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  Right to explanation

  In the regulation of algorithms, particularly artificial intelligence and its subfield of machine learning, a right to [an] explanation is a right to be given an explanation for an output of the algorithm. Such rights primarily refer to individual rights to be given an explanation for decisions that significantly affect an individual, particularly legally or financially. For example, a person who applies for a loan and is denied may ask for an explanation, which could be "Credit bureau X reports that you declared bankruptcy last year; this is the main factor in considering you too likely to default, and thus we will not give you the loan you applied for." Some such legal rights already exist, while the scope of a general "right to explanation" is a matter of ongoing debate. There have been arguments made that a "social right to explanation" is a crucial foundation for an information society, particularly as the institutions of that society will need to use digital technologies, artificial intelligence, machine learning. In other words, that the related automated decision making systems that use explainability would be more trustworthy and transparent. Without this right, which could be constituted both legally and through professional standards, the public will be left without much recourse to challenge the decisions of automated systems. == Examples == === Credit scoring in the United States === Under the Equal Credit Opportunity Act (Regulation B of the Code of Federal Regulations), Title 12, Chapter X, Part 1002, §1002.9, creditors are required to notify applicants who are denied credit with specific reasons for the detail. As detailed in §1002.9(b)(2): (2) Statement of specific reasons. The statement of reasons for adverse action required by paragraph (a)(2)(i) of this section must be specific and indicate the principal reason(s) for the adverse action. Statements that the adverse action was based on the creditor's internal standards or policies or that the applicant, joint applicant, or similar party failed to achieve a qualifying score on the creditor's credit scoring system are insufficient. The official interpretation of this section details what types of statements are acceptable. Creditors comply with this regulation by providing a list of reasons (generally at most 4, per interpretation of regulations), consisting of a numeric reason code (as identifier) and an associated explanation, identifying the main factors affecting a credit score. An example might be: 32: Balances on bankcard or revolving accounts too high compared to credit limits === European Union === The European Union General Data Protection Regulation (GDPR, enacted 2016, taking effect 2018) extends the automated decision-making rights in the 1995 Data Protection Directive to provide a legally disputed form of a right to an explanation, stated as such in Recital 71: "[the data subject should have] the right ... to obtain an explanation of the decision reached". In full: The data subject should have the right not to be subject to a decision, which may include a measure, evaluating personal aspects relating to him or her which is based solely on automated processing and which produces legal effects concerning him or her or similarly significantly affects him or her, such as automatic refusal of an online credit application or e-recruiting practices without any human intervention. ... In any case, such processing should be subject to suitable safeguards, which should include specific information to the data subject and the right to obtain human intervention, to express his or her point of view, to obtain an explanation of the decision reached after such assessment and to challenge the decision. However, the extent to which the regulations themselves provide a "right to explanation" is heavily debated. There are two main strands of criticism. There are significant legal issues with the right as found in Article 22 — as recitals are not binding, and the right to an explanation is not mentioned in the binding articles of the text, having been removed during the legislative process. In addition, there are significant restrictions on the types of automated decisions that are covered — which must be both "solely" based on automated processing, and have legal or similarly significant effects — which significantly limits the range of automated systems and decisions to which the right would apply. In particular, the right is unlikely to apply in many of the cases of algorithmic controversy that have been picked up in the media. The UK has also recently amended its implementation of Article 22. A second potential source of such a right has been pointed to in Article 15, the "right of access by the data subject". This restates a similar provision from the 1995 Data Protection Directive, allowing the data subject access to "meaningful information about the logic involved" in the same significant, solely automated decision-making, found in Article 22. Yet this too suffers from alleged challenges that relate to the timing of when this right can be drawn upon, as well as practical challenges that mean it may not be binding in many cases of public concern. Other EU legislative instruments contain explanation rights. The European Union's Artificial Intelligence Act provides in Article 86 a "[r]ight to explanation of individual decision-making" of certain high risk systems which produce significant, adverse effects to an individual's health, safety or fundamental rights. The right provides for "clear and meaningful explanations of the role of the AI system in the decision-making procedure and the main elements of the decision taken", although only applies to the extent other law does not provide such a right. The Digital Services Act in Article 27, and the Platform to Business Regulation in Article 5, both contain rights to have the main parameters of certain recommender systems to be made clear, although these provisions have been criticised as not matching the way that such systems work. The Platform Work Directive, which provides for regulation of automation in gig economy work as an extension of data protection law, further contains explanation provisions in Article 11, using the specific language of "explanation" in a binding article rather than a recital as is the case in the GDPR. Scholars note that remains uncertainty as to whether these provisions imply sufficiently tailored explanation in practice which will need to be resolved by courts. === France === In France the 2016 Loi pour une République numérique (Digital Republic Act or loi numérique) amends the country's administrative code to introduce a new provision for the explanation of decisions made by public sector bodies about individuals. It notes that where there is "a decision taken on the basis of an algorithmic treatment", the rules that define that treatment and its "principal characteristics" must be communicated to the citizen upon request, where there is not an exclusion (e.g. for national security or defence). These should include the following: the degree and the mode of contribution of the algorithmic processing to the decision- making; the data processed and its source; the treatment parameters, and where appropriate, their weighting, applied to the situation of the person concerned; the operations carried out by the treatment. Scholars have noted that this right, while limited to administrative decisions, goes beyond the GDPR right to explicitly apply to decision support rather than decisions "solely" based on automated processing, as well as provides a framework for explaining specific decisions. Indeed, the GDPR automated decision-making rights in the European Union, one of the places a "right to an explanation" has been sought within, find their origins in French law in the late 1970s. == Criticism == Some argue that a "right to explanation" is at best unnecessary, at worst harmful, and threatens to stifle innovation. Specific criticisms include: favoring human decisions over machine decisions, being redundant with existing laws, and focusing on process over outcome. Authors of study "Slave to the Algorithm? Why a 'Right to an Explanation' Is Probably Not the Remedy You Are Looking For" Lilian Edwards and Michael Veale argue that a right to explanation is not the solution to harms caused to stakeholders by algorithmic decisions. They also state that the right of explanation in the GDPR is narrowly defined, and is not compatible with how modern machine learning technologies are being developed. With these limitations, defining transparency within the context of algorithmic accountability remains a problem. For example, providing the source code of algorithms may not be sufficient and may create other problems in terms of privacy disclosures and the gaming of technical systems. To mitigate this issue, Edwards and Veale argue that an auditing system could be more effective, to allow auditors to loo

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  Markov model

  In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property. == Introduction == Andrey Andreyevich Markov (14 June 1856 – 20 July 1922) was a Russian mathematician best known for his work on stochastic processes. A primary subject of his research later became known as the Markov chain. There are four common Markov models used in different situations, depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made: == Markov chain == The simplest Markov model is the Markov chain. It models the state of a system with a random variable that changes through time. In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution. == Hidden Markov model == A hidden Markov model is a Markov chain for which the state is only partially observable or noisily observable. In other words, observations are related to the state of the system, but they are typically insufficient to precisely determine the state. Several well-known algorithms for hidden Markov models exist. For example, given a sequence of observations, the Viterbi algorithm will compute the most-likely corresponding sequence of states, the forward algorithm will compute the probability of the sequence of observations, and the Baum–Welch algorithm will estimate the starting probabilities, the transition function, and the observation function of a hidden Markov model. One common use is for speech recognition, where the observed data is the speech audio waveform and the hidden state is the spoken text. In this example, the Viterbi algorithm finds the most likely sequence of spoken words given the speech audio. == Markov decision process == A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards. == Partially observable Markov decision process == A partially observable Markov decision process (POMDP) is a Markov decision process in which the state of the system is only partially observed. POMDPs are known to be NP complete, but recent approximation techniques have made them useful for a variety of applications, such as controlling simple agents or robots. == Markov random field == A Markov random field, or Markov network, may be considered to be a generalization of a Markov chain in multiple dimensions. In a Markov chain, state depends only on the previous state in time, whereas in a Markov random field, each state depends on its neighbors in any of multiple directions. A Markov random field may be visualized as a field or graph of random variables, where the distribution of each random variable depends on the neighboring variables with which it is connected. More specifically, the joint distribution for any random variable in the graph can be computed as the product of the "clique potentials" of all the cliques in the graph that contain that random variable. Modeling a problem as a Markov random field is useful because it implies that the joint distributions at each vertex in the graph may be computed in this manner. == Hierarchical Markov models == Hierarchical Markov models can be applied to categorize human behavior at various levels of abstraction. For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model and the Abstract Hidden Markov Model. Both have been used for behavior recognition and certain conditional independence properties between different levels of abstraction in the model allow for faster learning and inference. == Tolerant Markov model == A Tolerant Markov model (TMM) is a probabilistic-algorithmic Markov chain model. It assigns the probabilities according to a conditioning context that considers the last symbol, from the sequence to occur, as the most probable instead of the true occurring symbol. A TMM can model three different natures: substitutions, additions or deletions. Successful applications have been efficiently implemented in DNA sequences compression. == Markov-chain forecasting models == Markov-chains have been used as a forecasting methods for several topics, for example price trends, wind power and solar irradiance. The Markov-chain forecasting models utilize a variety of different settings, from discretizing the time-series to hidden Markov-models combined with wavelets and the Markov-chain mixture distribution model (MCM).

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  Principal component analysis

  Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data are linearly transformed onto a new coordinate system such that the directions (principal components) capturing the largest variation in the data can be easily identified. The principal components of a collection of points in a real coordinate space are a sequence of p {\displaystyle p} unit vectors, where the i {\displaystyle i} -th vector is the direction of a line that best fits the data while being orthogonal to the first i − 1 {\displaystyle i-1} vectors. Here, a best-fitting line is defined as one that minimizes the average squared perpendicular distance from the points to the line. These directions (i.e., principal components) constitute an orthonormal basis in which different individual dimensions of the data are linearly uncorrelated. Many studies use the first two principal components in order to plot the data in two dimensions and to visually identify clusters of closely related data points. Principal component analysis has applications in many fields such as population genetics, microbiome studies, and atmospheric science. == Overview == When performing PCA, the first principal component of a set of p {\displaystyle p} variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the first component is removed, and we may proceed through p {\displaystyle p} iterations until all the variance is explained. PCA is most commonly used when many of the variables are highly correlated with each other and it is desirable to reduce their number to an independent set. The first principal component can equivalently be defined as a direction that maximizes the variance of the projected data. The i {\displaystyle i} -th principal component can be taken as a direction orthogonal to the first i − 1 {\displaystyle i-1} principal components that maximizes the variance of the projected data. For either objective, it can be shown that the principal components are eigenvectors of the data's covariance matrix. Thus, the principal components are often computed by eigendecomposition of the data covariance matrix or singular value decomposition of the data matrix. PCA is the simplest of the true eigenvector-based multivariate analyses and is closely related to factor analysis. Factor analysis typically incorporates more domain-specific assumptions about the underlying structure and solves eigenvectors of a slightly different matrix. PCA is also related to canonical correlation analysis (CCA). CCA defines coordinate systems that optimally describe the cross-covariance between two datasets while PCA defines a new orthogonal coordinate system that optimally describes variance in a single dataset. Robust and L1-norm-based variants of standard PCA have also been proposed. == History == PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in signal processing, the Hotelling transform in multivariate quality control, proper orthogonal decomposition (POD) in mechanical engineering, singular value decomposition (SVD) of X (invented in the last quarter of the 19th century), eigenvalue decomposition (EVD) of XTX in linear algebra, factor analysis (for a discussion of the differences between PCA and factor analysis see Ch. 7 of Jolliffe's Principal Component Analysis), Eckart–Young theorem (Harman, 1960), or empirical orthogonal functions (EOF) in meteorological science (Lorenz, 1956), empirical eigenfunction decomposition (Sirovich, 1987), quasiharmonic modes (Brooks et al., 1988), spectral decomposition in noise and vibration, and empirical modal analysis in structural dynamics. == Intuition == PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the values of each variable in the dataset on 0 by subtracting the mean of the variable's observed values from each of those values. These transformed values are used instead of the original observed values for each of the variables. Then, we compute the covariance matrix of the data and calculate the eigenvalues and corresponding eigenvectors of this covariance matrix. Then we must normalize each of the orthogonal eigenvectors to turn them into unit vectors. Once this is done, each of the mutually-orthogonal unit eigenvectors can be interpreted as an axis of the ellipsoid fitted to the data. This choice of basis will transform the covariance matrix into a diagonalized form, in which the diagonal elements represent the variance of each axis. The proportion of the variance that each eigenvector represents can be calculated by dividing the eigenvalue corresponding to that eigenvector by the sum of all eigenvalues. Biplots and scree plots (degree of explained variance) are used to interpret findings of the PCA. == Details == PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the second greatest variance on the second coordinate, and so on. Consider an n × p {\displaystyle n\times p} data matrix, X, with column-wise zero empirical mean (the sample mean of each column has been shifted to zero), where each of the n rows represents a different repetition of the experiment, and each of the p columns gives a particular kind of feature (say, the results from a particular sensor). Mathematically, the transformation is defined by a set of size l {\displaystyle l} (where l {\displaystyle l} is usually selected to be strictly less than p {\displaystyle p} to reduce dimensionality) of p {\displaystyle p} -dimensional vectors of weights or coefficients w ( k ) = ( w 1 , … , w p ) ( k ) {\displaystyle \mathbf {w} _{(k)}=(w_{1},\dots ,w_{p})_{(k)}} that map each row vector x ( i ) = ( x 1 , … , x p ) ( i ) {\displaystyle \mathbf {x} _{(i)}=(x_{1},\dots ,x_{p})_{(i)}} of X to a new vector of principal component scores t ( i ) = ( t 1 , … , t l ) ( i ) {\displaystyle \mathbf {t} _{(i)}=(t_{1},\dots ,t_{l})_{(i)}} , given by t k ( i ) = x ( i ) ⋅ w ( k ) f o r i = 1 , … , n k = 1 , … , l {\displaystyle {t_{k}}_{(i)}=\mathbf {x} _{(i)}\cdot \mathbf {w} _{(k)}\qquad \mathrm {for} \qquad i=1,\dots ,n\qquad k=1,\dots ,l} in such a way that the individual variables t 1 , … , t l {\displaystyle t_{1},\dots ,t_{l}} of t considered over the data set successively inherit the maximum possible variance from X, with each coefficient vector w constrained to be a unit vector. The above may equivalently be written in matrix form as T = X W {\displaystyle \mathbf {T} =\mathbf {X} \mathbf {W} } where T i k = t k ( i ) {\displaystyle {\mathbf {T} }_{ik}={t_{k}}_{(i)}} , X i j = x j ( i ) {\displaystyle {\mathbf {X} }_{ij}={x_{j}}_{(i)}} , and W j k = w j ( k ) {\displaystyle {\mathbf {W} }_{jk}={w_{j}}_{(k)}} . === First component === In order to maximize variance, the first weight vector w(1) thus has to satisfy w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( t 1 ) ( i ) 2 } = arg ⁡ max ‖ w ‖ = 1 { ∑ i ( x ( i ) ⋅ w ) 2 } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}(t_{1})_{(i)}^{2}\right\}=\arg \max _{\Vert \mathbf {w} \Vert =1}\,\left\{\sum _{i}\left(\mathbf {x} _{(i)}\cdot \mathbf {w} \right)^{2}\right\}} Equivalently, writing this in matrix form gives w ( 1 ) = arg ⁡ max ‖ w ‖ = 1 { ‖ X w ‖ 2 } = arg ⁡ max ‖ w ‖ = 1 { w T X T X w } {\displaystyle \mathbf {w} _{(1)}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\left\|\mathbf {Xw} \right\|^{2}\right\}=\arg \max _{\left\|\mathbf {w} \right\|=1}\left\{\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} \right\}} Since w(1) has been defined to be a unit vector, it equivalently also satisfies w ( 1 ) = arg ⁡ max { w T X T X w w T w } {\displaystyle \mathbf {w} _{(1)}=\arg \max \left\{{\frac {\mathbf {w} ^{\mathsf {T}}\mathbf {X} ^{\mathsf {T}}\mathbf {Xw} }{\mathbf {w} ^{\mathsf {T}}\mathbf {w} }}\right\}} The quantity to be maximised can be recognised as a Rayleigh quotient. A standard result for a positive semidefinite matrix such as XTX is that the quotient's maximum possible value is the largest eigenvalue of the matrix, which occurs when w is the corresponding eigenvector. With w(1) found, the first principal component of a data vector

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  Alternating decision tree

  An alternating decision tree (ADTree) is a machine learning method for classification. It generalizes decision trees and has connections to boosting. An ADTree consists of an alternation of decision nodes, which specify a predicate condition, and prediction nodes, which contain a single number. An instance is classified by an ADTree by following all paths for which all decision nodes are true, and summing any prediction nodes that are traversed. == History == ADTrees were introduced by Yoav Freund and Llew Mason. However, the algorithm as presented had several typographical errors. Clarifications and optimizations were later presented by Bernhard Pfahringer, Geoffrey Holmes and Richard Kirkby. Implementations are available in Weka and JBoost. == Motivation == Original boosting algorithms typically used either decision stumps or decision trees as weak hypotheses. As an example, boosting decision stumps creates a set of T {\displaystyle T} weighted decision stumps (where T {\displaystyle T} is the number of boosting iterations), which then vote on the final classification according to their weights. Individual decision stumps are weighted according to their ability to classify the data. Boosting a simple learner results in an unstructured set of T {\displaystyle T} hypotheses, making it difficult to infer correlations between attributes. Alternating decision trees introduce structure to the set of hypotheses by requiring that they build off a hypothesis that was produced in an earlier iteration. The resulting set of hypotheses can be visualized in a tree based on the relationship between a hypothesis and its "parent." Another important feature of boosted algorithms is that the data is given a different distribution at each iteration. Instances that are misclassified are given a larger weight while accurately classified instances are given reduced weight. == Alternating decision tree structure == An alternating decision tree consists of decision nodes and prediction nodes. Decision nodes specify a predicate condition. Prediction nodes contain a single number. ADTrees always have prediction nodes as both root and leaves. An instance is classified by an ADTree by following all paths for which all decision nodes are true and summing any prediction nodes that are traversed. This is different from binary classification trees such as CART (Classification and regression tree) or C4.5 in which an instance follows only one path through the tree. === Example === The following tree was constructed using JBoost on the spambase dataset (available from the UCI Machine Learning Repository). In this example, spam is coded as 1 and regular email is coded as −1. The following table contains part of the information for a single instance. The instance is scored by summing all of the prediction nodes through which it passes. In the case of the instance above, the score is calculated as The final score of 0.657 is positive, so the instance is classified as spam. The magnitude of the value is a measure of confidence in the prediction. The original authors list three potential levels of interpretation for the set of attributes identified by an ADTree: Individual nodes can be evaluated for their own predictive ability. Sets of nodes on the same path may be interpreted as having a joint effect The tree can be interpreted as a whole. Care must be taken when interpreting individual nodes as the scores reflect a re weighting of the data in each iteration. == Description of the algorithm == The inputs to the alternating decision tree algorithm are: A set of inputs ( x 1 , y 1 ) , … , ( x m , y m ) {\displaystyle (x_{1},y_{1}),\ldots ,(x_{m},y_{m})} where x i {\displaystyle x_{i}} is a vector of attributes and y i {\displaystyle y_{i}} is either -1 or 1. Inputs are also called instances. A set of weights w i {\displaystyle w_{i}} corresponding to each instance. The fundamental element of the ADTree algorithm is the rule. A single rule consists of a precondition, a condition, and two scores. A condition is a predicate of the form "attribute value." A precondition is simply a logical conjunction of conditions. Evaluation of a rule involves a pair of nested if statements: 1 if (precondition) 2 if (condition) 3 return score_one 4 else 5 return score_two 6 end if 7 else 8 return 0 9 end if Several auxiliary functions are also required by the algorithm: W + ( c ) {\displaystyle W_{+}(c)} returns the sum of the weights of all positively labeled examples that satisfy predicate c {\displaystyle c} W − ( c ) {\displaystyle W_{-}(c)} returns the sum of the weights of all negatively labeled examples that satisfy predicate c {\displaystyle c} W ( c ) = W + ( c ) + W − ( c ) {\displaystyle W(c)=W_{+}(c)+W_{-}(c)} returns the sum of the weights of all examples that satisfy predicate c {\displaystyle c} The algorithm is as follows: 1 function ad_tree 2 input Set of m training instances 3 4 wi = 1/m for all i 5 a = 1 2 ln W + ( t r u e ) W − ( t r u e ) {\displaystyle a={\frac {1}{2}}{\textrm {ln}}{\frac {W_{+}(true)}{W_{-}(true)}}} 6 R0 = a rule with scores a and 0, precondition "true" and condition "true." 7 P = { t r u e } {\displaystyle {\mathcal {P}}=\{true\}} 8 C = {\displaystyle {\mathcal {C}}=} the set of all possible conditions 9 for j = 1 … T {\displaystyle j=1\dots T} 10 p ∈ P , c ∈ C {\displaystyle p\in {\mathcal {P}},c\in {\mathcal {C}}} get values that minimize z = 2 ( W + ( p ∧ c ) W − ( p ∧ c ) + W + ( p ∧ ¬ c ) W − ( p ∧ ¬ c ) ) + W ( ¬ p ) {\displaystyle z=2\left({\sqrt {W_{+}(p\wedge c)W_{-}(p\wedge c)}}+{\sqrt {W_{+}(p\wedge \neg c)W_{-}(p\wedge \neg c)}}\right)+W(\neg p)} 11 P + = p ∧ c + p ∧ ¬ c {\displaystyle {\mathcal {P}}+=p\wedge c+p\wedge \neg c} 12 a 1 = 1 2 ln W + ( p ∧ c ) + 1 W − ( p ∧ c ) + 1 {\displaystyle a_{1}={\frac {1}{2}}{\textrm {ln}}{\frac {W_{+}(p\wedge c)+1}{W_{-}(p\wedge c)+1}}} 13 a 2 = 1 2 ln W + ( p ∧ ¬ c ) + 1 W − ( p ∧ ¬ c ) + 1 {\displaystyle a_{2}={\frac {1}{2}}{\textrm {ln}}{\frac {W_{+}(p\wedge \neg c)+1}{W_{-}(p\wedge \neg c)+1}}} 14 Rj = new rule with precondition p, condition c, and weights a1 and a2 15 w i = w i e − y i R j ( x i ) {\displaystyle w_{i}=w_{i}e^{-y_{i}R_{j}(x_{i})}} 16 end for 17 return set of Rj The set P {\displaystyle {\mathcal {P}}} grows by two preconditions in each iteration, and it is possible to derive the tree structure of a set of rules by making note of the precondition that is used in each successive rule. == Empirical results == Figure 6 in the original paper demonstrates that ADTrees are typically as robust as boosted decision trees and boosted decision stumps. Typically, equivalent accuracy can be achieved with a much simpler tree structure than recursive partitioning algorithms.

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  Circular thresholding

  Circular thresholding is an algorithm for automatic image threshold selection in image processing. Most threshold selection algorithms assume that the values (e.g. intensities) lie on a linear scale. However, some quantities such as hue and orientation are a circular quantity, and therefore require circular thresholding algorithms. The example shows that the standard linear version of Otsu's method when applied to the hue channel of an image of blood cells fails to correctly segment the large white blood cells (leukocytes). In contrast the white blood cells are correctly segmented by the circular version of Otsu's method. == Methods == There are a relatively small number of circular image threshold selection algorithms. The following examples are all based on Otsu's method for linear histograms: (Tseng, Li and Tung 1995) smooth the circular histogram, and apply Otsu's method. The histogram is cyclically rotated so that the selected threshold is shifted to zero. Otsu's method and histogram rotation are applied iteratively until several heuristics involving class size, threshold location, and class variance are satisfied. (Wu et al. 2006) smooth the circular histogram until it contains only two peaks. The histogram is cyclically rotated so that the midpoint between the peaks is shifted to zero. Otsu's method and histogram rotation are applied iteratively until convergence of the threshold. (Lai and Rosin 2014) applied Otsu's method to the circular histogram. For the two class circular thresholding task they showed that, for a histogram with an even number of bins, the optimal solution for Otsu's criterion of within-class variance is obtained when the histogram is split into two halves. Therefore the optimal solution can be efficiently obtained in linear rather than quadratic time. == References and further reading == D.-C. Tseng, Y.-F. Li, and C.-T. Tung, Circular histogram thresholding for color image segmentation in Proc. Int. Conf. Document Anal. Recognit., 1995, pp. 673–676. J. Wu, P. Zeng, Y. Zhou, and C. Olivier, A novel color image segmentation method and its application to white blood cell image analysis in Proc. Int. Conf. Signal Process., vol. 2. 2006, pp. 16–20. Y.K. Lai, P.L. Rosin, Efficient Circular Thresholding, IEEE Trans. on Image Processing 23(3), 992–1001 (2014). doi:10.1109/TIP.2013.2297014

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  Training, validation, and test data sets

  In machine learning, a common task is the study and construction of algorithms that can learn from and make predictions on data. Such algorithms function by making data-driven predictions or decisions, through building a mathematical model from input data. These input data used to build the model are usually divided into multiple data sets. In particular, three data sets are commonly used in different stages of the creation of the model: training, validation, and testing sets. The model is initially fit on a training data set, which is a set of examples used to fit the parameters (e.g. weights of connections between neurons in artificial neural networks) of the model. The model (e.g. a naive Bayes classifier) is trained on the training data set using a supervised learning method, for example using optimization methods such as gradient descent or stochastic gradient descent. In practice, the training data set often consists of pairs of an input vector (or scalar) and the corresponding output vector (or scalar), where the answer key is commonly denoted as the target (or label). The current model is run with the training data set and produces a result, which is then compared with the target, for each input vector in the training data set. Based on the result of the comparison and the specific learning algorithm being used, the parameters of the model are adjusted. The model fitting can include both variable selection and parameter estimation. Successively, the fitted model is used to predict the responses for the observations in a second data set called the validation data set. The validation data set provides an unbiased evaluation of a model fit on the training data set while tuning the model's hyperparameters (e.g. the number of hidden units—layers and layer widths—in a neural network). Validation data sets can be used for regularization by early stopping (stopping training when the error on the validation data set increases, as this is a sign of over-fitting to the training data set). This simple procedure is complicated in practice by the fact that the validation data set's error may fluctuate during training, producing multiple local minima. This complication has led to the creation of many ad-hoc rules for deciding when over-fitting has truly begun. Finally, the test data set is a data set used to provide an unbiased evaluation of a model fit on the training data set. When the data in the test data set has never been used (for example in cross-validation), the test data set is called a holdout data set. The term "validation set" is sometimes used instead of "test set" in some literature (e.g., if the original data set was partitioned into only two subsets, the test set might be referred to as the validation set). Deciding the sizes and strategies for data set division in training, test and validation sets is very dependent on the problem and data available. == Training data set == A training data set is a data set of examples used during the learning process and is used to fit the parameters (e.g., weights) of, for example, a classifier. For classification tasks, a supervised learning algorithm looks at the training data set to determine, or learn, the optimal combinations of variables that will generate a good predictive model. The goal is to produce a trained (fitted) model that generalizes well to new, unknown data. The fitted model is evaluated using “new” examples from the held-out data sets (validation and test data sets) to estimate the model’s accuracy in classifying new data. To reduce the risk of issues such as over-fitting, the examples in the validation and test data sets should not be used to train the model. Most approaches that search through training data for empirical relationships tend to overfit the data, meaning that they can identify and exploit apparent relationships in the training data that do not hold in general. When a training set is continuously expanded with new data, then this is incremental learning. == Validation data set == A validation data set is a data set of examples used to tune the hyperparameters (i.e. the architecture) of a model. It is sometimes also called the development set or the "dev set". An example of a hyperparameter for artificial neural networks includes the number of hidden units in each layer. It, as well as the testing set (as mentioned below), should follow the same probability distribution as the training data set. In order to avoid overfitting, when any classification parameter needs to be adjusted, it is necessary to have a validation data set in addition to the training and test data sets. For example, if the most suitable classifier for the problem is sought, the training data set is used to train the different candidate classifiers, the validation data set is used to compare their performances and decide which one to take and, finally, the test data set is used to obtain the performance characteristics such as accuracy, sensitivity, specificity, F-measure, and so on. The validation data set functions as a hybrid: it is training data used for testing, but neither as part of the low-level training nor as part of the final testing. The basic process of using a validation data set for model selection (as part of training data set, validation data set, and test data set) is: Since our goal is to find the network having the best performance on new data, the simplest approach to the comparison of different networks is to evaluate the error function using data which is independent of that used for training. Various networks are trained by minimization of an appropriate error function defined with respect to a training data set. The performance of the networks is then compared by evaluating the error function using an independent validation set, and the network having the smallest error with respect to the validation set is selected. This approach is called the hold out method. Since this procedure can itself lead to some overfitting to the validation set, the performance of the selected network should be confirmed by measuring its performance on a third independent set of data called a test set. An application of this process is in early stopping, where the candidate models are successive iterations of the same network, and training stops when the error on the validation set grows, choosing the previous model (the one with minimum error). == Test data set == A test data set is a data set that is independent of the training data set, but that follows the same probability distribution as the training data set. A test set is therefore a set of examples used only to assess the performance (i.e. generalization) of a specified classifier on unseen data. To do this, the model is used to predict classifications of examples in the test set. Those predictions are compared to the examples' true classifications to assess the model's accuracy. If a model fit to the training and validation data set also fits the test data set well, minimal overfitting has taken place (see figure below). A better fitting of the training or validation data sets as opposed to the test data set usually points to overfitting. In the scenario where a data set has a low number of samples, it is usually partitioned into a training set and a validation data set, where the model is trained on the training set and refined using the validation set to improve accuracy, but this approach will lead to overfitting. The holdout method can also be employed, where the test set is used at the end, after training on the training set. Other techniques, such as cross-validation and bootstrapping, are used on small data sets. The bootstrap method generates numerous simulated data sets of the same size by randomly sampling with replacement from the original data, allowing the random data points to serve as test sets for evaluating model performance. Cross-validation splits the data set into multiple folds, with a single sub-fold used as test data; the model is trained on the remaining folds, and all folds are cross-validated (with results averaged and models consolidated) to estimate final model performance. Note that some sources advise against using a single split, as it can lead to overfitting as well as biased model performance estimates. For this reason, data sets are split into three partitions: training, validation and test data sets. The standard machine learning practice is to train on the training set and tune hyperparameters using the validation set, where the validation process selects the model with the lowest validation loss, which is then tested on the test data set (normally held out) to assess the final model. The holdout method for the test set reduces computation by avoiding using the test set after each epoch. The test data set should never be used for validating the training model or fine-tuning hyperparameters, as it provides an accurate and honest evaluation of the model's final performance on unseen dat

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  Diffusion model

  In machine learning, diffusion models, also known as diffusion-based generative models or score-based generative models, are a class of latent variable generative models. A diffusion model consists of two major components: the forward diffusion process, and the reverse sampling process. The goal of diffusion models is to learn a diffusion process for a given dataset, such that the process can generate new elements that are distributed similarly as the original dataset. A diffusion model models data as generated by a diffusion process, whereby a new datum performs a random walk with drift through the space of all possible data. A trained diffusion model can be sampled in many ways, with different efficiency and quality. There are various equivalent formalisms, including Markov chains, denoising diffusion probabilistic models, noise conditioned score networks, and stochastic differential equations. They are typically trained using variational inference. The model responsible for denoising is typically called its "backbone". The backbone may be of any kind, but they are typically U-nets or transformers. As of 2024, diffusion models are mainly used for computer vision tasks, including image denoising, inpainting, super-resolution, image generation, and video generation. These typically involve training a neural network to sequentially denoise images blurred with Gaussian noise. The model is trained to reverse the process of adding noise to an image. After training to convergence, it can be used for image generation by starting with an image composed of random noise, and applying the network iteratively to denoise the image. Diffusion-based image generators have seen widespread commercial interest, such as Stable Diffusion and DALL-E. These models typically combine diffusion models with other models, such as text-encoders and cross-attention modules to allow text-conditioned generation. Other than computer vision, diffusion models have also found applications in natural language processing such as text generation and summarization, sound generation, and reinforcement learning. == Denoising diffusion model == === Non-equilibrium thermodynamics === Diffusion models were introduced in 2015 as a method to train a model that can sample from a highly complex probability distribution. They used techniques from non-equilibrium thermodynamics, especially diffusion. Consider, for example, how one might model the distribution of all naturally occurring photos. Each image is a point in the space of all images, and the distribution of naturally occurring photos is a "cloud" in space, which, by repeatedly adding noise to the images, diffuses out to the rest of the image space, until the cloud becomes all but indistinguishable from a Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . A model that can approximately undo the diffusion can then be used to sample from the original distribution. This is studied in "non-equilibrium" thermodynamics, as the starting distribution is not in equilibrium, unlike the final distribution. The equilibrium distribution is the Gaussian distribution N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with pdf ρ ( x ) ∝ e − 1 2 ‖ x ‖ 2 {\displaystyle \rho (x)\propto e^{-{\frac {1}{2}}\|x\|^{2}}} . This is just the Maxwell–Boltzmann distribution of particles in a potential well V ( x ) = 1 2 ‖ x ‖ 2 {\displaystyle V(x)={\frac {1}{2}}\|x\|^{2}} at temperature 1. The initial distribution, being very much out of equilibrium, would diffuse towards the equilibrium distribution, making biased random steps that are a sum of pure randomness (like a Brownian walker) and gradient descent down the potential well. The randomness is necessary: if the particles were to undergo only gradient descent, then they will all fall to the origin, collapsing the distribution. === Denoising Diffusion Probabilistic Model (DDPM) === The 2020 paper proposed the Denoising Diffusion Probabilistic Model (DDPM), which improves upon the previous method by variational inference. ==== Forward diffusion ==== To present the model, some notation is required. β 1 , . . . , β T ∈ ( 0 , 1 ) {\displaystyle \beta _{1},...,\beta _{T}\in (0,1)} are fixed constants. α t := 1 − β t {\displaystyle \alpha _{t}:=1-\beta _{t}} α ¯ t := α 1 ⋯ α t {\displaystyle {\bar {\alpha }}_{t}:=\alpha _{1}\cdots \alpha _{t}} σ t := 1 − α ¯ t {\displaystyle \sigma _{t}:={\sqrt {1-{\bar {\alpha }}_{t}}}} σ ~ t := σ t − 1 σ t β t {\displaystyle {\tilde {\sigma }}_{t}:={\frac {\sigma _{t-1}}{\sigma _{t}}}{\sqrt {\beta _{t}}}} μ ~ t ( x t , x 0 ) := α t ( 1 − α ¯ t − 1 ) x t + α ¯ t − 1 ( 1 − α t ) x 0 σ t 2 {\displaystyle {\tilde {\mu }}_{t}(x_{t},x_{0}):={\frac {{\sqrt {\alpha _{t}}}(1-{\bar {\alpha }}_{t-1})x_{t}+{\sqrt {{\bar {\alpha }}_{t-1}}}(1-\alpha _{t})x_{0}}{\sigma _{t}^{2}}}} N ( μ , Σ ) {\displaystyle {\mathcal {N}}(\mu ,\Sigma )} is the normal distribution with mean μ {\displaystyle \mu } and variance Σ {\displaystyle \Sigma } , and N ( x | μ , Σ ) {\displaystyle {\mathcal {N}}(x|\mu ,\Sigma )} is the probability density at x {\displaystyle x} . A vertical bar denotes conditioning. A forward diffusion process starts at some starting point x 0 ∼ q {\displaystyle x_{0}\sim q} , where q {\displaystyle q} is the probability distribution to be learned, then repeatedly adds noise to it by x t = 1 − β t x t − 1 + β t z t {\displaystyle x_{t}={\sqrt {1-\beta _{t}}}x_{t-1}+{\sqrt {\beta _{t}}}z_{t}} where z 1 , . . . , z T {\displaystyle z_{1},...,z_{T}} are IID (Independent and identically distributed random variables) samples from N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The coefficients 1 − β t {\displaystyle {\sqrt {1-\beta _{t}}}} and β t {\displaystyle {\sqrt {\beta _{t}}}} ensure that Var ( X t ) = I {\displaystyle {\mbox{Var}}(X_{t})=I} assuming that Var ( X 0 ) = I {\displaystyle {\mbox{Var}}(X_{0})=I} . The values of β t {\displaystyle \beta _{t}} are chosen such that for any starting distribution of x 0 {\displaystyle x_{0}} , if it has finite second moment, then lim t → ∞ x t | x 0 {\displaystyle \lim _{t\to \infty }x_{t}|x_{0}} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . The entire diffusion process then satisfies q ( x 0 : T ) = q ( x 0 ) q ( x 1 | x 0 ) ⋯ q ( x T | x T − 1 ) = q ( x 0 ) N ( x 1 | α 1 x 0 , β 1 I ) ⋯ N ( x T | α T x T − 1 , β T I ) {\displaystyle q(x_{0:T})=q(x_{0})q(x_{1}|x_{0})\cdots q(x_{T}|x_{T-1})=q(x_{0}){\mathcal {N}}(x_{1}|{\sqrt {\alpha _{1}}}x_{0},\beta _{1}I)\cdots {\mathcal {N}}(x_{T}|{\sqrt {\alpha _{T}}}x_{T-1},\beta _{T}I)} or ln ⁡ q ( x 0 : T ) = ln ⁡ q ( x 0 ) − ∑ t = 1 T 1 2 β t ‖ x t − 1 − β t x t − 1 ‖ 2 + C {\displaystyle \ln q(x_{0:T})=\ln q(x_{0})-\sum _{t=1}^{T}{\frac {1}{2\beta _{t}}}\|x_{t}-{\sqrt {1-\beta _{t}}}x_{t-1}\|^{2}+C} where C {\displaystyle C} is a normalization constant and often omitted. In particular, we note that x 1 : T | x 0 {\displaystyle x_{1:T}|x_{0}} is a Gaussian process, which affords us considerable freedom in reparameterization. For example, by standard manipulation with Gaussian process, x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} x t − 1 | x t , x 0 ∼ N ( μ ~ t ( x t , x 0 ) , σ ~ t 2 I ) {\displaystyle x_{t-1}|x_{t},x_{0}\sim {\mathcal {N}}({\tilde {\mu }}_{t}(x_{t},x_{0}),{\tilde {\sigma }}_{t}^{2}I)} In particular, notice that for large t {\displaystyle t} , the variable x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} converges to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} . That is, after a long enough diffusion process, we end up with some x T {\displaystyle x_{T}} that is very close to N ( 0 , I ) {\displaystyle {\mathcal {N}}(0,I)} , with all traces of the original x 0 ∼ q {\displaystyle x_{0}\sim q} gone. For example, since x t | x 0 ∼ N ( α ¯ t x 0 , σ t 2 I ) {\displaystyle x_{t}|x_{0}\sim N\left({\sqrt {{\bar {\alpha }}_{t}}}x_{0},\sigma _{t}^{2}I\right)} we can sample x t | x 0 {\displaystyle x_{t}|x_{0}} directly "in one step", instead of going through all the intermediate steps x 1 , x 2 , . . . , x t − 1 {\displaystyle x_{1},x_{2},...,x_{t-1}} . ==== Backward diffusion ==== The key idea of DDPM is to use a neural network parametrized by θ {\displaystyle \theta } . The network takes in two arguments x t , t {\displaystyle x_{t},t} , and outputs a vector μ θ ( x t , t ) {\displaystyle \mu _{\theta }(x_{t},t)} and a matrix Σ θ ( x t , t ) {\displaystyle \Sigma _{\theta }(x_{t},t)} , such that each step in the forward diffusion process can be approximately undone by x t − 1 ∼ N ( μ θ ( x t , t ) , Σ θ ( x t , t ) ) {\displaystyle x_{t-1}\sim {\mathcal {N}}(\mu _{\theta }(x_{t},t),\Sigma _{\theta }(x_{t},t))} . This then gives us a backward diffusion process p θ {\displaystyle p_{\theta }} defined by p θ ( x T ) = N ( x T | 0 , I ) {\displaystyle p_{\theta }(x

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  Softplus

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

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  Cryptee

  Cryptee is a privacy focused client-side encrypted and cross-platform productivity suite and data storage service. == History == Cryptee was founded in 2017, by John Ozbay, a cybersecurity researcher, commenter, and activist, to exclusively focus on providing a secure document editing service similar to Google Docs and Photos for everyone, with a particular focus on victims and survivors of domestic abuse, journalists and reporters. == Software == Users can write personal documents, notes, journals, store images, videos, and various kinds of other files. The source code of Cryptee is open source and publicly available to allow anyone to audit the service with ease, and help identify errors or potential vulnerabilities in a public and transparent manner. Cryptee has a few key features that differentiate it from other services in the industry, such as its Ghost Folders and Ghost Albums features, built specifically with victims and survivors of domestic abuse, journalists and reporters in mind. Cryptee allows users to hide (ghost) folders for plausible deniability also as known as deniable encryption in the field of cryptography and steganography, and ensure privacy even under coercion. === Features === Cryptee Docs' features include: To-do lists, Markdown support, KaTeX math and file attachments. cross-platform accessible, as it is a progressive web app. Bulk transfer from other note taking apps such as Evernote. Encrypted PDF and print-accurate (A4 and U.S. Letter paper-sized) text editing. Ability to edit docx files Cryptee Photos' features include: Ability to create slideshows. Ability to store original quality of photos. Ability to tag photos for organization. === Commercial strategy === The company's commercial strategy is focused on offering to its users an open source and transparent Photo Storage, Document Editor and Cloud Storage services without trackers or advertisements as it seeks to compete with Google Docs, Google Photos and similar services through its offerings. === Privacy === Cryptee utilizes zero-access storage to safe-keep all users' sensitive digital belongings. == Advocacy == === Lockdown mode === In July 2022, to fortify iPhones against the Pegasus Spyware, Apple announced a new, upcoming Lockdown Mode feature in iOS 16, welcomed by many experts. In the following weeks after Apple's announcement, in August 2022, the Founder and CEO of Cryptee, and privacy activist John Ozbay published their research detailing shortcoming of Apple's Lockdown Mode. They demonstrated that enabling Lockdown Mode makes it possible for all websites and online ads to be able to detect if users have Lockdown Mode enabled or not. This was due to the fact that disabling web fonts (an attack surface) was detectable by websites. === Confrontations against Apple === ==== On PWAs ==== In February 2024, Apple announced plans to kill progressive web apps on iOS devices in the EU, claiming it was to comply with the Digital Markets Act (DMA). The announcement was criticized as anti-competitive by many in the tech industry, including by Tim Sweeney, the CEO of Epic Games. In response, Cryptee started working together with Open Web Advocacy (OWA), an international not-for-profit digital rights group to advocate for the future of the open web, promote web browser choice on mobile operating systems through challenging Apple's anti-competitive third party browser engine ban, and to champion the use and equality of progressive web apps over native apps, by reaching out to the European Union's Digital Markets Act (DMA) team. To better understand the consequences of Apple's decision to kill web apps, the EU announced that they "seek to investigate Apple over cutting off web apps", and that they sent "requests for information to Apple and to app developers, who can provide useful information for our assessment". Apart from sending a response to the EU, Cryptee, along with the OWA, launched an open letter to Tim Cook, which in 48 hours, got thousands of signatories including European Parliament Members Karen Melchior and Patrick Breyer; and thousands of other developers and organizations from over 100 countries. Consequently, 24 hours later, Apple backed off, and reversed course on its plan to cut off progressive web apps in the EU. ==== Ozbay's representations ==== Following the events, eventually on March 18, 2024, Founder and CEO of Cryptee John Ozbay represented the Open Web Advocacy group in European Union's Digital Markets Act (DMA) hearing for Apple. At the hearing, OWA confronted Apple, accused Apple of "maliciously intending to undermine user choice", and stated that there was no defense for Apple's behavior. In response, according to the tech news outlet Ars Technica, Apple's spokesperson "seemed to dodge Ozbay's question". ==== Cooperation with the EU ==== Within a week of the hearing, the European Union announced a DMA non-compliance investigation against Apple and United States' Department of Justice filed an antitrust lawsuit against Apple. A few months later, on June 27, 2024, Cryptee, in cooperation with EDRi — an international advocacy group, along with Article 19 — a British international human rights organization, Privacy International, F-Droid, Free Software Foundation Europe, Guardian Project and others have submitted a comprehensive analysis to the European Commission about how Apple's plans to comply with the Digital Markets Act are insufficient. == Reviews == In a 2018 article, Wall Street Journal's MarketWatch reviewed Cryptee, articulating the fact that Cryptee offers zero-access storage for photos, files, documents and notes, and pointed out that: "Being based in Estonia puts Cryptee outside the “14 eyes jurisdiction,” an international surveillance alliance of European Union and North American countries, making it less likely it will be targeted with demands for data". In addition, the review highlighted Cryptee's Ghost Folders feature which ensures privacy even under coercion. In a 2019 article, Reclaim The Net named Cryptee as one of the "5 great privacy-focused Evernote alternatives to keep your notes safe", underlining that: "When it comes to security, this app is state of the art." and that "When making this app, the developers thought about every aspect of security and have taken every precaution to make it as secure as possible.". The review further underscored Cryptee's open-source nature, its strong encryption, and easy migration features. In a 2021 article, The Verge reviewed Cryptee, pointing out that Cryptee, based out of Europe, is one of the main photo storage service alternatives to Google Photos, and that it's their recommendation for users who are "concerned about privacy and like the idea of encryption" as Cryptee "offers to keep all your photos encrypted using AES-256". In a 2024 article, Beebom, enlisted Cryptee as one of the "7 best iCloud Photos Alternatives for iPhone and iPad", complimenting Cryptee's simplicity, its use of encryption to safeguard users' photos against hacking by not storing any unencrypted data. The article also provided further attention to Cryptee's additional features such as such as Ghost Albums, slideshows, easy-to-use drag and drop uploads, tagging and users' ability to store original-quality photos on Cryptee, concluding that Cryptee is "a safe bet if you are on the lookout for a privacy-centric iCloud Photos alternative".

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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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  Elastic map

  Elastic maps provide a tool for nonlinear dimensionality reduction. By their construction, they are a system of elastic springs embedded in the data space. This system approximates a low-dimensional manifold. The elastic coefficients of this system allow the switch from completely unstructured k-means clustering (zero elasticity) to the estimators located closely to linear PCA manifolds (for high bending and low stretching modules). With some intermediate values of the elasticity coefficients, this system effectively approximates non-linear principal manifolds. This approach is based on a mechanical analogy between principal manifolds, that are passing through "the middle" of the data distribution, and elastic membranes and plates. The method was developed by A.N. Gorban, A.Y. Zinovyev and A.A. Pitenko in 1996–1998. == Energy of elastic map == Let S {\displaystyle {\mathcal {S}}} be a data set in a finite-dimensional Euclidean space. Elastic map is represented by a set of nodes w j {\displaystyle {\bf {w}}_{j}} in the same space. Each datapoint s ∈ S {\displaystyle s\in {\mathcal {S}}} has a host node, namely the closest node w j {\displaystyle {\bf {w}}_{j}} (if there are several closest nodes then one takes the node with the smallest number). The data set S {\displaystyle {\mathcal {S}}} is divided into classes K j = { s | w j is a host of s } {\displaystyle K_{j}=\{s\ |\ {\bf {w}}_{j}{\mbox{ is a host of }}s\}} . The approximation energy D is the distortion D = 1 2 ∑ j = 1 k ∑ s ∈ K j ‖ s − w j ‖ 2 {\displaystyle D={\frac {1}{2}}\sum _{j=1}^{k}\sum _{s\in K_{j}}\|s-{\bf {w}}_{j}\|^{2}} , which is the energy of the springs with unit elasticity which connect each data point with its host node. It is possible to apply weighting factors to the terms of this sum, for example to reflect the standard deviation of the probability density function of any subset of data points { s i } {\displaystyle \{s_{i}\}} . On the set of nodes an additional structure is defined. Some pairs of nodes, ( w i , w j ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j})} , are connected by elastic edges. Call this set of pairs E {\displaystyle E} . Some triplets of nodes, ( w i , w j , w k ) {\displaystyle ({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})} , form bending ribs. Call this set of triplets G {\displaystyle G} . The stretching energy is U E = 1 2 λ ∑ ( w i , w j ) ∈ E ‖ w i − w j ‖ 2 {\displaystyle U_{E}={\frac {1}{2}}\lambda \sum _{({\bf {w}}_{i},{\bf {w}}_{j})\in E}\|{\bf {w}}_{i}-{\bf {w}}_{j}\|^{2}} , The bending energy is U G = 1 2 μ ∑ ( w i , w j , w k ) ∈ G ‖ w i − 2 w j + w k ‖ 2 {\displaystyle U_{G}={\frac {1}{2}}\mu \sum _{({\bf {w}}_{i},{\bf {w}}_{j},{\bf {w}}_{k})\in G}\|{\bf {w}}_{i}-2{\bf {w}}_{j}+{\bf {w}}_{k}\|^{2}} , where λ {\displaystyle \lambda } and μ {\displaystyle \mu } are the stretching and bending moduli respectively. The stretching energy is sometimes referred to as the membrane, while the bending energy is referred to as the thin plate term. For example, on the 2D rectangular grid the elastic edges are just vertical and horizontal edges (pairs of closest vertices) and the bending ribs are the vertical or horizontal triplets of consecutive (closest) vertices. The total energy of the elastic map is thus U = D + U E + U G . {\displaystyle U=D+U_{E}+U_{G}.} The position of the nodes { w j } {\displaystyle \{{\bf {w}}_{j}\}} is determined by the mechanical equilibrium of the elastic map, i.e. its location is such that it minimizes the total energy U {\displaystyle U} . == Expectation-maximization algorithm == For a given splitting of dataset S {\displaystyle {\mathcal {S}}} in classes K j {\displaystyle K_{j}} , minimization of the quadratic functional U {\displaystyle U} is a linear problem with the sparse matrix of coefficients. Therefore, similar to principal component analysis or k-means, a splitting method is used: For given { w j } {\displaystyle \{{\bf {w}}_{j}\}} find { K j } {\displaystyle \{K_{j}\}} ; For given { K j } {\displaystyle \{K_{j}\}} minimize U {\displaystyle U} and find { w j } {\displaystyle \{{\bf {w}}_{j}\}} ; If no change, terminate. This expectation-maximization algorithm guarantees a local minimum of U {\displaystyle U} . For improving the approximation various additional methods are proposed. For example, the softening strategy is used. This strategy starts with a rigid grids (small length, small bending and large elasticity modules λ {\displaystyle \lambda } and μ {\displaystyle \mu } coefficients) and finishes with soft grids (small λ {\displaystyle \lambda } and μ {\displaystyle \mu } ). The training goes in several epochs, each epoch with its own grid rigidness. Another adaptive strategy is growing net: one starts from a small number of nodes and gradually adds new nodes. Each epoch goes with its own number of nodes. == Applications == Most important applications of the method and free software are in bioinformatics for exploratory data analysis and visualisation of multidimensional data, for data visualisation in economics, social and political sciences, as an auxiliary tool for data mapping in geographic informational systems and for visualisation of data of various nature. The method is applied in quantitative biology for reconstructing the curved surface of a tree leaf from a stack of light microscopy images. This reconstruction is used for quantifying the geodesic distances between trichomes and their patterning, which is a marker of the capability of a plant to resist to pathogenes. Recently, the method is adapted as a support tool in the decision process underlying the selection, optimization, and management of financial portfolios. The method of elastic maps has been systematically tested and compared with several machine learning methods on the applied problem of identification of the flow regime of a gas-liquid flow in a pipe. There are various regimes: Single phase water or air flow, Bubbly flow, Bubbly-slug flow, Slug flow, Slug-churn flow, Churn flow, Churn-annular flow, and Annular flow. The simplest and most common method used to identify the flow regime is visual observation. This approach is, however, subjective and unsuitable for relatively high gas and liquid flow rates. Therefore, the machine learning methods are proposed by many authors. The methods are applied to differential pressure data collected during a calibration process. The method of elastic maps provided a 2D map, where the area of each regime is represented. The comparison with some other machine learning methods is presented in Table 1 for various pipe diameters and pressure. Here, ANN stands for the backpropagation artificial neural networks, SVM stands for the support vector machine, SOM for the self-organizing maps. The hybrid technology was developed for engineering applications. In this technology, elastic maps are used in combination with Principal Component Analysis (PCA), Independent Component Analysis (ICA) and backpropagation ANN. The textbook provides a systematic comparison of elastic maps and self-organizing maps (SOMs) in applications to economic and financial decision-making.

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  Kubeflow

  Kubeflow is an open-source platform for machine learning and MLOps on Kubernetes introduced by Google. The different stages in a typical machine learning lifecycle are represented with different software components in Kubeflow, including model development (Kubeflow Notebooks), model training (Kubeflow Pipelines, Kubeflow Training Operator), model serving (KServe), and automated machine learning (Katib). Each component of Kubeflow can be deployed separately, and it is not a requirement to deploy every component. == History == The Kubeflow project was first announced at KubeCon + CloudNativeCon North America 2017 by Google engineers David Aronchick, Jeremy Lewi, and Vishnu Kannan to address a perceived lack of flexible options for building production-ready machine learning systems. The project has also stated it began as a way for Google to open-source how they ran TensorFlow internally. The first release of Kubeflow (Kubeflow 0.1) was announced at KubeCon + CloudNativeCon Europe 2018. Kubeflow 1.0 was released in March 2020 via a public blog post announcing that many Kubeflow components were graduating to a "stable status", indicating they were now ready for production usage. In October 2022, Google announced that the Kubeflow project had applied to join the Cloud Native Computing Foundation. In July 2023, the foundation voted to accept Kubeflow as an incubating stage project. == Components == === Kubeflow Notebooks for model development === Machine learning models are developed in the notebooks component called Kubeflow Notebooks. The component runs web-based development environments inside a Kubernetes cluster, with native support for Jupyter Notebook, Visual Studio Code, and RStudio. === Kubeflow Pipelines for model training === Once developed, models are trained in the Kubeflow Pipelines component. The component acts as a platform for building and deploying portable, scalable machine learning workflows based on Docker containers. Google Cloud Platform has adopted the Kubeflow Pipelines DSL within its Vertex AI Pipelines product. === Kubeflow Training Operator for model training === For certain machine learning models and libraries, the Kubeflow Training Operator component provides Kubernetes custom resources support. The component runs distributed or non-distributed TensorFlow, PyTorch, Apache MXNet, XGBoost, and MPI training jobs on Kubernetes. === KServe for model serving === The KServe component (previously named KFServing) provides Kubernetes custom resources for serving machine learning models on arbitrary frameworks including TensorFlow, XGBoost, scikit-learn, PyTorch, and ONNX. KServe was developed collaboratively by Google, IBM, Bloomberg, NVIDIA, and Seldon. Publicly disclosed adopters of KServe include Bloomberg, Gojek, the Wikimedia Foundation, and others. === Katib for automated machine learning === Lastly, Kubeflow includes a component for automated training and development of machine learning models, the Katib component. It is described as a Kubernetes-native project and features hyperparameter tuning, early stopping, and neural architecture search. == Release timeline ==

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  Network Abstraction Layer

  The Network Abstraction Layer (NAL) is a part of the H.264/AVC and HEVC video coding standards. The main goal of the NAL is the provision of a "network-friendly" video representation addressing "conversational" (video telephony) and "non conversational" (storage, broadcast, or streaming) applications. NAL has achieved a significant improvement in application flexibility relative to prior video coding standards. == Introduction == An increasing number of services and growing popularity of high definition TV are creating greater needs for higher coding efficiency. Moreover, other transmission media such as cable modem, xDSL, or UMTS offer much lower data rates than broadcast channels, and enhanced coding efficiency can enable the transmission of more video channels or higher quality video representations within existing digital transmission capacities. Video coding for telecommunication applications has diversified from ISDN and T1/E1 service to embrace PSTN, mobile wireless networks, and LAN/Internet network delivery. Throughout this evolution, continued efforts have been made to maximize coding efficiency while dealing with the diversification of network types and their characteristic formatting and loss/error robustness requirements. The H.264/AVC and HEVC standards are designed for technical solutions including areas like broadcasting (over cable, satellite, cable modem, DSL, terrestrial, etc.) interactive or serial storage on optical and magnetic devices, conversational services, video-on-demand or multimedia streaming, multimedia messaging services, etc. Moreover, new applications may be deployed over existing and future networks. This raises the question about how to handle this variety of applications and networks. To address this need for flexibility and customizability, the design covers a NAL that formats the Video Coding Layer (VCL) representation of the video and provides header information in a manner appropriate for conveyance by a variety of transport layers or storage media. The NAL is designed in order to provide "network friendliness" to enable simple and effective customization of the use of VCL for a broad variety of systems. The NAL facilitates the ability to map VCL data to transport layers such as: RTP/IP for any kind of real-time wire-line and wireless Internet services. File formats, e.g., ISO MP4 for storage and MMS. H.32X for wireline and wireless conversational services. MPEG-2 systems for broadcasting services, etc. The full degree of customization of the video content to fit the needs of each particular application is outside the scope of the video coding standardization effort, but the design of the NAL anticipates a variety of such mappings. Some key concepts of the NAL are NAL units, byte stream, and packet formats uses of NAL units, parameter sets, and access units. A short description of these concepts is given below. == NAL units == The coded video data is organized into NAL units, each of which is effectively a packet that contains an integer number of bytes. The first byte of each H.264/AVC NAL unit is a header byte that contains an indication of the type of data in the NAL unit. For HEVC the header was extended to two bytes. All the remaining bytes contain payload data of the type indicated by the header. The NAL unit structure definition specifies a generic format for use in both packet-oriented and bitstream-oriented transport systems, and a series of NAL units generated by an encoder is referred to as a NAL unit stream. == NAL Units in Byte-Stream Format Use == Some systems require delivery of the entire or partial NAL unit stream as an ordered stream of bytes or bits within which the locations of NAL unit boundaries need to be identifiable from patterns within the coded data itself. For use in such systems, the H.264/AVC and HEVC specifications define a byte stream format. In the byte stream format, each NAL unit is prefixed by a specific pattern of three bytes called a start code prefix. The boundaries of the NAL unit can then be identified by searching the coded data for the unique start code prefix pattern. The use of emulation prevention bytes guarantees that start code prefixes are unique identifiers of the start of a new NAL unit. A small amount of additional data (one byte per video picture) is also added to allow decoders that operate in systems that provide streams of bits without alignment to byte boundaries to recover the necessary alignment from the data in the stream. Additional data can also be inserted in the byte stream format that allows expansion of the amount of data to be sent and can aid in achieving more rapid byte alignment recovery, if desired. == NAL Units in Packet-Transport System Use == In other systems (e.g., IP/RTP systems), the coded data is carried in packets that are framed by the system transport protocol, and identification of the boundaries of NAL units within the packets can be established without use of start code prefix patterns. In such systems, the inclusion of start code prefixes in the data would be a waste of data carrying capacity, so instead the NAL units can be carried in data packets without start code prefixes. == VCL and Non-VCL NAL Units == NAL units are classified into VCL and non-VCL NAL units. VCL NAL units contain the data that represents the values of the samples in the video pictures. Non-VCL NAL units contain any associated additional information such as parameter sets (important header data that can apply to a large number of VCL NAL units) and supplemental enhancement information (timing information and other supplemental data that may enhance usability of the decoded video signal but are not necessary for decoding the values of the samples in the video pictures). == Parameter Sets == A parameter set contains shared configuration data that is carried in non-VCL NAL units. Parameter sets are typically reused when decoding many coded pictures within a video sequence. Each VCL NAL unit references a picture parameter set (PPS), which in turn references a sequence parameter set (SPS). There are two types of parameter sets: Sequence parameter set (SPS), which specifies mostly constant configuration such as resolution, bit depth, or chroma format. (For a concrete implementation, see FFmpeg's SPS struct.) Picture parameter set (PPS), which applies on top of an SPS, and specifies configuration such as QP offsets. (For a concrete implementation, see FFmpeg's PPS struct.) The sequence and picture parameter-set mechanism decouples the transmission of infrequently changing information from the transmission of coded representations of the values of the samples in the video pictures. Each VCL NAL unit contains an identifier that refers to the content of the relevant picture parameter set and each picture parameter set contains an identifier that refers to the content of the relevant sequence parameter set. In this manner, a small amount of data (the identifier) can be used to refer to a larger amount of information (the parameter set) without repeating that information within each VCL NAL unit. Sequence and picture parameter sets can be sent well ahead of the VCL NAL units that they apply to, and can be repeated to provide robustness against data loss. In some applications, parameter sets may be sent within the channel that carries the VCL NAL units (termed "in-band" transmission). In other applications, it can be advantageous to convey the parameter sets "out-of-band" using a more reliable transport mechanism than the video channel itself. == Access Units == A set of NAL units in a specified form is referred to as an access unit. The decoding of each access unit results in one decoded picture. Each access unit contains a set of VCL NAL units that together compose a primary coded picture. It may also be prefixed with an access unit delimiter to aid in locating the start of the access unit. Some supplemental enhancement information containing data such as picture timing information may also precede the primary coded picture. The primary coded picture consists of a set of VCL NAL units consisting of slices or slice data partitions that represent the samples of the video picture. Following the primary coded picture may be some additional VCL NAL units that contain redundant representations of areas of the same video picture. These are referred to as redundant coded pictures, and are available for use by a decoder in recovering from loss or corruption of the data in the primary coded pictures. Decoders are not required to decode redundant coded pictures if they are present. Finally, if the coded picture is the last picture of a coded video sequence (a sequence of pictures that is independently decodable and uses only one sequence parameter set), an end of sequence NAL unit may be present to indicate the end of the sequence; and if the coded picture is the last coded picture in the entire NAL unit stream, an end of stream NAL unit may be present to

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  Logic learning machine

  Logic learning machine (LLM) is a machine learning method based on the generation of intelligible rules. LLM is an efficient implementation of the Switching Neural Network (SNN) paradigm, developed by Marco Muselli, Senior Researcher at the Italian National Research Council CNR-IEIIT in Genoa. LLM has been employed in many different sectors, including the field of medicine (orthopedic patient classification, DNA micro-array analysis and Clinical Decision Support Systems), financial services and supply chain management. == History == The Switching Neural Network approach was developed in the 1990s to overcome the drawbacks of the most commonly used machine learning methods. In particular, black box methods, such as multilayer perceptron and support vector machine, had good accuracy but could not provide deep insight into the studied phenomenon. On the other hand, decision trees were able to describe the phenomenon but often lacked accuracy. Switching Neural Networks made use of Boolean algebra to build sets of intelligible rules able to obtain very good performance. In 2014, an efficient version of Switching Neural Network was developed and implemented in the Rulex suite with the name Logic Learning Machine. Also, an LLM version devoted to regression problems was developed. == General == Like other machine learning methods, LLM uses data to build a model able to perform a good forecast about future behaviors. LLM starts from a table including a target variable (output) and some inputs and generates a set of rules that return the output value y {\displaystyle y} corresponding to a given configuration of inputs. A rule is written in the form: if premise then consequence where consequence contains the output value whereas premise includes one or more conditions on the inputs. According to the input type, conditions can have different forms: for categorical variables the input value must be in a given subset: x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} . for ordered variables the condition is written as an inequality or an interval: x 2 ≤ α {\displaystyle x_{2}\leq \alpha } or β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } A possible rule is therefore in the form if x 1 ∈ { A , B , C , . . . } {\displaystyle x_{1}\in \{A,B,C,...\}} AND x 2 ≤ α {\displaystyle x_{2}\leq \alpha } AND β ≤ x 3 ≤ γ {\displaystyle \beta \leq x_{3}\leq \gamma } then y = y ¯ {\displaystyle y={\bar {y}}} == Types == According to the output type, different versions of the Logic Learning Machine have been developed: Logic Learning Machine for classification, when the output is a categorical variable, which can assume values in a finite set Logic Learning Machine for regression, when the output is an integer or real number.

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  Genetic operator

  A genetic operator is an operator used in evolutionary algorithms (EA) to guide the algorithm towards a solution to a given problem. There are three main types of operators (mutation, crossover and selection), which must work in conjunction with one another in order for the algorithm to be successful. Genetic operators are used to create and maintain genetic diversity (mutation operator), combine existing solutions (also known as chromosomes) into new solutions (crossover) and select between solutions (selection). The classic representatives of evolutionary algorithms include genetic algorithms, evolution strategies, genetic programming and evolutionary programming. In his book discussing the use of genetic programming for the optimization of complex problems, computer scientist John Koza has also identified an 'inversion' or 'permutation' operator; however, the effectiveness of this operator has never been conclusively demonstrated and this operator is rarely discussed in the field of genetic programming. For combinatorial problems, however, these and other operators tailored to permutations are frequently used by other EAs. Mutation (or mutation-like) operators are said to be unary operators, as they only operate on one chromosome at a time. In contrast, crossover operators are said to be binary operators, as they operate on two chromosomes at a time, combining two existing chromosomes into one new chromosome. == Operators == Genetic variation is a necessity for the process of evolution. Genetic operators used in evolutionary algorithms are analogous to those in the natural world: survival of the fittest, or selection; reproduction (crossover, also called recombination); and mutation. === Selection === Selection operators give preference to better candidate solutions (chromosomes), allowing them to pass on their 'genes' to the next generation (iteration) of the algorithm. The best solutions are determined using some form of objective function (also known as a 'fitness function' in evolutionary algorithms), before being passed to the crossover operator. Different methods for choosing the best solutions exist, for example, fitness proportionate selection and tournament selection. A further or the same selection operator is used to determine the individuals for being selected to form the next parental generation. The selection operator may also ensure that the best solution(s) from the current generation always become(s) a member of the next generation without being altered; this is known as elitism or elitist selection. === Crossover === Crossover is the process of taking more than one parent solutions (chromosomes) and producing a child solution from them. By recombining portions of good solutions, the evolutionary algorithm is more likely to create a better solution. As with selection, there are a number of different methods for combining the parent solutions, including the edge recombination operator (ERO) and the 'cut and splice crossover' and 'uniform crossover' methods. The crossover method is often chosen to closely match the chromosome's representation of the solution; this may become particularly important when variables are grouped together as building blocks, which might be disrupted by a non-respectful crossover operator. Similarly, crossover methods may be particularly suited to certain problems; the ERO is considered a good option for solving the travelling salesman problem. === Mutation === The mutation operator encourages genetic diversity amongst solutions and attempts to prevent the evolutionary algorithm converging to a local minimum by stopping the solutions becoming too close to one another. In mutating the current pool of solutions, a given solution may change between slightly and entirely from the previous solution. By mutating the solutions, an evolutionary algorithm can reach an improved solution solely through the mutation operator. Again, different methods of mutation may be used; these range from a simple bit mutation (flipping random bits in a binary string chromosome with some low probability) to more complex mutation methods in which genes in the solution are changed, for example by adding a random value from the Gaussian distribution to the current gene value. As with the crossover operator, the mutation method is usually chosen to match the representation of the solution within the chromosome. == Combining operators == While each operator acts to improve the solutions produced by the evolutionary algorithm working individually, the operators must work in conjunction with each other for the algorithm to be successful in finding a good solution. Using the selection operator on its own will tend to fill the solution population with copies of the best solution from the population. If the selection and crossover operators are used without the mutation operator, the algorithm will tend to converge to a local minimum, that is, a good but sub-optimal solution to the problem. Using the mutation operator on its own leads to a random walk through the search space. Only by using all three operators together can the evolutionary algorithm become a noise-tolerant global search algorithm, yielding good solutions to the problem at hand.

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