AI Art Modifier

AI Art Modifier — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

• 

  Lawbot

  Lawbots are a broad class of customer-facing legal AI applications that are used to automate specific legal tasks, such as document automation and legal research. The terms robot lawyer and lawyer bot are used as synonyms to lawbot. A robot lawyer or a robo-lawyer refers to a legal AI application that can perform tasks that are typically done by paralegals or young associates at law firms. However, there is some debate on the correctness of the term. Some commentators say that legal AI is technically speaking neither a lawyer nor a robot and should not be referred to as such. Other commentators believe that the term can be misleading and note that the robot lawyer of the future will not be one all-encompassing application but a collection of specialized bots for various tasks. Lawbots use various artificial intelligence techniques or other intelligent systems to limit humans' direct ongoing involvement in certain steps of a legal matter. The user interfaces on lawbots vary from smart searches and step-by-step forms to chatbots. Consumer and enterprise-facing lawbot solutions often do not require direct supervision from a legal professional. Depending on the task, some client-facing solutions used at law firms operate under an attorney supervision. == Levels of autonomy == The following levels of autonomy (LoA) are suggested for automated AI legal reasoning: Level 0 (LoA0): No automation for AI legal reasoning Level 1 (LoA1): Simple assistance automation Level 2 (LoA2): Advanced assistance automation Level 3 (LoA3): Semi-autonomous automation Level 4 (LoA4): Domain automation Level 5 (LoA5): Fully-autonomous automation Level 6 (LoA6): Superhuman automation == Examples == Some legal AI solutions are developed and marketed directly to the customers or consumers, whereas other applications are tools for the attorneys at law firms. There are already hundreds of legal AI solutions that operate in multitude of ways varying in sophistication and dependence on scripted algorithms. One notable legal technology chatbot application is DoNotPay. It had started off as an app for contesting parking tickets, but has since expanded to include features that help users with many different types of legal issues, ranging from consumer protection to immigration rights and other social issues. == Impact on the legal industry == In the 2016 report, Deloitte estimated that more than 110,000 law jobs in just the United Kingdom alone could disappear within the next twenty years due to automation. This change could result in the creation of more highly skilled jobs and in the reduction of paralegal and temporary positions. Deloitte's report asserts that "there is significant potential for high-skilled roles that involve repetitive processes to be automated by smart and self-learning algorithms". According to Lawyers to Engage, between 22% of a lawyer’s work and 35% of a legal assistant’s work can be automated in the US. Top law schools like Harvard have already begun to integrate Artificial Intelligence into the curriculum. Legal tech start-up companies have begun developing applications that assist law firms with completing low-risk legal processes. These applications can enable lawyers to focus on more work that requires their specific expertise. The automation of processes like contract reviewing, enforcement of negotiations (smart contracts) and client intake (expert systems) allows law firms to streamline their procedures and improve efficiency. In addition, automation benefits small-to-medium law firms that do not have the resources to utilize junior talent on such routine tasks. The increase of law firms utilizing automated applications could result into legal tech becoming a necessity in the industry. Digital Reason CEO, Tim Estes, stated that those who refuse the opportunity to integrate AI in their workflow are “most at risk.” In 2018, Forbes reported a 713% increase in investments in legal tech. This rapid growth is reflective of law firms beginning to “cede business to… new model legal providers… that meld technological, business and legal expertise.” == Access to law and justice == It has been widely estimated for at least the last generation that all the programs and resources devoted to ensuring access to justice address only 20% of the civil legal needs of low-income people in the United States. Drawing on this experience, in late 2011, the U.S. government-funded Legal Services Corporation decided to convene a summit of leaders to explore how best to use technology in the access-to-justice community. The group adopted a mission for The Summit on the Use of Technology to Expand Access to Justice (Summit) consistent with the magnitude of the challenge: "to explore the potential of technology to move the United States toward providing some form of effective assistance to 100% of persons otherwise unable to afford an attorney for dealing with essential civil legal needs". In April 2017, joined by Microsoft and Pro Bono Net, the Legal Services Corporation (LSC) announced a pilot program to develop online, statewide legal portals to direct individuals with civil legal needs to the most appropriate forms of assistance. == Technological limitations == Current research in subjects such as computational privacy, explainable machine learning, Bayesian deep learning, knowledge-intensive machine learning, and transfer learning reveals that we do not yet have the technology to enable Level 4 to 6 AI lawbots. In 2023, OpenLaw began developing a model called Law Bot, which interacts in a conversational way as an attorney. The dialogue format makes it possible for Law Bot to answer follow-up questions, challenge incorrect premises, and reject inappropriate requests. Currently, they try to ensure it is in full compliance with all laws and regulations while conducting further beta testing before releasing it to the general public.

  Read more →
• 

  Dominance-based rough set approach

  The dominance-based rough set approach (DRSA) is an extension of rough set theory for multi-criteria decision analysis (MCDA), introduced by Greco, Matarazzo and Słowiński. The main change compared to the classical rough sets is the substitution for the indiscernibility relation by a dominance relation, which permits one to deal with inconsistencies typical to consideration of criteria and preference-ordered decision classes. == Multicriteria classification (sorting) == Multicriteria classification (sorting) is one of the problems considered within MCDA and can be stated as follows: given a set of objects evaluated by a set of criteria (attributes with preference-order domains), assign these objects to some pre-defined and preference-ordered decision classes, such that each object is assigned to exactly one class. Due to the preference ordering, improvement of evaluations of an object on the criteria should not worsen its class assignment. The sorting problem is very similar to the problem of classification, however, in the latter, the objects are evaluated by regular attributes and the decision classes are not necessarily preference ordered. The problem of multicriteria classification is also referred to as ordinal classification problem with monotonicity constraints and often appears in real-life application when ordinal and monotone properties follow from the domain knowledge about the problem. As an illustrative example, consider the problem of evaluation in a high school. The director of the school wants to assign students (objects) to three classes: bad, medium and good (notice that class good is preferred to medium and medium is preferred to bad). Each student is described by three criteria: level in Physics, Mathematics and Literature, each taking one of three possible values bad, medium and good. Criteria are preference-ordered and improving the level from one of the subjects should not result in worse global evaluation (class). As a more serious example, consider classification of bank clients, from the viewpoint of bankruptcy risk, into classes safe and risky. This may involve such characteristics as "return on equity (ROE)", "return on investment (ROI)" and "return on sales (ROS)". The domains of these attributes are not simply ordered but involve a preference order since, from the viewpoint of bank managers, greater values of ROE, ROI or ROS are better for clients being analysed for bankruptcy risk . Thus, these attributes are criteria. Neglecting this information in knowledge discovery may lead to wrong conclusions. == Data representation == === Decision table === In DRSA, data are often presented using a particular form of decision table. Formally, a DRSA decision table is a 4-tuple S = ⟨ U , Q , V , f ⟩ {\displaystyle S=\langle U,Q,V,f\rangle } , where U {\displaystyle U\,\!} is a finite set of objects, Q {\displaystyle Q\,\!} is a finite set of criteria, V = ⋃ q ∈ Q V q {\displaystyle V=\bigcup {}_{q\in Q}V_{q}} where V q {\displaystyle V_{q}\,\!} is the domain of the criterion q {\displaystyle q\,\!} and f : U × Q → V {\displaystyle f\colon U\times Q\to V} is an information function such that f ( x , q ) ∈ V q {\displaystyle f(x,q)\in V_{q}} for every ( x , q ) ∈ U × Q {\displaystyle (x,q)\in U\times Q} . The set Q {\displaystyle Q\,\!} is divided into condition criteria (set C ≠ ∅ {\displaystyle C\neq \emptyset } ) and the decision criterion (class) d {\displaystyle d\,\!} . Notice, that f ( x , q ) {\displaystyle f(x,q)\,\!} is an evaluation of object x {\displaystyle x\,\!} on criterion q ∈ C {\displaystyle q\in C} , while f ( x , d ) {\displaystyle f(x,d)\,\!} is the class assignment (decision value) of the object. An example of decision table is shown in Table 1 below. === Outranking relation === It is assumed that the domain of a criterion q ∈ Q {\displaystyle q\in Q} is completely preordered by an outranking relation ⪰ q {\displaystyle \succeq _{q}} ; x ⪰ q y {\displaystyle x\succeq _{q}y} means that x {\displaystyle x\,\!} is at least as good as (outranks) y {\displaystyle y\,\!} with respect to the criterion q {\displaystyle q\,\!} . Without loss of generality, we assume that the domain of q {\displaystyle q\,\!} is a subset of reals, V q ⊆ R {\displaystyle V_{q}\subseteq \mathbb {R} } , and that the outranking relation is a simple order between real numbers ≥ {\displaystyle \geq \,\!} such that the following relation holds: x ⪰ q y ⟺ f ( x , q ) ≥ f ( y , q ) {\displaystyle x\succeq _{q}y\iff f(x,q)\geq f(y,q)} . This relation is straightforward for gain-type ("the more, the better") criterion, e.g. company profit. For cost-type ("the less, the better") criterion, e.g. product price, this relation can be satisfied by negating the values from V q {\displaystyle V_{q}\,\!} . === Decision classes and class unions === Let T = { 1 , … , n } {\displaystyle T=\{1,\ldots ,n\}\,\!} . The domain of decision criterion, V d {\displaystyle V_{d}\,\!} consist of n {\displaystyle n\,\!} elements (without loss of generality we assume V d = T {\displaystyle V_{d}=T\,\!} ) and induces a partition of U {\displaystyle U\,\!} into n {\displaystyle n\,\!} classes Cl = { C l t , t ∈ T } {\displaystyle {\textbf {Cl}}=\{Cl_{t},t\in T\}} , where C l t = { x ∈ U : f ( x , d ) = t } {\displaystyle Cl_{t}=\{x\in U\colon f(x,d)=t\}} . Each object x ∈ U {\displaystyle x\in U} is assigned to one and only one class C l t , t ∈ T {\displaystyle Cl_{t},t\in T} . The classes are preference-ordered according to an increasing order of class indices, i.e. for all r , s ∈ T {\displaystyle r,s\in T} such that r ≥ s {\displaystyle r\geq s\,\!} , the objects from C l r {\displaystyle Cl_{r}\,\!} are strictly preferred to the objects from C l s {\displaystyle Cl_{s}\,\!} . For this reason, we can consider the upward and downward unions of classes, defined respectively, as: C l t ≥ = ⋃ s ≥ t C l s C l t ≤ = ⋃ s ≤ t C l s t ∈ T {\displaystyle Cl_{t}^{\geq }=\bigcup _{s\geq t}Cl_{s}\qquad Cl_{t}^{\leq }=\bigcup _{s\leq t}Cl_{s}\qquad t\in T} == Main concepts == === Dominance === We say that x {\displaystyle x\,\!} dominates y {\displaystyle y\,\!} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted by x D p y {\displaystyle xD_{p}y\,\!} , if x {\displaystyle x\,\!} is better than y {\displaystyle y\,\!} on every criterion from P {\displaystyle P\,\!} , x ⪰ q y , ∀ q ∈ P {\displaystyle x\succeq _{q}y,\,\forall q\in P} . For each P ⊆ C {\displaystyle P\subseteq C} , the dominance relation D P {\displaystyle D_{P}\,\!} is reflexive and transitive, i.e. it is a partial pre-order. Given P ⊆ C {\displaystyle P\subseteq C} and x ∈ U {\displaystyle x\in U} , let D P + ( x ) = { y ∈ U : y D p x } {\displaystyle D_{P}^{+}(x)=\{y\in U\colon yD_{p}x\}} D P − ( x ) = { y ∈ U : x D p y } {\displaystyle D_{P}^{-}(x)=\{y\in U\colon xD_{p}y\}} represent P-dominating set and P-dominated set with respect to x ∈ U {\displaystyle x\in U} , respectively. === Rough approximations === The key idea of the rough set philosophy is approximation of one knowledge by another knowledge. In DRSA, the knowledge being approximated is a collection of upward and downward unions of decision classes and the "granules of knowledge" used for approximation are P-dominating and P-dominated sets. The P-lower and the P-upper approximation of C l t ≥ , t ∈ T {\displaystyle Cl_{t}^{\geq },t\in T} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted as P _ ( C l t ≥ ) {\displaystyle {\underline {P}}(Cl_{t}^{\geq })} and P ¯ ( C l t ≥ ) {\displaystyle {\overline {P}}(Cl_{t}^{\geq })} , respectively, are defined as: P _ ( C l t ≥ ) = { x ∈ U : D P + ( x ) ⊆ C l t ≥ } {\displaystyle {\underline {P}}(Cl_{t}^{\geq })=\{x\in U\colon D_{P}^{+}(x)\subseteq Cl_{t}^{\geq }\}} P ¯ ( C l t ≥ ) = { x ∈ U : D P − ( x ) ∩ C l t ≥ ≠ ∅ } {\displaystyle {\overline {P}}(Cl_{t}^{\geq })=\{x\in U\colon D_{P}^{-}(x)\cap Cl_{t}^{\geq }\neq \emptyset \}} Analogously, the P-lower and the P-upper approximation of C l t ≤ , t ∈ T {\displaystyle Cl_{t}^{\leq },t\in T} with respect to P ⊆ C {\displaystyle P\subseteq C} , denoted as P _ ( C l t ≤ ) {\displaystyle {\underline {P}}(Cl_{t}^{\leq })} and P ¯ ( C l t ≤ ) {\displaystyle {\overline {P}}(Cl_{t}^{\leq })} , respectively, are defined as: P _ ( C l t ≤ ) = { x ∈ U : D P − ( x ) ⊆ C l t ≤ } {\displaystyle {\underline {P}}(Cl_{t}^{\leq })=\{x\in U\colon D_{P}^{-}(x)\subseteq Cl_{t}^{\leq }\}} P ¯ ( C l t ≤ ) = { x ∈ U : D P + ( x ) ∩ C l t ≤ ≠ ∅ } {\displaystyle {\overline {P}}(Cl_{t}^{\leq })=\{x\in U\colon D_{P}^{+}(x)\cap Cl_{t}^{\leq }\neq \emptyset \}} Lower approximations group the objects which certainly belong to class union C l t ≥ {\displaystyle Cl_{t}^{\geq }} (respectively C l t ≤ {\displaystyle Cl_{t}^{\leq }} ). This certainty comes from the fact, that object x ∈ U {\displaystyle x\in U} belongs to the lower approximation P _ ( C l t ≥ ) {\displaystyle {\underline {P}}(Cl_{t}^{\geq })} (respectively P _ ( C l t ≤ ) {\displaystyle {\underl

  Read more →
• 

  Dynamic Bayesian network

  A dynamic Bayesian network (DBN) is a Bayesian network (BN) which relates variables to each other over adjacent time steps. == History == A dynamic Bayesian network (DBN) is often called a "two-timeslice" BN (2TBN) because it says that at any point in time T, the value of a variable can be calculated from the internal regressors and the immediate prior value (time T-1). DBNs were developed by Paul Dagum in the early 1990s at Stanford University's Section on Medical Informatics. Dagum developed DBNs to unify and extend traditional linear state-space models such as Kalman filters, linear and normal forecasting models such as ARMA and simple dependency models such as hidden Markov models into a general probabilistic representation and inference mechanism for arbitrary nonlinear and non-normal time-dependent domains. Today, DBNs are common in robotics, and have shown potential for a wide range of data mining applications. For example, they have been used in speech recognition, digital forensics, protein sequencing, and bioinformatics. DBN is a generalization of hidden Markov models and Kalman filters. DBNs are conceptually related to probabilistic Boolean networks and can, similarly, be used to model dynamical systems at steady-state.

  Read more →
• 

  Dimensionality reduction

  Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable. Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Linear approaches can be further divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate step to facilitate other analyses. == Feature selection == The process of feature selection aims to find a suitable subset of the input variables (features, or attributes) for the task at hand. The three strategies are: the filter strategy (e.g., information gain), the wrapper strategy (e.g., accuracy-guided search), and the embedded strategy (features are added or removed while building the model based on prediction errors). Data analysis such as regression or classification can be done in the reduced space more accurately than in the original space. == Feature projection == Feature projection (also called feature extraction) transforms the data from the high-dimensional space to a space of fewer dimensions. The data transformation may be linear, as in principal component analysis (PCA), but many nonlinear dimensionality reduction techniques also exist. For multidimensional data, tensor representation can be used in dimensionality reduction through multilinear subspace learning. === Principal component analysis (PCA) === The main linear technique for dimensionality reduction, principal component analysis, performs a linear mapping of the data to a lower-dimensional space in such a way that the variance of the data in the low-dimensional representation is maximized. In practice, the covariance (and sometimes the correlation) matrix of the data is constructed and the eigenvectors on this matrix are computed. The eigenvectors that correspond to the largest eigenvalues (the principal components) can now be used to reconstruct a large fraction of the variance of the original data. Moreover, the first few eigenvectors can often be interpreted in terms of the large-scale physical behavior of the system, because they often contribute the vast majority of the system's energy, especially in low-dimensional systems. Still, this must be proved on a case-by-case basis as not all systems exhibit this behavior. The original space (with dimension of the number of points) has been reduced (with data loss, but hopefully retaining the most important variance) to the space spanned by a few eigenvectors. === Non-negative matrix factorization (NMF) === NMF decomposes a non-negative matrix to the product of two non-negative ones, which has been a promising tool in fields where only non-negative signals exist, such as astronomy. NMF is well known since the multiplicative update rule by Lee & Seung, which has been continuously developed: the inclusion of uncertainties, the consideration of missing data and parallel computation, sequential construction which leads to the stability and linearity of NMF, as well as other updates including handling missing data in digital image processing. With a stable component basis during construction, and a linear modeling process, sequential NMF is able to preserve the flux in direct imaging of circumstellar structures in astronomy, as one of the methods of detecting exoplanets, especially for the direct imaging of circumstellar discs. In comparison with PCA, NMF does not remove the mean of the matrices, which leads to physical non-negative fluxes; therefore NMF is able to preserve more information than PCA as demonstrated by Ren et al. === Kernel PCA === Principal component analysis can be employed in a nonlinear way by means of the kernel trick. The resulting technique is capable of constructing nonlinear mappings that maximize the variance in the data. The resulting technique is called kernel PCA. === Graph-based kernel PCA === Other prominent nonlinear techniques include manifold learning techniques such as Isomap, locally linear embedding (LLE), Hessian LLE, Laplacian eigenmaps, and methods based on tangent space analysis. These techniques assume that the high-dimensional input data lies near a low-dimensional manifold embedded in the ambient space, and construct a low-dimensional representation using a cost function that retains local properties of the data; they can be viewed as defining a graph-based kernel for Kernel PCA. More recently, techniques have been proposed that, instead of defining a fixed kernel, try to learn the kernel using semidefinite programming. The most prominent example of such a technique is maximum variance unfolding (MVU). The central idea of MVU is to exactly preserve all pairwise distances between nearest neighbors (in the inner product space) while maximizing the distances between points that are not nearest neighbors. An alternative approach to neighborhood preservation is through the minimization of a cost function that measures differences between distances in the input and output spaces. Important examples of such techniques include: classical multidimensional scaling, which is identical to PCA; Isomap, which uses geodesic distances in the data space; diffusion maps, which use diffusion distances in the data space; t-distributed stochastic neighbor embedding (t-SNE), which minimizes the divergence between distributions over pairs of points; and curvilinear component analysis. A different approach to nonlinear dimensionality reduction is through the use of autoencoders, a special kind of feedforward neural networks with a bottleneck hidden layer. The training of deep encoders is typically performed using a greedy layer-wise pre-training (e.g., using a stack of restricted Boltzmann machines) that is followed by a finetuning stage based on backpropagation. === Linear discriminant analysis (LDA) === Linear discriminant analysis (LDA) is a generalization of Fisher's linear discriminant, a method used in statistics, pattern recognition, and machine learning to find a linear combination of features that characterizes or separates two or more classes of objects or events. === Generalized discriminant analysis (GDA) === GDA deals with nonlinear discriminant analysis using kernel function operator. The underlying theory is close to the support-vector machines (SVM) insofar as the GDA method provides a mapping of the input vectors into high-dimensional feature space. Similar to LDA, the objective of GDA is to find a projection for the features into a lower dimensional space by maximizing the ratio of between-class scatter to within-class scatter. === Autoencoder === Autoencoders can be used to learn nonlinear dimension reduction functions and codings together with an inverse function from the coding to the original representation. === t-SNE === T-distributed Stochastic Neighbor Embedding (t-SNE) is a nonlinear dimensionality reduction technique useful for the visualization of high-dimensional datasets. It is not recommended for use in analysis such as clustering or outlier detection since it does not necessarily preserve densities or distances well. === UMAP === Uniform manifold approximation and projection (UMAP) is a nonlinear dimensionality reduction technique. Visually, it is similar to t-SNE, but it assumes that the data is uniformly distributed on a locally connected Riemannian manifold and that the Riemannian metric is locally constant or approximately locally constant. == Dimension reduction == For high-dimensional datasets, dimension reduction is usually performed prior to applying a k-nearest neighbors (k-NN) algorithm in order to mitigate the curse of dimensionality. Feature extraction and dimension reduction can be combined in one step, using principal component analysis (PCA), linear discriminant analysis (LDA), canonical correlation analysis (CCA), or non-negative matrix factorization (NMF) techniques to pre-process the data, followed by clustering via k-NN on feature vectors in a reduced-dimension space. In machine learning, this process is also called low-dimensional embedding. For high-dimensional datasets (e.g., when performing similarity search on live video streams, DNA data, or high-dimensional time series), running a fast approximate k-NN search using locality-sensitive hashing, random projection, "sketches", or other high-dimensional similarity search techniques from the VLDB conference toolbox may be the only fe

  Read more →
• 

  Autocommit

  In the context of data management, autocommit is a mode of operation of a database connection. Each individual database interaction (i.e., each SQL statement) submitted through the database connection in autocommit mode will be executed in its own transaction that is implicitly committed. A SQL statement executed in autocommit mode cannot be rolled back. Autocommit mode incurs per-statement transaction overhead and can often lead to undesirable performance or resource utilization impact on the database. Nonetheless, in systems such as Microsoft SQL Server, as well as connection technologies such as ODBC and Microsoft OLE DB, autocommit mode is the default for all statements that change data, in order to ensure that individual statements will conform to the ACID (atomicity-consistency-isolation-durability) properties of transactions. The alternative to autocommit mode (non-autocommit) means that the SQL client application itself is responsible for ending transactions explicitly via the commit or rollback SQL commands. Non-autocommit mode enables grouping of multiple data manipulation SQL commands into a single atomic transaction. Some DBMS (e.g. MariaDB) force autocommit for every DDL statement, even in non-autocommit mode. In this case, before each DDL statement, previous DML statements in transaction are autocommitted. Each DDL statement is executed in its own new autocommit transaction.

  Read more →
• 

  Win–stay, lose–switch

  In psychology, game theory, statistics, and machine learning, win–stay, lose–switch (also win–stay, lose–shift or Pavlov, named after Ivan Pavlov) is a heuristic learning strategy used to model learning in decision situations. It was first invented as an improvement over randomization in bandit problems. It was later applied to the prisoner's dilemma in order to model the evolution of altruism. In most versions, it starts either with a cooperate, then proceeds as always, or starts with a "probe" of cooperate-defect-cooperate to determine the other player's strategy. A mutual cooperation is regarded as a win. The learning rule bases its decision only on the outcome of the previous play. Outcomes are divided into successes (wins) and failures (losses). If the play on the previous round resulted in a success, then the agent plays the same strategy on the next round. Alternatively, if the play resulted in a failure the agent switches to another action. A large-scale empirical study of players of the game rock, paper, scissors shows that a variation of this strategy is adopted by real-world players of the game, instead of the Nash equilibrium strategy of choosing entirely at random between the three options.

  Read more →
• 

  Multilinear subspace learning

  Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality reduction can be performed on a data tensor that contains a collection of observations that have been vectorized, or observations that are treated as matrices and concatenated into a data tensor. Here are some examples of data tensors whose observations are vectorized or whose observations are matrices concatenated into data tensor images (2D/3D), video sequences (3D/4D), and hyperspectral cubes (3D/4D). The mapping from a high-dimensional vector space to a set of lower dimensional vector spaces is a multilinear projection. When observations are retained in the same organizational structure as matrices or higher order tensors, their representations are computed by performing linear projections into the column space, row space and fiber space. Multilinear subspace learning algorithms are higher-order generalizations of linear subspace learning methods such as principal component analysis (PCA), independent component analysis (ICA), linear discriminant analysis (LDA) and canonical correlation analysis (CCA). == Background == Multilinear methods may be causal in nature and perform causal inference, or they may be simple regression methods from which no causal conclusion are drawn. Linear subspace learning algorithms are traditional dimensionality reduction techniques that are well suited for datasets that are the result of varying a single causal factor. Unfortunately, they often become inadequate when dealing with datasets that are the result of multiple causal factors. . Multilinear subspace learning can be applied to observations whose measurements were vectorized and organized into a data tensor for causally aware dimensionality reduction. These methods may also be employed in reducing horizontal and vertical redundancies irrespective of the causal factors when the observations are treated as a "matrix" (ie. a collection of independent column/row observations) and concatenated into a tensor. == Algorithms == === Multilinear principal component analysis === Historically, multilinear principal component analysis has been referred to as "M-mode PCA", a terminology which was coined by Peter Kroonenberg. In 2005, Vasilescu and Terzopoulos introduced the Multilinear PCA terminology as a way to better differentiate between multilinear tensor decompositions that computed 2nd order statistics associated with each data tensor mode, and subsequent work on Multilinear Independent Component Analysis that computed higher order statistics for each tensor mode. MPCA is an extension of PCA. === Multilinear independent component analysis === Multilinear independent component analysis is an extension of ICA. === Multilinear linear discriminant analysis === Multilinear extension of LDA TTP-based: Discriminant Analysis with Tensor Representation (DATER) TTP-based: General tensor discriminant analysis (GTDA) TVP-based: Uncorrelated Multilinear Discriminant Analysis (UMLDA) === Multilinear canonical correlation analysis === Multilinear extension of CCA TTP-based: Tensor Canonical Correlation Analysis (TCCA) TVP-based: Multilinear Canonical Correlation Analysis (MCCA) TVP-based: Bayesian Multilinear Canonical Correlation Analysis (BMTF) A TTP is a direct projection of a high-dimensional tensor to a low-dimensional tensor of the same order, using N projection matrices for an Nth-order tensor. It can be performed in N steps with each step performing a tensor-matrix multiplication (product). The N steps are exchangeable. This projection is an extension of the higher-order singular value decomposition (HOSVD) to subspace learning. Hence, its origin is traced back to the Tucker decomposition in 1960s. A TVP is a direct projection of a high-dimensional tensor to a low-dimensional vector, which is also referred to as the rank-one projections. As TVP projects a tensor to a vector, it can be viewed as multiple projections from a tensor to a scalar. Thus, the TVP of a tensor to a P-dimensional vector consists of P projections from the tensor to a scalar. The projection from a tensor to a scalar is an elementary multilinear projection (EMP). In EMP, a tensor is projected to a point through N unit projection vectors. It is the projection of a tensor on a single line (resulting a scalar), with one projection vector in each mode. Thus, the TVP of a tensor object to a vector in a P-dimensional vector space consists of P EMPs. This projection is an extension of the canonical decomposition, also known as the parallel factors (PARAFAC) decomposition. === Typical approach in MSL === There are N sets of parameters to be solved, one in each mode. The solution to one set often depends on the other sets (except when N=1, the linear case). Therefore, the suboptimal iterative procedure in is followed. Initialization of the projections in each mode For each mode, fixing the projection in all the other mode, and solve for the projection in the current mode. Do the mode-wise optimization for a few iterations or until convergence. This is originated from the alternating least square method for multi-way data analysis. == Code == MATLAB Tensor Toolbox by Sandia National Laboratories. The MPCA algorithm written in Matlab (MPCA+LDA included). The UMPCA algorithm written in Matlab (data included). The UMLDA algorithm written in Matlab (data included). == Tensor data sets == 3D gait data (third-order tensors): 128x88x20(21.2M); 64x44x20(9.9M); 32x22x10(3.2M);

  Read more →
• 

  Proximal policy optimization

  Proximal policy optimization (PPO) is a reinforcement learning (RL) algorithm for training an intelligent agent. Specifically, it is a policy gradient method, often used for deep RL when the policy network is very large. == History == The predecessor to PPO, Trust Region Policy Optimization (TRPO), was published in 2015. It addressed the instability issue of another algorithm, the Deep Q-Network (DQN), by using the trust region method to limit the KL divergence between the old and new policies. However, TRPO uses the Hessian matrix (a matrix of second derivatives) to enforce the trust region, but the Hessian is inefficient for large-scale problems. PPO was published in 2017. It was essentially an approximation of TRPO that does not require computing the Hessian. The KL divergence constraint was approximated by simply clipping the policy gradient. Since 2018, PPO was the default RL algorithm at OpenAI. PPO has been applied to many areas, such as controlling a robotic arm, beating professional players at Dota 2 (OpenAI Five), and playing Atari games. == TRPO == TRPO, the predecessor of PPO, is an on-policy algorithm. It can be used for environments with either discrete or continuous action spaces. The pseudocode is as follows: Input: initial policy parameters θ 0 {\textstyle \theta _{0}} , initial value function parameters ϕ 0 {\textstyle \phi _{0}} Hyperparameters: KL-divergence limit δ {\textstyle \delta } , backtracking coefficient α {\textstyle \alpha } , maximum number of backtracking steps K {\textstyle K} for k = 0 , 1 , 2 , … {\textstyle k=0,1,2,\ldots } do Collect set of trajectories D k = { τ i } {\textstyle {\mathcal {D}}_{k}=\left\{\tau _{i}\right\}} by running policy π k = π ( θ k ) {\textstyle \pi _{k}=\pi \left(\theta _{k}\right)} in the environment. Compute rewards-to-go R ^ t {\textstyle {\hat {R}}_{t}} . Compute advantage estimates, A ^ t {\textstyle {\hat {A}}_{t}} (using any method of advantage estimation) based on the current value function V ϕ k {\textstyle V_{\phi _{k}}} . Estimate policy gradient as g ^ k = 1 | D k | ∑ τ ∈ D k ∑ t = 0 T ∇ θ log ⁡ π θ ( a t ∣ s t ) | θ k A ^ t {\displaystyle {\hat {g}}_{k}=\left.{\frac {1}{\left|{\mathcal {D}}_{k}\right|}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\nabla _{\theta }\log \pi _{\theta }\left(a_{t}\mid s_{t}\right)\right|_{\theta _{k}}{\hat {A}}_{t}} Use the conjugate gradient algorithm to compute x ^ k ≈ H ^ k − 1 g ^ k {\displaystyle {\hat {x}}_{k}\approx {\hat {H}}_{k}^{-1}{\hat {g}}_{k}} where H ^ k {\textstyle {\hat {H}}_{k}} is the Hessian of the sample average KL-divergence. Update the policy by backtracking line search with θ k + 1 = θ k + α j 2 δ x ^ k T H ^ k x ^ k x ^ k {\displaystyle \theta _{k+1}=\theta _{k}+\alpha ^{j}{\sqrt {\frac {2\delta }{{\hat {x}}_{k}^{T}{\hat {H}}_{k}{\hat {x}}_{k}}}}{\hat {x}}_{k}} where j ∈ { 0 , 1 , 2 , … K } {\textstyle j\in \{0,1,2,\ldots K\}} is the smallest value which improves the sample loss and satisfies the sample KL-divergence constraint. Fit value function by regression on mean-squared error: ϕ k + 1 = arg ⁡ min ϕ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T ( V ϕ ( s t ) − R ^ t ) 2 {\displaystyle \phi _{k+1}=\arg \min _{\phi }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\left(V_{\phi }\left(s_{t}\right)-{\hat {R}}_{t}\right)^{2}} typically via some gradient descent algorithm. == PPO == The pseudocode is as follows: Input: initial policy parameters θ 0 {\textstyle \theta _{0}} , initial value function parameters ϕ 0 {\textstyle \phi _{0}} for k = 0 , 1 , 2 , … {\textstyle k=0,1,2,\ldots } do Collect set of trajectories D k = { τ i } {\textstyle {\mathcal {D}}_{k}=\left\{\tau _{i}\right\}} by running policy π k = π ( θ k ) {\textstyle \pi _{k}=\pi \left(\theta _{k}\right)} in the environment. Compute rewards-to-go R ^ t {\textstyle {\hat {R}}_{t}} . Compute advantage estimates, A ^ t {\textstyle {\hat {A}}_{t}} (using any method of advantage estimation) based on the current value function V ϕ k {\textstyle V_{\phi _{k}}} . Update the policy by maximizing the PPO-Clip objective: θ k + 1 = arg ⁡ max θ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T min ( π θ ( a t ∣ s t ) π θ k ( a t ∣ s t ) A π θ k ( s t , a t ) , g ( ϵ , A π θ k ( s t , a t ) ) ) {\displaystyle \theta _{k+1}=\arg \max _{\theta }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\min \left({\frac {\pi _{\theta }\left(a_{t}\mid s_{t}\right)}{\pi _{\theta _{k}}\left(a_{t}\mid s_{t}\right)}}A^{\pi _{\theta _{k}}}\left(s_{t},a_{t}\right),\quad g\left(\epsilon ,A^{\pi _{\theta _{k}}}\left(s_{t},a_{t}\right)\right)\right)} typically via stochastic gradient ascent with Adam. Fit value function by regression on mean-squared error: ϕ k + 1 = arg ⁡ min ϕ 1 | D k | T ∑ τ ∈ D k ∑ t = 0 T ( V ϕ ( s t ) − R ^ t ) 2 {\displaystyle \phi _{k+1}=\arg \min _{\phi }{\frac {1}{\left|{\mathcal {D}}_{k}\right|T}}\sum _{\tau \in {\mathcal {D}}_{k}}\sum _{t=0}^{T}\left(V_{\phi }\left(s_{t}\right)-{\hat {R}}_{t}\right)^{2}} typically via some gradient descent algorithm. Like all policy gradient methods, PPO is used for training an RL agent whose actions are determined by a differentiable policy function by gradient ascent. Intuitively, a policy gradient method takes small policy update steps, so the agent can reach higher and higher rewards in expectation. Policy gradient methods may be unstable: A step size that is too big may direct the policy in a suboptimal direction, thus having little possibility of recovery; a step size that is too small lowers the overall efficiency. To solve the instability, PPO implements a clip function that constrains the policy update of an agent from being too large, so that larger step sizes may be used without negatively affecting the gradient ascent process. === Basic concepts === To begin the PPO training process, the agent is set in an environment to perform actions based on its current input. In the early phase of training, the agent can freely explore solutions and keep track of the result. Later, with a certain amount of transition samples and policy updates, the agent will select an action to take by randomly sampling from the probability distribution P ( A | S ) {\displaystyle P(A|S)} generated by the policy network. The actions that are most likely to be beneficial will have the highest probability of being selected from the random sample. After an agent arrives at a different scenario (a new state) by acting, it is rewarded with a positive reward or a negative reward. The objective of an agent is to maximize the cumulative reward signal across sequences of states, known as episodes. === Policy gradient laws: the advantage function === The advantage function (denoted as A {\displaystyle A} ) is central to PPO, as it tries to answer the question of whether a specific action of the agent is better or worse than some other possible action in a given state. By definition, the advantage function is an estimate of the relative value for a selected action. If the output of this function is positive, it means that the action in question is better than the average return, so the possibilities of selecting that specific action will increase. The opposite is true for a negative advantage output. The advantage function can be defined as A = Q − V {\displaystyle A=Q-V} , where Q {\displaystyle Q} is the discounted sum of rewards (the total weighted reward for the completion of an episode) and V {\displaystyle V} is the baseline estimate. Since the advantage function is calculated after the completion of an episode, the program records the outcome of the episode. Therefore, calculating advantage is essentially an unsupervised learning problem. The baseline estimate comes from the value function that outputs the expected discounted sum of an episode starting from the current state. In the PPO algorithm, the baseline estimate will be noisy (with some variance), as it also uses a neural network, like the policy function itself. With Q {\displaystyle Q} and V {\displaystyle V} computed, the advantage function is calculated by subtracting the baseline estimate from the actual discounted return. If A > 0 {\displaystyle A>0} , the actual return of the action is better than the expected return from experience; if A < 0 {\displaystyle A<0} , the actual return is worse. === Ratio function === In PPO, the ratio function ( r t {\displaystyle r_{t}} ) calculates the probability of selecting action a {\displaystyle a} in state s {\displaystyle s} given the current policy network, divided by the previous probability under the old policy. In other words: If r t ( θ ) > 1 {\displaystyle r_{t}(\theta )>1} , where θ {\displaystyle \theta } are the policy network parameters, then selecting action a {\displaystyle a} in state s {\displaystyle s} is more likely based on the current policy than the previous policy. If 0 ≤ r t ( θ ) < 1 {\displaystyle 0\leq r_{t}(\theta )<1} , then selecting actio

  Read more →
• 

  Instance (computer science)

  In computer science, an instance or token (from metalogic and metamathematics) is a specific occurrence of a software element that is based on a type definition. When created, an occurrence is said to have been instantiated, and both the creation process and the result of creation are called instantiation. == Examples == Chat AI instance In chat-based AI systems, an assistant can be invoked across many independent conversation sessions (often called a thread), each with its own message history. A specific execution of the assistant over that session may be represented as a run (an execution on a thread). Class instance In object-oriented programming, an object created from a class type. Each instance of a class shares the class-defined structure and behavior but has its own identity and state. Procedural instance In some contexts (including Simula), each procedure call can be viewed as an instance of that procedure—an activation with its own parameters and local variables. Computer instance In cloud computing and virtualization, an instance commonly refers to a provisioned virtual machine or virtual server with an allocated combination of compute, memory, network, and storage resources. Polygonal model In computer graphics, a model may be instanced so it can be drawn multiple times with different transforms and parameters, improving performance by reusing shared geometry data. Program instance In a POSIX-oriented operating system, a running process is an instance of a program. It can be instantiated via system calls such as fork() and exec(). Each executing process is an instance of a program it has been instantiated from.

  Read more →
• 

  Stochastic variance reduction

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

  Read more →
• 

  Lattice Miner

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

  Read more →
• 

  Crossover (evolutionary algorithm)

  Crossover in evolutionary algorithms and evolutionary computation, also called recombination, is a genetic operator used to combine the genetic information of two parents to generate new offspring. It is one way to stochastically generate new solutions from an existing population, and is analogous to the crossover that happens during sexual reproduction in biology. New solutions can also be generated by cloning an existing solution, which is analogous to asexual reproduction. Newly generated solutions may be mutated before being added to the population. The aim of recombination is to transfer good characteristics from two different parents to one child. Different algorithms in evolutionary computation may use different data structures to store genetic information, and each genetic representation can be recombined with different crossover operators. Typical data structures that can be recombined with crossover are bit arrays, vectors of real numbers, or trees. The list of operators presented below is by no means complete and serves mainly as an exemplary illustration of this dyadic genetic operator type. More operators and more details can be found in the literature. == Crossover for binary arrays == Traditional genetic algorithms store genetic information in a chromosome represented by a bit array. Crossover methods for bit arrays are popular and an illustrative example of genetic recombination. === One-point crossover === A point on both parents' chromosomes is picked randomly, and designated a 'crossover point'. Bits to the right of that point are swapped between the two parent chromosomes. This results in two offspring, each carrying some genetic information from both parents. === Two-point and k-point crossover === In two-point crossover, two crossover points are picked randomly from the parent chromosomes. The bits in between the two points are swapped between the parent organisms. Two-point crossover is equivalent to performing two single-point crossovers with different crossover points. This strategy can be generalized to k-point crossover for any positive integer k, picking k crossover points. === Uniform crossover === In uniform crossover, typically, each bit is chosen from either parent with equal probability. Other mixing ratios are sometimes used, resulting in offspring which inherit more genetic information from one parent than the other. In a uniform crossover, we don’t divide the chromosome into segments, rather we treat each gene separately. In this, we essentially flip a coin for each chromosome to decide whether or not it will be included in the off-spring. == Crossover for integer or real-valued genomes == For the crossover operators presented above and for most other crossover operators for bit strings, it holds that they can also be applied accordingly to integer or real-valued genomes whose genes each consist of an integer or real-valued number. Instead of individual bits, integer or real-valued numbers are then simply copied into the child genome. The offspring lie on the remaining corners of the hyperbody spanned by the two parents P 1 = ( 1.5 , 6 , 8 ) {\displaystyle P_{1}=(1.5,6,8)} and P 2 = ( 7 , 2 , 1 ) {\displaystyle P_{2}=(7,2,1)} , as exemplified in the accompanying image for the three-dimensional case. === Discrete recombination === If the rules of the uniform crossover for bit strings are applied during the generation of the offspring, this is also called discrete recombination. === Intermediate recombination === In this recombination operator, the allele values of the child genome a i {\displaystyle a_{i}} are generated by mixing the alleles of the two parent genomes a i , P 1 {\displaystyle a_{i,P_{1}}} and a i , P 2 {\displaystyle a_{i,P_{2}}} : α i = α i , P 1 ⋅ β i + α i , P 2 ⋅ ( 1 − β i ) w i t h β i ∈ [ − d , 1 + d ] {\displaystyle \alpha _{i}=\alpha _{i,P_{1}}\cdot \beta _{i}+\alpha _{i,P_{2}}\cdot \left(1-\beta _{i}\right)\quad {\mathsf {with}}\quad \beta _{i}\in \left[-d,1+d\right]} randomly equally distributed per gene i {\displaystyle i} The choice of the interval [ − d , 1 + d ] {\displaystyle [-d,1+d]} causes that besides the interior of the hyperbody spanned by the allele values of the parent genes additionally a certain environment for the range of values of the offspring is in question. A value of 0.25 {\displaystyle 0.25} is recommended for d {\displaystyle d} to counteract the tendency to reduce the allele values that otherwise exists at d = 0 {\displaystyle d=0} . The adjacent figure shows for the two-dimensional case the range of possible new alleles of the two exemplary parents P 1 = ( 3 , 6 ) {\displaystyle P_{1}=(3,6)} and P 2 = ( 9 , 2 ) {\displaystyle P_{2}=(9,2)} in intermediate recombination. The offspring of discrete recombination C 1 {\displaystyle C_{1}} and C 2 {\displaystyle C_{2}} are also plotted. Intermediate recombination satisfies the arithmetic calculation of the allele values of the child genome required by virtual alphabet theory. Discrete and intermediate recombination are used as a standard in the evolution strategy. == Crossover for permutations == For combinatorial tasks, permutations are usually used that are specifically designed for genomes that are themselves permutations of a set. The underlying set is usually a subset of N {\displaystyle \mathbb {N} } or N 0 {\displaystyle \mathbb {N} _{0}} . If 1- or n-point or uniform crossover for integer genomes is used for such genomes, a child genome may contain some values twice and others may be missing. This can be remedied by genetic repair, e.g. by replacing the redundant genes in positional fidelity for missing ones from the other child genome. In order to avoid the generation of invalid offspring, special crossover operators for permutations have been developed which fulfill the basic requirements of such operators for permutations, namely that all elements of the initial permutation are also present in the new one and only the order is changed. It can be distinguished between combinatorial tasks, where all sequences are admissible, and those where there are constraints in the form of inadmissible partial sequences. A well-known representative of the first task type is the traveling salesman problem (TSP), where the goal is to visit a set of cities exactly once on the shortest tour. An example of the constrained task type is the scheduling of multiple workflows. Workflows involve sequence constraints on some of the individual work steps. For example, a thread cannot be cut until the corresponding hole has been drilled in a workpiece. Such problems are also called order-based permutations. In the following, two crossover operators are presented as examples, the partially mapped crossover (PMX) motivated by the TSP and the order crossover (OX1) designed for order-based permutations. A second offspring can be produced in each case by exchanging the parent chromosomes. === Partially mapped crossover (PMX) === The PMX operator was designed as a recombination operator for TSP like Problems. The explanation of the procedure is illustrated by an example: === Order crossover (OX1) === The order crossover goes back to Davis in its original form and is presented here in a slightly generalized version with more than two crossover points. It transfers information about the relative order from the second parent to the offspring. First, the number and position of the crossover points are determined randomly. The resulting gene sequences are then processed as described below: Among other things, order crossover is well suited for scheduling multiple workflows, when used in conjunction with 1- and n-point crossover. === Further crossover operators for permutations === Over time, a large number of crossover operators for permutations have been proposed, so the following list is only a small selection. For more information, the reader is referred to the literature. cycle crossover (CX) order-based crossover (OX2) position-based crossover (POS) edge recombination voting recombination (VR) alternating-positions crossover (AP) maximal preservative crossover (MPX) merge crossover (MX) sequential constructive crossover operator (SCX) The usual approach to solving TSP-like problems by genetic or, more generally, evolutionary algorithms, presented earlier, is either to repair illegal descendants or to adjust the operators appropriately so that illegal offspring do not arise in the first place. Alternatively, Riazi suggests the use of a double chromosome representation, which avoids illegal offspring.

  Read more →
• 

  Procreate (software)

  Procreate is a raster graphics editor app for digital painting developed and published by the Australian company Savage Interactive for iOS and iPadOS. It was launched on the App Store in 2011. == Versions == === Procreate === Procreate for iPad was first released in 2011 by the Tasmanian software company Savage Interactive. In June 2013, Savage launched Procreate 2 in conjunction with iOS 7, adding new features such as higher resolution capabilities and more brush options. In 2016, Procreate became one of the top ten best-selling iPad apps on the App Store. In 2018, Procreate became the overall best selling iPad app. With iOS 26, Procreate adapted Liquid Glass into its software. As of March 2026, the most recent version of Procreate for the iPad is 5.4.9. === Procreate Pocket === Procreate Pocket was released to the App Store in December 2014. In 2018, Savage launched Procreate Pocket 2.0 to the App Store. In December 2018, Procreate Pocket received Apple's "App of the Year" award. As of September 2025, the most recent version of Procreate Pocket (for the iPhone) is 4.0.15. === Procreate Dreams === Procreate Dreams, their more recent app focused on 2D animation, was released on the App Store on November 22, 2023. While the application is commended for its intuitive interface and accessibility, some reviewers have noted that it may lack some key animations features, such as reference layers. In June 2024, Procreate Dreams received the 2024 Apple Design Award for Innovation. In December 2025, Savage Interactive released Procreate Dreams 2, a long awaited update and redesign to Procreate Dreams. == Features == The current versions of Procreate use Valkyrie, a proprietary graphics engine to allow customisable brush options and importing brushes from Adobe Photoshop. Procreate offers known features like layers, masks, and blending mode. Its biggest standout compared to other professional drawing software is its simple UI and comparatively easy learning curve. The app also allows for animation. Savage expanded upon Procreate's animation features with a companion app dedicated to 2D animation called Procreate Dreams, released in November 2023. On August 2024, Procreate announced that it would not be incorporating generative artificial intelligence into its software. Savage offers a free internet forum called Procreate Discussions in which users can ask for help, suggest ideas, and share user-generated content on the marketplace or the resources board. == Notable users == Concept artist Doug Chiang creates robot, vehicle, and creature designs for Star Wars in Procreate. Professional artists have also used Procreate to create the posters for Stranger Things, Logan, and Blade Runner 2049, as well as several covers for The New Yorker. It has also been professionally adopted at Marvel Comics, DC Comics, Disney Animation, and Pixar.

  Read more →
• 

  Multifactor dimensionality reduction

  Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable. MDR was designed specifically to identify nonadditive interactions among discrete variables that influence a binary outcome and is considered a nonparametric and model-free alternative to traditional statistical methods such as logistic regression. The basis of the MDR method is a constructive induction or feature engineering algorithm that converts two or more variables or attributes to a single attribute. This process of constructing a new attribute changes the representation space of the data. The end goal is to create or discover a representation that facilitates the detection of nonlinear or nonadditive interactions among the attributes such that prediction of the class variable is improved over that of the original representation of the data. == Illustrative example == Consider the following simple example using the exclusive OR (XOR) function. XOR is a logical operator that is commonly used in data mining and machine learning as an example of a function that is not linearly separable. The table below represents a simple dataset where the relationship between the attributes (X1 and X2) and the class variable (Y) is defined by the XOR function such that Y = X1 XOR X2. Table 1 A machine learning algorithm would need to discover or approximate the XOR function in order to accurately predict Y using information about X1 and X2. An alternative strategy would be to first change the representation of the data using constructive induction to facilitate predictive modeling. The MDR algorithm would change the representation of the data (X1 and X2) in the following manner. MDR starts by selecting two attributes. In this simple example, X1 and X2 are selected. Each combination of values for X1 and X2 are examined and the number of times Y=1 and/or Y=0 is counted. In this simple example, Y=1 occurs zero times and Y=0 occurs once for the combination of X1=0 and X2=0. With MDR, the ratio of these counts is computed and compared to a fixed threshold. Here, the ratio of counts is 0/1 which is less than our fixed threshold of 1. Since 0/1 < 1 we encode a new attribute (Z) as a 0. When the ratio is greater than one we encode Z as a 1. This process is repeated for all unique combinations of values for X1 and X2. Table 2 illustrates our new transformation of the data. Table 2 The machine learning algorithm now has much less work to do to find a good predictive function. In fact, in this very simple example, the function Y = Z has a classification accuracy of 1. A nice feature of constructive induction methods such as MDR is the ability to use any data mining or machine learning method to analyze the new representation of the data. Decision trees, neural networks, or a naive Bayes classifier could be used in combination with measures of model quality such as balanced accuracy and mutual information. == Machine learning with MDR == As illustrated above, the basic constructive induction algorithm in MDR is very simple. However, its implementation for mining patterns from real data can be computationally complex. As with any machine learning algorithm there is always concern about overfitting. That is, machine learning algorithms are good at finding patterns in completely random data. It is often difficult to determine whether a reported pattern is an important signal or just chance. One approach is to estimate the generalizability of a model to independent datasets using methods such as cross-validation. Models that describe random data typically don't generalize. Another approach is to generate many random permutations of the data to see what the data mining algorithm finds when given the chance to overfit. Permutation testing makes it possible to generate an empirical p-value for the result. Replication in independent data may also provide evidence for an MDR model but can be sensitive to difference in the data sets. These approaches have all been shown to be useful for choosing and evaluating MDR models. An important step in a machine learning exercise is interpretation. Several approaches have been used with MDR including entropy analysis and pathway analysis. Tips and approaches for using MDR to model gene-gene interactions have been reviewed. == Extensions to MDR == Numerous extensions to MDR have been introduced. These include family-based methods, fuzzy methods, covariate adjustment, odds ratios, risk scores, survival methods, robust methods, methods for quantitative traits, and many others. == Applications of MDR == MDR has mostly been applied to detecting gene-gene interactions or epistasis in genetic studies of common human diseases such as atrial fibrillation, autism, bladder cancer, breast cancer, cardiovascular disease, hypertension, obesity, pancreatic cancer, prostate cancer and tuberculosis. It has also been applied to other biomedical problems such as the genetic analysis of pharmacology outcomes. A central challenge is the scaling of MDR to big data such as that from genome-wide association studies (GWAS). Several approaches have been used. One approach is to filter the features prior to MDR analysis. This can be done using biological knowledge through tools such as BioFilter. It can also be done using computational tools such as ReliefF. Another approach is to use stochastic search algorithms such as genetic programming to explore the search space of feature combinations. Yet another approach is a brute-force search using high-performance computing. == Implementations == www.epistasis.org provides an open-source and freely-available MDR software package. An R package for MDR. An sklearn-compatible Python implementation. An R package for Model-Based MDR. MDR in Weka. Generalized MDR.

  Read more →
• 

  Unique negative dimension

  Unique negative dimension (UND) is a complexity measure for the model of learning from positive examples. The unique negative dimension of a class C {\displaystyle C} of concepts is the size of the maximum subclass D ⊆ C {\displaystyle D\subseteq C} such that for every concept c ∈ D {\displaystyle c\in D} , we have ∩ ( D ∖ { c } ) ∖ c {\displaystyle \cap (D\setminus \{c\})\setminus c} is nonempty. This concept was originally proposed by M. Gereb-Graus in "Complexity of learning from one-side examples", Technical Report TR-20-89, Harvard University Division of Engineering and Applied Science, 1989.

  Read more →