movies are most central in the implicit 5-stars graph? == Gremlin graph traversal machine == Gremlin is a virtual machine composed of an instruction set as well as an execution engine. An analogy is drawn between Gremlin and Java. === Gremlin steps (instruction set) === The following traversal is a Gremlin traversal in the Gremlin-Java8 dialect. The Gremlin language (i.e. the fluent-style of expressing a graph traversal) can be represented in any host language that supports function composition and function nesting. Due to this simple requirement, there exists various Gremlin dialects including Gremlin-Groovy, Gremlin-Scala, Gremlin-Clojure, etc. The above Gremlin-Java8 traversal is ultimately compiled down to a step sequence called a traversal. A string representation of the traversal above provided below. The steps are the primitives of the Gremlin graph traversal machine. They are the parameterized instructions that the machine ultimately executes. The Gremlin instruction set is approximately 30 steps. These steps are sufficient to provide general purpose computing and what is typically required to express the common motifs of any graph traversal query. Given that Gremlin is a language, an instruction set, and a virtual machine, it is possible to design another traversal language that compiles to the Gremlin traversal machine (analogous to how Scala compiles to the JVM). For instance, the popular SPARQL graph pattern match language can be compiled to execute on the Gremlin machine. The following SPARQL query would compile to In Gremlin-Java8, the SPARQL query above would be represented as below and compile to the identical Gremlin step sequence (i.e. traversal). === Gremlin Machine (virtual machine) === The Gremlin graph traversal machine can execute on a single machine or across a multi-machine compute cluster. Execution agnosticism allows Gremlin to run over both graph databases (OLTP) and graph processors (OLAP).

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  Lattice Miner

  Lattice Miner is a formal concept analysis software tool for the construction, visualization and manipulation of concept lattices. It allows the generation of formal concepts and association rules as well as the transformation of formal contexts via apposition, subposition, reduction and object/attribute generalization, and the manipulation of concept lattices via approximation, projection and selection. Lattice Miner allows also the drawing of nested line diagrams. == Introduction == Formal concept analysis (FCA) is a branch of applied mathematics based on the formalization of concept and concept hierarchy and mainly used as a framework for conceptual clustering and rule mining. Over the last two decades, a collection of tools have emerged to help FCA users visualize and analyze concept lattices. They range from the earliest DOS-based implementations (e.g., ConImp and GLAD) to more recent implementations in Java like ToscanaJ, Galicia, ConExp and Coron. A main issue in the development of FCA tools is to visualize large concept lattices and provide efficient mechanisms to highlight patterns (e.g., concepts, associations) that could be relevant to the user. The initial objective of the FCA tool called Lattice Miner was to focus on visualization mechanisms for the representation of concept lattices, including nested line diagrams. Later on, many other interesting features were integrated into the tool. == Functional architecture of Lattice Miner == Lattice Miner is a Java-based platform whose functions are articulated around a core. The Lattice Miner core provides all low-level operations and structures for the representation and manipulation of contexts, lattices and association rules. Mainly, the core of Lattice Miner consists of three modules: context, concept and association rule modules. The user interface offers a context editor and concept lattice manipulator to assist the user in a set of tasks. The architecture of Lattice Miner is open and modular enough to allow the integration of new features and facilities in each one of its components. === Context module === The context module offers all the basic operations and structures to manipulate binary and valued contexts as well as context decomposition to produce nested line diagrams. Basic context operations include apposition, subposition, generalization, clarification, reduction as well as the complementary context computation. The module provides also the arrow relations (for context reduction and decomposition) [2]. The tool has an input LMB format and recognizes the binary format SLF found in Galicia and the format CEX produced by ConExp. === Concept module === The main function of the concept module is to generate the concepts of the current binary context and construct the corresponding lattice and nested structure (see Figures 2 and 3). It provides the user with basic operators such as projection, selection, and exact search as well as advanced features like pair approximation. Some known algorithms are included in this module such as Bordat’s procedure, Godin’s algorithm and NextClosure algorithm. The approximation feature implemented in Lattice Miner is based on the following idea: given a pair (X,Y) where X ⊆ G, and Y ⊆ M, is there a set of formal concepts (Ai,Bi) which are “close to” (X,Y)? To answer this question, The tool starts to identify the type of couple that the pair (X,Y) represents. It can be a formal concept, a protoconcept, a semiconcept or a preconcept. In the last case, the approximation is given by the interval [(X",X′),(Y′,Y")] and highlighted in the line diagram. === Association rule module === This module includes procedures for computing the (stem) Guigues–Duquenne base using NextClosure algorithm [3], as well as the generic and informative bases. Implications with negation can be obtained using the apposition of a context and its complementary. This module embeds also procedures for the computation of a non-redundant family C of implications and the closure of a set Y of attributes for the given implication set C. === User interface === The initial objective of Lattice Miner was to focus on lattice drawing and visualization either as a flat or nested structure by taking into account the cognitive process of human beings and known principles for lattice drawing (e.g., reducing the number of edge intersections, ensuring diagram symmetry). Some well-known visualization techniques were implemented such as focus & context and fisheye view. The basic idea behind focus & context visualization paradigm is to allow a viewer to see key (important) objects in full detail in the foreground (focus) while at the same time an overview of all the surrounding information (context) remains available in the background. Lattice Miner translates the focus & context paradigm into clear and blurred elements while the size of nodes and the intensity of their color were used to indicate their importance. Various forms of highlighting, labelling and animation are also provided. In order to better handle the display of large lattices, nested line diagrams are offered in the tool. Figure 3 shows the third level of the nested line diagram corresponding to the binary context of Figure 1 where three levels of nesting are defined. Each one of the inner nodes of this diagram represents a combination of attributes from the previous two (outer) levels. Real inner concepts (see the node on the left hand-side of the diagram) are identified by colored nodes while void elements are in grey color. Each node of levels 1 and 2 can be expanded to exhibit its internal line diagram. Both flat and nested diagrams can be saved as an image. Simple (flat) lattices can also be saved as an XML format file.

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  Multi-surface method

  The multi-surface method (MSM) is a form of decision making using the concept of piecewise-linear separability of datasets to categorize data. == Introduction == Two datasets are linearly separable if their convex hulls do not intersect. The method may be formulated as a feedforward neural network with weights that are trained via linear programming. Comparisons between neural networks trained with the MSM versus backpropagation show MSM is better able to classify data. The decision problem associated linear program for the MSM is NP-complete. == Mathematical formulation == Given two finite disjoint point sets A , B ∈ R n {\displaystyle {\mathcal {A,B}}\in \mathbb {R} ^{n}} , find a discriminant, f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } such that f ( A ) > 0 , f ( B ) ≤ 0 {\displaystyle f({\mathcal {A}})>0,f({\mathcal {B}})\leq 0} . If the intersection of convex hulls of the two sets is the empty set, then it is possible to use a single linear program to obtain a linear discriminant of the form, f ( x ) = c x + γ {\displaystyle f(x)=cx+\gamma } . Usually, in real applications, the sets' convex hulls do intersect, and a (often non-convex) piecewise-linear discriminant can be used, through the use of several linear programs.

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  Open Syllabus Project

  The Open Syllabus Project (OSP) is an online open-source platform that catalogs and analyzes millions of college syllabi. Founded by researchers from the American Assembly at Columbia University, the OSP has amassed the most extensive collection of searchable syllabi. Since its beta launch in 2016, the OSP has collected over 7 million course syllabi from over 80 countries, primarily by scraping publicly accessible university websites. The project is directed by Joe Karaganis. == History == The OSP was formed by a group of data scientists, sociologists, and digital-humanities researchers at the American Assembly, a public-policy institute based at Columbia University. The OSP was partly funded by the Sloan Foundation and the Arcadia Fund. Joe Karaganis, former vice-president of the American Assembly, serves as the project director of the OSP. The project builds on prior attempts to archive syllabi, such as H-Net, MIT OpenCourseWare, and historian Dan Cohen's defunct Syllabus Finder website (Cohen now sits on the OSP's advisory board). The OSP became a non-profit and independent of the American Assembly in November 2019. In January 2016, the OSP launched a beta version of their "Syllabus Explorer," which they had collected data for since 2013. The Syllabus Explorer allows users to browse and search texts from over one million college course syllabi. The OSP launched a more comprehensive version 2.0 of the Syllabus Explorer in July 2019. The newer version includes an interactive visualization that displays texts as dots on a knowledge map. As of 2022, the OSP has collected over 7 million course syllabi. The Syllabus Explorer represents the "largest collection of searchable syllabi ever amassed." == Methodology == The OSP has collected syllabi data from over 80 countries dating to 2000. The syllabi stem from over 4,000 worldwide institutions. Most of the OSP's data originates from the United States. Canada, Australia, and the U.K also have large datasets. The OSP primarily collects syllabi by scraping publicly accessible university websites. The OSP also allows syllabi submissions from faculty, students, and administrators. The OSP developers use machine learning and natural language processing to extract metadata from such syllabi. Since only metadata is collected, no individual syllabus or personal identifying information is found in the OSP database. The OSP classifies the syllabi into 62 subject fields – corresponding to the U.S. Department of Education's Classification of Instructional Programs (CIP). Additionally, the OSP assigns each text a "teaching score" from 0–100. This score represents the text's percentile rank among citations in the total citation count and is a numerical indicator of the relative frequency of which a particular work is taught. The OSP also has data on which texts are most likely to be assigned together. The developers behind the OSP admit that the database is incomplete and likely contains "a fair number of errors." Karaganis estimates that 80–100 million syllabi exist in the United States alone. The OSP is unable to access syllabi behind private course-management software like Blackboard. == Notable findings == === Anthropology === Using data from the OSP, anthropologist Laurence Ralph uncovered that black anthropologists are "woefully under-represented in (if not erased from) most anthropology syllabi." Black authors wrote less than 1 percent of the top 1,000 assigned works. === Economics === The database indicates Greg Mankiw is the most frequently cited author for college economics courses. === English literature === The OSP found that Mary Shelley's Frankenstein was the most widely taught novel in college courses. Additionally, the majority of novels published after 1945 taught in English classes were historical fiction. === Female writers === The most read female writer on college campuses is Kate L. Turabian for her A Manual for Writers of Research Papers, Theses, and Dissertations . Turabian is followed by Diana Hacker, Toni Morrison, Jane Austen, and Virginia Woolf. === Film === The most assigned film according to the OSP is the 1929 Soviet documentary film, Man with a Movie Camera. English filmmaker Alfred Hitchcock is the most assigned director in college courses. === History === Historians George Brown Tindall and David Emory Shi's America: A Narrative History is the number one assigned textbook for history, followed by Anne Moody's memoir, Coming of Age in Mississippi. === Philosophy === The most assigned texts in the field of philosophy include Aristotle's Nicomachean Ethics, John Stuart Mill's Utilitarianism, and Plato's Republic. Plato's Republic was also the second most assigned text in universities in the English-speaking world (only behind Strunk and White's Elements of Style). === Physics === David Halliday's et al. Fundamentals of Physics is the number one ranked physics textbook in the OSP's database. === Political science === Data from the OSP indicates that the dominant political science texts are written almost exclusively by white men and scholars based in the West. In the top 200 most-frequently assigned works, 15 are authored by at least one woman. === Public administration === American president Woodrow Wilson's article "The Study of Administration" was the most frequently assigned text in public affairs and administration syllabi. == Reception == According to William Germano et al., the OSP is a "fascinating resource but is also prone to misrepresenting or at least distracting us from the most important business of a syllabus: communicating with students." Historian William Caferro remarks that the OSP is a "tacit experience of sharing, but a useful one." English professor Bart Beaty writes that, "Despite the many reservations about the completeness of its data, the OSP provides a rare opportunity for scholars to move beyond the anecdotal in discussions of canon-formation in teaching." Media theorist Elizabeth Losh opines that "big data approaches", like the OSP, may "raise troubling questions for instructors about informed consent, pedagogical privacy, and quantified metrics."

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  Stochastic variance reduction

  (Stochastic) variance reduction is an algorithmic approach to minimizing functions that can be decomposed into finite sums. By exploiting the finite sum structure, variance reduction techniques are able to achieve convergence rates that are impossible to achieve with methods that treat the objective as an infinite sum, as in the classical Stochastic approximation setting. Variance reduction approaches are widely used for training machine learning models such as logistic regression and support vector machines as these problems have finite-sum structure and uniform conditioning that make them ideal candidates for variance reduction. == Finite sum objectives == A function f {\displaystyle f} is considered to have finite sum structure if it can be decomposed into a summation or average: f ( x ) = 1 n ∑ i = 1 n f i ( x ) , {\displaystyle f(x)={\frac {1}{n}}\sum _{i=1}^{n}f_{i}(x),} where the function value and derivative of each f i {\displaystyle f_{i}} can be queried independently. Although variance reduction methods can be applied for any positive n {\displaystyle n} and any f i {\displaystyle f_{i}} structure, their favorable theoretical and practical properties arise when n {\displaystyle n} is large compared to the condition number of each f i {\displaystyle f_{i}} , and when the f i {\displaystyle f_{i}} have similar (but not necessarily identical) Lipschitz smoothness and strong convexity constants. The finite sum structure should be contrasted with the stochastic approximation setting which deals with functions of the form f ( θ ) = E ξ ⁡ [ F ( θ , ξ ) ] {\textstyle f(\theta )=\operatorname {E} _{\xi }[F(\theta ,\xi )]} which is the expected value of a function depending on a random variable ξ {\textstyle \xi } . Any finite sum problem can be optimized using a stochastic approximation algorithm by using F ( ⋅ , ξ ) = f ξ {\displaystyle F(\cdot ,\xi )=f_{\xi }} . == Rapid Convergence == Stochastic variance reduced methods without acceleration are able to find a minima of f {\displaystyle f} within accuracy ϵ > {\displaystyle \epsilon >} , i.e. f ( x ) − f ( x ∗ ) ≤ ϵ {\displaystyle f(x)-f(x_{})\leq \epsilon } in a number of steps of the order: O ( ( L μ + n ) log ⁡ ( 1 ϵ ) ) . {\displaystyle O\left(\left({\frac {L}{\mu }}+n\right)\log \left({\frac {1}{\epsilon }}\right)\right).} The number of steps depends only logarithmically on the level of accuracy required, in contrast to the stochastic approximation framework, where the number of steps O ( L / ( μ ϵ ) ) {\displaystyle O{\bigl (}L/(\mu \epsilon ){\bigr )}} required grows proportionally to the accuracy required. Stochastic variance reduction methods converge almost as fast as the gradient descent method's O ( ( L / μ ) log ⁡ ( 1 / ϵ ) ) {\displaystyle O{\bigl (}(L/\mu )\log(1/\epsilon ){\bigr )}} rate, despite using only a stochastic gradient, at a 1 / n {\displaystyle 1/n} lower cost than gradient descent. Accelerated methods in the stochastic variance reduction framework achieve even faster convergence rates, requiring only O ( ( n L μ + n ) log ⁡ ( 1 ϵ ) ) {\displaystyle O\left(\left({\sqrt {\frac {nL}{\mu }}}+n\right)\log \left({\frac {1}{\epsilon }}\right)\right)} steps to reach ϵ {\displaystyle \epsilon } accuracy, potentially n {\displaystyle {\sqrt {n}}} faster than non-accelerated methods. Lower complexity bounds. for the finite sum class establish that this rate is the fastest possible for smooth strongly convex problems. == Approaches == Variance reduction approaches fall within four main categories: table averaging methods, full-gradient snapshot methods, recursive estimator methods (e.g., SARAH), and dual methods. Each category contains methods designed for dealing with convex, non-smooth, and non-convex problems, each differing in hyper-parameter settings and other algorithmic details. === SAGA === In the SAGA method, the prototypical table averaging approach, a table of size n {\displaystyle n} is maintained that contains the last gradient witnessed for each f i {\displaystyle f_{i}} term, which we denote g i {\displaystyle g_{i}} . At each step, an index i {\displaystyle i} is sampled, and a new gradient ∇ f i ( x k ) {\displaystyle \nabla f_{i}(x_{k})} is computed. The iterate x k {\displaystyle x_{k}} is updated with: x k + 1 = x k − γ [ ∇ f i ( x k ) − g i + 1 n ∑ i = 1 n g i ] , {\displaystyle x_{k+1}=x_{k}-\gamma \left[\nabla f_{i}(x_{k})-g_{i}+{\frac {1}{n}}\sum _{i=1}^{n}g_{i}\right],} and afterwards table entry i {\displaystyle i} is updated with g i = ∇ f i ( x k ) {\displaystyle g_{i}=\nabla f_{i}(x_{k})} . SAGA is among the most popular of the variance reduction methods due to its simplicity, easily adaptable theory, and excellent performance. It is the successor of the SAG method, improving on its flexibility and performance. === SVRG === The stochastic variance reduced gradient method (SVRG), the prototypical snapshot method, uses a similar update except instead of using the average of a table it instead uses a full-gradient that is reevaluated at a snapshot point x ~ {\displaystyle {\tilde {x}}} at regular intervals of m ≥ n {\displaystyle m\geq n} iterations. The update becomes: x k + 1 = x k − γ [ ∇ f i ( x k ) − ∇ f i ( x ~ ) + ∇ f ( x ~ ) ] , {\displaystyle x_{k+1}=x_{k}-\gamma [\nabla f_{i}(x_{k})-\nabla f_{i}({\tilde {x}})+\nabla f({\tilde {x}})],} This approach requires two stochastic gradient evaluations per step, one to compute ∇ f i ( x k ) {\displaystyle \nabla f_{i}(x_{k})} and one to compute ∇ f i ( x ~ ) , {\displaystyle \nabla f_{i}({\tilde {x}}),} where-as table averaging approaches need only one. Despite the high computational cost, SVRG is popular as its simple convergence theory is highly adaptable to new optimization settings. It also has lower storage requirements than tabular averaging approaches, which make it applicable in many settings where tabular methods can not be used. === SARAH === The SARAH (stochastic recursive gradient) method maintains a recursive estimator of the gradient rather than storing a table of past gradients (as in SAGA) or computing periodic full-gradient snapshots (as in SVRG). At the start of an inner loop, a full gradient is computed at a reference point x ~ {\displaystyle {\tilde {x}}} : v 0 = ∇ f ( x ~ ) {\displaystyle v_{0}=\nabla f({\tilde {x}})} . For inner iterations, with a sampled index i k {\displaystyle i_{k}} , the gradient estimator and iterate are updated by: v k = ∇ f i k ( x k ) − ∇ f i k ( x k − 1 ) + v k − 1 , x k + 1 = x k − γ v k . {\displaystyle v_{k}=\nabla f_{i_{k}}(x_{k})-\nabla f_{i_{k}}(x_{k-1})+v_{k-1},\qquad x_{k+1}=x_{k}-\gamma v_{k}.} This recursion requires two component-gradient evaluations per step ∇ f i k ( x k ) {\displaystyle \nabla f_{i_{k}}(x_{k})} and ∇ f i k ( x k − 1 ) {\displaystyle \nabla f_{i_{k}}(x_{k-1})} but does not need to store per-sample gradients, resulting in lower memory cost than table-averaging methods. SARAH admits linear convergence for strongly convex functions and has been extended to more general nonconvex and composite problems. === SDCA === Exploiting the dual representation of the objective leads to another variance reduction approach that is particularly suited to finite-sums where each term has a structure that makes computing the convex conjugate f i ∗ , {\displaystyle f_{i}^{},} or its proximal operator tractable. The standard SDCA method considers finite sums that have additional structure compared to generic finite sum setting: f ( x ) = 1 n ∑ i = 1 n f i ( x T v i ) + λ 2 ‖ x ‖ 2 , {\displaystyle f(x)={\frac {1}{n}}\sum _{i=1}^{n}f_{i}(x^{T}v_{i})+{\frac {\lambda }{2}}\|x\|^{2},} where each f i {\displaystyle f_{i}} is 1 dimensional and each v i {\displaystyle v_{i}} is a data point associated with f i {\displaystyle f_{i}} . SDCA solves the dual problem: max α ∈ R n − 1 n ∑ i = 1 n f i ∗ ( − α i ) − λ 2 ‖ 1 λ n ∑ i = 1 n α i v i ‖ 2 , {\displaystyle \max _{\alpha \in \mathbb {R} ^{n}}-{\frac {1}{n}}\sum _{i=1}^{n}f_{i}^{}(-\alpha _{i})-{\frac {\lambda }{2}}\left\|{\frac {1}{\lambda n}}\sum _{i=1}^{n}\alpha _{i}v_{i}\right\|^{2},} by a stochastic coordinate ascent procedure, where at each step the objective is optimized with respect to a randomly chosen coordinate α i {\displaystyle \alpha _{i}} , leaving all other coordinates the same. An approximate primal solution x {\displaystyle x} can be recovered from the α {\displaystyle \alpha } values: x = 1 λ n ∑ i = 1 n α i v i {\displaystyle x={\frac {1}{\lambda n}}\sum _{i=1}^{n}\alpha _{i}v_{i}} . This method obtains similar theoretical rates of convergence to other stochastic variance reduced methods, while avoiding the need to specify a step-size parameter. It is fast in practice when λ {\displaystyle \lambda } is large, but significantly slower than the other approaches when λ {\displaystyle \lambda } is small. == Accelerated approaches == Accelerated variance reduction methods are built upon the standard methods above. The earliest approaches make use of proximal operators t

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  Variational autoencoder

  In machine learning, a variational autoencoder (VAE) is an artificial neural network architecture introduced by Diederik P. Kingma and Max Welling in 2013. It is part of the families of probabilistic graphical models and variational Bayesian methods. In addition to being seen as an autoencoder neural network architecture, variational autoencoders can also be studied within the mathematical formulation of variational Bayesian methods, connecting a neural encoder network to its decoder through a probabilistic latent space (for example, as a multivariate Gaussian distribution) that corresponds to the parameters of a variational distribution. Thus, the encoder maps each point (such as an image) from a large complex dataset into a distribution within the latent space, rather than to a single point in that space. The decoder has the opposite function, which is to map from the latent space to the input space, again according to a distribution (although in practice, noise is rarely added during the decoding stage). By mapping a point to a distribution instead of a single point, the network can avoid overfitting the training data. Both networks are typically trained together with the usage of the reparameterization trick, although the variance of the noise model can be learned separately. Although this type of model was initially designed for unsupervised learning, its effectiveness has been proven for semi-supervised learning and supervised learning. == Overview of architecture and operation == A variational autoencoder is a generative model with a prior and noise distribution respectively. Usually such models are trained using the expectation-maximization meta-algorithm (e.g. probabilistic PCA, (spike & slab) sparse coding). Such a scheme optimizes a lower bound of the data likelihood, which is usually computationally intractable, and in doing so requires the discovery of q-distributions, or variational posteriors. These q-distributions are normally parameterized for each individual data point in a separate optimization process. However, variational autoencoders use a neural network as an amortized approach to jointly optimize across data points. In that way, the same parameters are reused for multiple data points, which can result in massive memory savings. The first neural network takes as input the data points themselves, and outputs parameters for the variational distribution. As it maps from a known input space to the low-dimensional latent space, it is called the encoder. The decoder is the second neural network of this model. It is a function that maps from the latent space to the input space, e.g. as the means of the noise distribution. It is possible to use another neural network that maps to the variance, however this can be omitted for simplicity. In such a case, the variance can be optimized with gradient descent. To optimize this model, one needs to know two terms: the "reconstruction error", and the Kullback–Leibler divergence (KL-D). Both terms are derived from the free energy expression of the probabilistic model, and therefore differ depending on the noise distribution and the assumed prior of the data, here referred to as p-distribution. For example, a standard VAE task such as IMAGENET is typically assumed to have a gaussianly distributed noise; however, tasks such as binarized MNIST require a Bernoulli noise. The KL-D from the free energy expression maximizes the probability mass of the q-distribution that overlaps with the p-distribution, which unfortunately can result in mode-seeking behaviour. The "reconstruction" term is the remainder of the free energy expression, and requires a sampling approximation to compute its expectation value. More recent approaches replace Kullback–Leibler divergence (KL-D) with various statistical distances, see "Statistical distance VAE variants" below. == Formulation == From the point of view of probabilistic modeling, one wants to maximize the likelihood of the data x {\displaystyle x} by their chosen parameterized probability distribution p θ ( x ) = p ( x | θ ) {\displaystyle p_{\theta }(x)=p(x|\theta )} . This distribution is usually chosen to be a Gaussian N ( x | μ , σ ) {\displaystyle N(x|\mu ,\sigma )} which is parameterized by μ {\displaystyle \mu } and σ {\displaystyle \sigma } respectively, and as a member of the exponential family it is easy to work with as a noise distribution. Simple distributions are easy enough to maximize, however distributions where a prior is assumed over the latents z {\displaystyle z} results in intractable integrals. Let us find p θ ( x ) {\displaystyle p_{\theta }(x)} via marginalizing over z {\displaystyle z} . p θ ( x ) = ∫ z p θ ( x , z ) d z , {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x,z})\,dz,} where p θ ( x , z ) {\displaystyle p_{\theta }({x,z})} represents the joint distribution under p θ {\displaystyle p_{\theta }} of the observable data x {\displaystyle x} and its latent representation or encoding z {\displaystyle z} . According to the chain rule, the equation can be rewritten as p θ ( x ) = ∫ z p θ ( x | z ) p θ ( z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }({x|z})p_{\theta }(z)\,dz} In the vanilla variational autoencoder, z {\displaystyle z} is usually taken to be a finite-dimensional vector of real numbers, and p θ ( x | z ) {\displaystyle p_{\theta }({x|z})} to be a Gaussian distribution. Then p θ ( x ) {\displaystyle p_{\theta }(x)} is a mixture of Gaussian distributions. It is now possible to define the set of the relationships between the input data and its latent representation as Prior p θ ( z ) {\displaystyle p_{\theta }(z)} Likelihood p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} Posterior p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} Unfortunately, the computation of p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} is expensive and in most cases intractable. To speed up the calculus to make it feasible, it is necessary to introduce a further function to approximate the posterior distribution as q ϕ ( z | x ) ≈ p θ ( z | x ) {\displaystyle q_{\phi }({z|x})\approx p_{\theta }({z|x})} with ϕ {\displaystyle \phi } defined as the set of real values that parametrize q {\displaystyle q} . This is sometimes called amortized inference, since by "investing" in finding a good q ϕ {\displaystyle q_{\phi }} , one can later infer z {\displaystyle z} from x {\displaystyle x} quickly without doing any integrals. In this way, the problem is to find a good probabilistic autoencoder, in which the conditional likelihood distribution p θ ( x | z ) {\displaystyle p_{\theta }(x|z)} is computed by the probabilistic decoder, and the approximated posterior distribution q ϕ ( z | x ) {\displaystyle q_{\phi }(z|x)} is computed by the probabilistic encoder. Parametrize the encoder as E ϕ {\displaystyle E_{\phi }} , and the decoder as D θ {\displaystyle D_{\theta }} . == Evidence lower bound (ELBO) == Like many deep learning approaches that use gradient-based optimization, VAEs require a differentiable loss function to update the network weights through backpropagation. For variational autoencoders, the idea is to jointly optimize the generative model parameters θ {\displaystyle \theta } to reduce the reconstruction error between the input and the output, and ϕ {\displaystyle \phi } to make q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} as close as possible to p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . As reconstruction loss, mean squared error and cross entropy are often used. The Kullback–Leibler divergence D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) {\displaystyle D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))} can be used as a loss function to squeeze q ϕ ( z | x ) {\displaystyle q_{\phi }({z|x})} under p θ ( z | x ) {\displaystyle p_{\theta }(z|x)} . This divergence loss expands to D K L ( q ϕ ( z | x ) ∥ p θ ( z | x ) ) = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( z | x ) ] = E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x ) p θ ( x , z ) ] = ln ⁡ p θ ( x ) + E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ q ϕ ( z | x ) p θ ( x , z ) ] . {\displaystyle {\begin{aligned}D_{KL}(q_{\phi }({z|x})\parallel p_{\theta }({z|x}))&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }(z|x)}{p_{\theta }(z|x)}}\right]\\&=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})p_{\theta }(x)}{p_{\theta }(x,z)}}\right]\\&=\ln p_{\theta }(x)+\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {q_{\phi }({z|x})}{p_{\theta }(x,z)}}\right].\end{aligned}}} Now, define the evidence lower bound (ELBO): L θ , ϕ ( x ) := E z ∼ q ϕ ( ⋅ | x ) [ ln ⁡ p θ ( x , z ) q ϕ ( z | x ) ] = ln ⁡ p θ ( x ) − D K L ( q ϕ ( ⋅ | x ) ∥ p θ ( ⋅ | x ) ) {\displaystyle L_{\theta ,\phi }(x):=\mathbb {E} _{z\sim q_{\phi }(\cdot |x)}\left[\ln {\frac {p_{\theta }(x,z)}{q_{\phi }({z|x})}}\right]=\ln p_{\theta }(x)-D_{KL}(q_{\phi }({\cdot |x})\parallel p_{\theta }({\cdot |x}))} Maximizing the ELBO θ ∗ , ϕ ∗ = argmax θ , ϕ L θ , ϕ ( x ) {\dis

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