AI Chatbot Robot

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  Device-independent pixel

  A device-independent pixel (also: density-independent pixel, dip, dp) is a unit of length. A typical use is to allow mobile device software to scale the display of information and user interaction to different screen sizes. The abstraction allows an application to work in pixels as a measurement, while the underlying graphics system converts the abstract pixel measurements of the application into real pixel measurements appropriate to the particular device. For example, on the Android operating system a device-independent pixel is equivalent to one physical pixel on a 160 dpi screen, while the Windows Presentation Foundation specifies one device-independent pixel as equivalent to 1/96th of an inch. As dp is a physical unit it has an absolute value which can be measured in traditional units, e.g. for Android devices 1 dp equals 1/160 of inch or 0.15875 mm. While traditional pixels only refer to the display of information, device-independent pixels may also be used to measure user input such as input on a touch screen device.

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  Stochastic gradient descent

  Stochastic gradient descent (often abbreviated SGD) is an iterative method for optimizing an objective function with suitable smoothness properties (e.g. differentiable or subdifferentiable). It can be regarded as a stochastic approximation of gradient descent optimization, since it replaces the actual gradient (calculated from the entire data set) by an estimate thereof (calculated from a randomly selected subset of the data). Especially in high-dimensional optimization problems this reduces the very high computational burden, achieving faster iterations in exchange for a lower convergence rate. The basic idea behind stochastic approximation can be traced back to the Robbins–Monro algorithm of the 1950s. Today, stochastic gradient descent has become an important optimization method in machine learning. == Background == Both statistical estimation and machine learning consider the problem of minimizing an objective function that has the form of a sum: Q ( w ) = 1 n ∑ i = 1 n Q i ( w ) , {\displaystyle Q(w)={\frac {1}{n}}\sum _{i=1}^{n}Q_{i}(w),} where the parameter w {\displaystyle w} that minimizes Q ( w ) {\displaystyle Q(w)} is to be estimated. Each summand function Q i {\displaystyle Q_{i}} is typically associated with the i {\displaystyle i} -th observation in the data set (used for training). In classical statistics, sum-minimization problems arise in least squares and in maximum-likelihood estimation (for independent observations). The general class of estimators that arise as minimizers of sums are called M-estimators. However, in statistics, it has been long recognized that requiring even local minimization is too restrictive for some problems of maximum-likelihood estimation. Therefore, contemporary statistical theorists often consider stationary points of the likelihood function (or zeros of its derivative, the score function, and other estimating equations). The sum-minimization problem also arises for empirical risk minimization. There, Q i ( w ) {\displaystyle Q_{i}(w)} is the value of the loss function at i {\displaystyle i} -th example, and Q ( w ) {\displaystyle Q(w)} is the empirical risk. When used to minimize the above function, a standard (or "batch") gradient descent method would perform the following iterations: w := w − η ∇ Q ( w ) = w − η n ∑ i = 1 n ∇ Q i ( w ) . {\displaystyle w:=w-\eta \,\nabla Q(w)=w-{\frac {\eta }{n}}\sum _{i=1}^{n}\nabla Q_{i}(w).} The step size is denoted by η {\displaystyle \eta } (sometimes called the learning rate in machine learning) and here " := {\displaystyle :=} " denotes the update of a variable in the algorithm. In many cases, the summand functions have a simple form that enables inexpensive evaluations of the sum-function and the sum gradient. For example, in statistics, one-parameter exponential families allow economical function-evaluations and gradient-evaluations. However, in other cases, evaluating the sum-gradient may require expensive evaluations of the gradients from all summand functions. When the training set is enormous and no simple formulas exist, evaluating the sums of gradients becomes very expensive, because evaluating the gradient requires evaluating all the summand functions' gradients. To economize on the computational cost at every iteration, stochastic gradient descent samples a subset of summand functions at every step. This is very effective in the case of large-scale machine learning problems. == Iterative method == In stochastic (or "on-line") gradient descent, the true gradient of Q ( w ) {\displaystyle Q(w)} is approximated by a gradient at a single sample: w := w − η ∇ Q i ( w ) . {\displaystyle w:=w-\eta \,\nabla Q_{i}(w).} As the algorithm sweeps through the training set, it performs the above update for each training sample. Several passes can be made over the training set until the algorithm converges. If this is done, the data can be shuffled for each pass to prevent cycles. Typical implementations may use an adaptive learning rate so that the algorithm converges. In pseudocode, stochastic gradient descent can be presented as : A compromise between computing the true gradient and the gradient at a single sample is to compute the gradient against more than one training sample (called a "mini-batch") at each step. This can perform significantly better than "true" stochastic gradient descent described, because the code can make use of vectorization libraries rather than computing each step separately as was first shown in where it was called "the bunch-mode back-propagation algorithm". It may also result in smoother convergence, as the gradient computed at each step is averaged over more training samples. The convergence of stochastic gradient descent has been analyzed using the theories of convex minimization and of stochastic approximation. Briefly, when the learning rates η {\displaystyle \eta } decrease with an appropriate rate, and subject to relatively mild assumptions, stochastic gradient descent converges almost surely to a global minimum when the objective function is convex or pseudoconvex, and otherwise converges almost surely to a local minimum. This is in fact a consequence of the Robbins–Siegmund theorem. == Linear regression == Suppose we want to fit a straight line y ^ = w 1 + w 2 x {\displaystyle {\hat {y}}=w_{1}+w_{2}x} to a training set with observations ( ( x 1 , y 1 ) , ( x 2 , y 2 ) … , ( x n , y n ) ) {\displaystyle ((x_{1},y_{1}),(x_{2},y_{2})\ldots ,(x_{n},y_{n}))} and corresponding estimated responses ( y ^ 1 , y ^ 2 , … , y ^ n ) {\displaystyle ({\hat {y}}_{1},{\hat {y}}_{2},\ldots ,{\hat {y}}_{n})} using least squares. The objective function to be minimized is Q ( w ) = ∑ i = 1 n Q i ( w ) = ∑ i = 1 n ( y ^ i − y i ) 2 = ∑ i = 1 n ( w 1 + w 2 x i − y i ) 2 . {\displaystyle Q(w)=\sum _{i=1}^{n}Q_{i}(w)=\sum _{i=1}^{n}\left({\hat {y}}_{i}-y_{i}\right)^{2}=\sum _{i=1}^{n}\left(w_{1}+w_{2}x_{i}-y_{i}\right)^{2}.} The last line in the above pseudocode for this specific problem will become: [ w 1 w 2 ] ← [ w 1 w 2 ] − η [ ∂ ∂ w 1 ( w 1 + w 2 x i − y i ) 2 ∂ ∂ w 2 ( w 1 + w 2 x i − y i ) 2 ] = [ w 1 w 2 ] − η [ 2 ( w 1 + w 2 x i − y i ) 2 x i ( w 1 + w 2 x i − y i ) ] . {\displaystyle {\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}\leftarrow {\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}-\eta {\begin{bmatrix}{\frac {\partial }{\partial w_{1}}}(w_{1}+w_{2}x_{i}-y_{i})^{2}\\{\frac {\partial }{\partial w_{2}}}(w_{1}+w_{2}x_{i}-y_{i})^{2}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\end{bmatrix}}-\eta {\begin{bmatrix}2(w_{1}+w_{2}x_{i}-y_{i})\\2x_{i}(w_{1}+w_{2}x_{i}-y_{i})\end{bmatrix}}.} Note that in each iteration or update step, the gradient is only evaluated at a single x i {\displaystyle x_{i}} . This is the key difference between stochastic gradient descent and batched gradient descent. In general, given a linear regression y ^ = ∑ k ∈ 1 : m w k x k {\displaystyle {\hat {y}}=\sum _{k\in 1:m}w_{k}x_{k}} problem, stochastic gradient descent behaves differently when m < n {\displaystyle m

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  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

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  Cultural algorithm

  Cultural algorithms (CA) are a branch of evolutionary computation where there is a knowledge component that is called the belief space in addition to the population component. In this sense, cultural algorithms can be seen as an extension to a conventional genetic algorithm. Cultural algorithms were introduced by Reynolds (see references). == Belief space == The belief space of a cultural algorithm is divided into distinct categories. These categories represent different domains of knowledge that the population has of the search space. The belief space is updated after each iteration by the best individuals of the population. The best individuals can be selected using a fitness function that assesses the performance of each individual in population much like in genetic algorithms. === List of belief space categories === Normative knowledge A collection of desirable value ranges for the individuals in the population component e.g. acceptable behavior for the agents in population. Domain specific knowledge Information about the domain of the cultural algorithm problem is applied to. Situational knowledge Specific examples of important events - e.g. successful/unsuccessful solutions Temporal knowledge History of the search space - e.g. the temporal patterns of the search process Spatial knowledge Information about the topography of the search space == Population == The population component of the cultural algorithm is approximately the same as that of the genetic algorithm. == Communication protocol == Cultural algorithms require an interface between the population and belief space. The best individuals of the population can update the belief space via the update function. Also, the knowledge categories of the belief space can affect the population component via the influence function. The influence function can affect population by altering the genome or the actions of the individuals. == Pseudocode for cultural algorithms == Initialize population space (choose initial population) Initialize belief space (e.g. set domain specific knowledge and normative value-ranges) Repeat until termination condition is met Perform actions of the individuals in population space Evaluate each individual by using the fitness function Select the parents to reproduce a new generation of offspring Let the belief space alter the genome of the offspring by using the influence function Update the belief space by using the accept function (this is done by letting the best individuals to affect the belief space) == Applications == Various optimization problems Social simulation Real-parameter optimization

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  Autonomous agent

  An autonomous agent is an artificial intelligence (AI) system that can perform complex tasks independently. == Definitions == There are various definitions of autonomous agent. According to Brustoloni (1991): "Autonomous agents are systems capable of autonomous, purposeful action in the real world." According to Maes (1995): "Autonomous agents are computational systems that inhabit some complex dynamic environment, sense and act autonomously in this environment, and by doing so realize a set of goals or tasks for which they are designed." Franklin and Graesser (1997) review different definitions and propose their definition: "An autonomous agent is a system situated within and a part of an environment that senses that environment and acts on it, over time, in pursuit of its own agenda and so as to effect what it senses in the future." They explain that: "Humans and some animals are at the high end of being an agent, with multiple, conflicting drives, multiples senses, multiple possible actions, and complex sophisticated control structures. At the low end, with one or two senses, a single action, and an absurdly simple control structure we find a thermostat." == Agent appearance == Lee et al. (2015) post safety issue from how the combination of external appearance and internal autonomous agent have impact on human reaction about autonomous vehicles. Their study explores the human-like appearance agent and high level of autonomy are strongly correlated with social presence, intelligence, safety and trustworthiness. In specific, appearance impacts most on affective trust while autonomy impacts most on both affective and cognitive domain of trust where cognitive trust is characterized by knowledge-based factors and affective trust is largely emotion driven. == Applications == Agentic AI systems: Advanced AI agents that can scope out projects and complete them with necessary tools, representing a significant evolution from simple task-oriented systems. Internet of things (IoT) Integration: Autonomous agents increasingly interact with IoT devices, enabling smart home systems, industrial monitoring, and urban infrastructure management. Collaborative software development: Tools like Cognition AI's Devin aim to create autonomous software engineers capable of complex reasoning, planning, and completing engineering tasks requiring thousands of decisions. Enterprise automation: Business process automation platforms like Salesforce's Agentforce provide autonomous bots for various service functions. == Challenges and considerations == Uncertainty and incomplete information: Autonomous agents must make decisions with limited or uncertain information about their environment and future states. Integration complexity: Incorporating autonomous agents into existing systems and workflows can be technically challenging and resource-intensive. Scalability: As systems become more complex and more agents are used, maintaining coordination and avoiding conflicts becomes increasingly difficult. Trust: Research has shown the combination of external appearance and internal autonomous capabilities significantly impacts human reactions and trust. Lee et al. (2015) found that human-like appearance and high levels of autonomy are strongly correlated with social presence, intelligence, safety, and trustworthiness perceptions. Specifically, appearance impacts affective trust most significantly, while autonomy affects both affective and cognitive trust domains, where affective trust is emotionally driven, and cognitive trust is characterized by knowledge-based factors. Vulnerability to manipulation: Researchers from Harvard, MIT and other educational institutions found that AI agents could become vulnerable to manipulation and could perform detrimental actions in the process of being helpful. == Ethical and regulatory concerns == Accountability: Determining responsibility when autonomous agents make incorrect or harmful decisions remains a complex issue. Privacy and security: autonomous agents often require access to sensitive data, raising concerns about data protection and system security.

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  Ordination (statistics)

  Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space. In the ordination space, quantities that are near each other share attributes (i.e., are similar to some degree), and dissimilar objects are farther from each other. Such relationships between the objects, on each of several axes or latent variables, are then characterized numerically and/or graphically in a biplot. The first ordination method, principal components analysis, was suggested by Karl Pearson in 1901. == Methods == Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analysis, and redundancy analysis, belong to the first group). The second group includes some distance-based methods such as non-metric multidimensional scaling, and machine learning methods such as T-distributed stochastic neighbor embedding and nonlinear dimensionality reduction. The third group includes model-based ordination methods, which can be considered as multivariate extensions of Generalized Linear Models. Model-based ordination methods are more flexible in their application than classical ordination methods, so that it is for example possible to include random-effects. Unlike in the aforementioned two groups, there is no (implicit or explicit) distance measure in the ordination. Instead, a distribution needs to be specified for the responses as is typical for statistical models. These and other assumptions, such as the assumed mean-variance relationship, can be validated with the use of residual diagnostics, unlike in other ordination methods. == Applications == Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics.

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  Modern Hopfield network

  Modern Hopfield networks (also known as Dense Associative Memories) are generalizations of the classical Hopfield networks that break the linear scaling relationship between the number of input features and the number of stored memories. This is achieved by introducing stronger non-linearities (either in the energy function or neurons’ activation functions) leading to super-linear (even an exponential) memory storage capacity as a function of the number of feature neurons. The network still requires a sufficient number of hidden neurons. The key theoretical idea behind the modern Hopfield networks is to use an energy function and an update rule that is more sharply peaked around the stored memories in the space of neuron’s configurations compared to the classical Hopfield network. == Classical Hopfield networks == Hopfield networks are recurrent neural networks with dynamical trajectories converging to fixed point attractor states and described by an energy function. The state of each model neuron i {\textstyle i} is defined by a time-dependent variable V i {\displaystyle V_{i}} , which can be chosen to be either discrete or continuous. A complete model describes the mathematics of how the future state of activity of each neuron depends on the known present or previous activity of all the neurons. In the original Hopfield model of associative memory, the variables were binary, and the dynamics were described by a one-at-a-time update of the state of the neurons. An energy function quadratic in the V i {\displaystyle V_{i}} was defined, and the dynamics consisted of changing the activity of each single neuron i {\displaystyle i} only if doing so would lower the total energy of the system. This same idea was extended to the case of V i {\displaystyle V_{i}} being a continuous variable representing the output of neuron i {\displaystyle i} , and V i {\displaystyle V_{i}} being a monotonic function of an input current. The dynamics became expressed as a set of first-order differential equations for which the "energy" of the system always decreased. The energy in the continuous case has one term which is quadratic in the V i {\displaystyle V_{i}} (as in the binary model), and a second term which depends on the gain function (neuron's activation function). While having many desirable properties of associative memory, both of these classical systems suffer from a small memory storage capacity, which scales linearly with the number of input features. == Discrete variables == A simple example of the Modern Hopfield network can be written in terms of binary variables V i {\displaystyle V_{i}} that represent the active V i = + 1 {\displaystyle V_{i}=+1} and inactive V i = − 1 {\displaystyle V_{i}=-1} state of the model neuron i {\displaystyle i} . E = − ∑ μ = 1 N mem F ( ∑ i = 1 N f ξ μ i V i ) {\displaystyle E=-\sum \limits _{\mu =1}^{N_{\text{mem}}}F{\Big (}\sum \limits _{i=1}^{N_{f}}\xi _{\mu i}V_{i}{\Big )}} In this formula the weights ξ μ i {\textstyle \xi _{\mu i}} represent the matrix of memory vectors (index μ = 1... N mem {\displaystyle \mu =1...N_{\text{mem}}} enumerates different memories, and index i = 1... N f {\displaystyle i=1...N_{f}} enumerates the content of each memory corresponding to the i {\displaystyle i} -th feature neuron), and the function F ( x ) {\displaystyle F(x)} is a rapidly growing non-linear function. The update rule for individual neurons (in the asynchronous case) can be written in the following form V i ( t + 1 ) = sign ⁡ [ ∑ μ = 1 N mem ( F ( ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) − F ( − ξ μ i + ∑ j ≠ i ξ μ j V j ( t ) ) ) ] {\displaystyle V_{i}^{(t+1)}=\operatorname {sign} {\bigg [}\sum \limits _{\mu =1}^{N_{\text{mem}}}{\bigg (}F{\Big (}\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}-F{\Big (}-\xi _{\mu i}+\sum \limits _{j\neq i}\xi _{\mu j}V_{j}^{(t)}{\Big )}{\bigg )}{\bigg ]}} which states that in order to calculate the updated state of the i {\textstyle i} -th neuron the network compares two energies: the energy of the network with the i {\displaystyle i} -th neuron in the ON state and the energy of the network with the i {\displaystyle i} -th neuron in the OFF state, given the states of the remaining neuron. The updated state of the i {\displaystyle i} -th neuron selects the state that has the lowest of the two energies. In the limiting case when the non-linear energy function is quadratic F ( x ) = x 2 {\displaystyle F(x)=x^{2}} these equations reduce to the familiar energy function and the update rule for the classical binary Hopfield network. The memory storage capacity of these networks can be calculated for random binary patterns. For the power energy function F ( x ) = x n {\displaystyle F(x)=x^{n}} the maximal number of memories that can be stored and retrieved from this network without errors is given by N mem max ≈ 1 2 ( 2 n − 3 ) ! ! N f n − 1 ln ⁡ ( N f ) {\displaystyle N_{\text{mem}}^{\max }\approx {\frac {1}{2(2n-3)!!}}{\frac {N_{f}^{n-1}}{\ln(N_{f})}}} For an exponential energy function F ( x ) = e x {\textstyle F(x)=e^{x}} the memory storage capacity is exponential in the number of feature neurons N mem max ≈ 2 N f / 2 {\displaystyle N_{\text{mem}}^{\max }\approx 2^{N_{f}/2}} == Continuous variables == Modern Hopfield networks or Dense Associative Memories can be best understood in continuous variables and continuous time. Consider the network architecture, shown in Fig.1, and the equations for the neurons' state evolutionwhere the currents of the feature neurons are denoted by x i {\textstyle x_{i}} , and the currents of the memory neurons are denoted by h μ {\displaystyle h_{\mu }} ( h {\displaystyle h} stands for hidden neurons). There are no synaptic connections among the feature neurons or the memory neurons. A matrix ξ μ i {\displaystyle \xi _{\mu i}} denotes the strength of synapses from a feature neuron i {\displaystyle i} to the memory neuron μ {\displaystyle \mu } . The synapses are assumed to be symmetric, so that the same value characterizes a different physical synapse from the memory neuron μ {\displaystyle \mu } to the feature neuron i {\displaystyle i} . The outputs of the memory neurons and the feature neurons are denoted by f μ {\displaystyle f_{\mu }} and g i {\displaystyle g_{i}} , which are non-linear functions of the corresponding currents. In general these outputs can depend on the currents of all the neurons in that layer so that f μ = f ( { h μ } ) {\displaystyle f_{\mu }=f(\{h_{\mu }\})} and g i = g ( { x i } ) {\textstyle g_{i}=g(\{x_{i}\})} . It is convenient to define these activation function as derivatives of the Lagrangian functions for the two groups of neuronsThis way the specific form of the equations for neuron's states is completely defined once the Lagrangian functions are specified. Finally, the time constants for the two groups of neurons are denoted by τ f {\displaystyle \tau _{f}} and τ h {\displaystyle \tau _{h}} , I i {\displaystyle I_{i}} is the input current to the network that can be driven by the presented data. General systems of non-linear differential equations can have many complicated behaviors that can depend on the choice of the non-linearities and the initial conditions. For Hopfield networks, however, this is not the case - the dynamical trajectories always converge to a fixed point attractor state. This property is achieved because these equations are specifically engineered so that they have an underlying energy function The terms grouped into square brackets represent a Legendre transform of the Lagrangian function with respect to the states of the neurons. If the Hessian matrices of the Lagrangian functions are positive semi-definite, the energy function is guaranteed to decrease on the dynamical trajectory This property makes it possible to prove that the system of dynamical equations describing temporal evolution of neurons' activities will eventually reach a fixed point attractor state. In certain situations one can assume that the dynamics of hidden neurons equilibrates at a much faster time scale compared to the feature neurons, τ h ≪ τ f {\textstyle \tau _{h}\ll \tau _{f}} . In this case the steady state solution of the second equation in the system (1) can be used to express the currents of the hidden units through the outputs of the feature neurons. This makes it possible to reduce the general theory (1) to an effective theory for feature neurons only. The resulting effective update rules and the energies for various common choices of the Lagrangian functions are shown in Fig.2. In the case of log-sum-exponential Lagrangian function the update rule (if applied once) for the states of the feature neurons is the attention mechanism commonly used in many modern AI systems (see Ref. for the derivation of this result from the continuous time formulation). == Relationship to classical Hopfield network with continuous variables == Classical formulation of continuous Hopfield networks can be understood as a

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  Multispectral pattern recognition

  Multispectral remote sensing is the collection and analysis of reflected, emitted, or back-scattered energy from an object or an area of interest in multiple bands of regions of the electromagnetic spectrum (Jensen, 2005). Subcategories of multispectral remote sensing include hyperspectral, in which hundreds of bands are collected and analyzed, and ultraspectral remote sensing where many hundreds of bands are used (Logicon, 1997). The main purpose of multispectral imaging is the potential to classify the image using multispectral classification. This is a much faster method of image analysis than is possible by human interpretation. == Multispectral remote sensing systems == Remote sensing systems gather data via instruments typically carried on satellites in orbit around the Earth. The remote sensing scanner detects the energy that radiates from the object or area of interest. This energy is recorded as an analog electrical signal and converted into a digital value though an A-to-D conversion. There are several multispectral remote sensing systems that can be categorized in the following way: === Multispectral imaging using discrete detectors and scanning mirrors === Landsat Multispectral Scanner (MSS) Landsat Thematic Mapper (TM) NOAA Geostationary Operational Environmental Satellite (GOES) NOAA Advanced Very High Resolution Radiometer (AVHRR) NASA and ORBIMAGE, Inc., Sea-viewing Wide field-of-view Sensor (SeaWiFS) Daedalus, Inc., Aircraft Multispectral Scanner (AMS) NASA Airborne Terrestrial Applications Sensor (ATLAS) === Multispectral imaging using linear arrays === SPOT 1, 2, and 3 High Resolution Visible (HRV) sensors and Spot 4 and 5 High Resolution Visible Infrared (HRVIR) and vegetation sensor Indian Remote Sensing System (IRS) Linear Imaging Self-scanning Sensor (LISS) Space Imaging, Inc. (IKONOS) Digital Globe, Inc. (QuickBird) ORBIMAGE, Inc. (OrbView-3) ImageSat International, Inc. (EROS A1) NASA Terra Advanced Spaceborne Thermal Emission and Reflection Radiometer (ASTER) NASA Terra Multiangle Imaging Spectroradiometer (MISR) === Imaging spectrometry using linear and area arrays === NASA Jet Propulsion Laboratory Airborne Visible/Infrared Imaging Spectrometer (AVIRIS) Compact Airborne Spectrographic Imager 3 (CASI 3) NASA Terra Moderate Resolution Imaging Spectrometer (MODIS) NASA Earth Observer (EO-1) Advanced Land Imager (ALI), Hyperion, and LEISA Atmospheric Corrector (LAC) === Satellite analog and digital photographic systems === Russian SPIN-2 TK-350, and KVR-1000 NASA Space Shuttle and International Space Station Imagery == Multispectral classification methods == A variety of methods can be used for the multispectral classification of images: Algorithms based on parametric and nonparametric statistics that use ratio-and interval-scaled data and nonmetric methods that can also incorporate nominal scale data (Duda et al., 2001), Supervised or unsupervised classification logic, Hard or soft (fuzzy) set classification logic to create hard or fuzzy thematic output products, Per-pixel or object-oriented classification logic, and Hybrid approaches == Supervised classification == In this classification method, the identity and location of some of the land-cover types are obtained beforehand from a combination of fieldwork, interpretation of aerial photography, map analysis, and personal experience. The analyst would locate sites that have similar characteristics to the known land-cover types. These areas are known as training sites because the known characteristics of these sites are used to train the classification algorithm for eventual land-cover mapping of the remainder of the image. Multivariate statistical parameters (means, standard deviations, covariance matrices, correlation matrices, etc.) are calculated for each training site. All pixels inside and outside of the training sites are evaluated and allocated to the class with the more similar characteristics. === Classification scheme === The first step in the supervised classification method is to identify the land-cover and land-use classes to be used. Land-cover refers to the type of material present on the site (e.g. water, crops, forest, wet land, asphalt, and concrete). Land-use refers to the modifications made by people to the land cover (e.g. agriculture, commerce, settlement). All classes should be selected and defined carefully to properly classify remotely sensed data into the correct land-use and/or land-cover information. To achieve this purpose, it is necessary to use a classification system that contains taxonomically correct definitions of classes. If a hard classification is desired, the following classes should be used: Mutually exclusive: there is not any taxonomic overlap of any classes (i.e., rain forest and evergreen forest are distinct classes). Exhaustive: all land-covers in the area have been included. Hierarchical: sub-level classes (e.g., single-family residential, multiple-family residential) are created, allowing that these classes can be included in a higher category (e.g., residential). Some examples of hard classification schemes are: American Planning Association Land-Based Classification System United States Geological Survey Land-use/Land-cover Classification System for Use with Remote Sensor Data U.S. Department of the Interior Fish and Wildlife Service U.S. National Vegetation and Classification System International Geosphere-Biosphere Program IGBP Land Cover Classification System === Training sites === Once the classification scheme is adopted, the image analyst may select training sites in the image that are representative of the land-cover or land-use of interest. If the environment where the data was collected is relatively homogeneous, the training data can be used. If different conditions are found in the site, it would not be possible to extend the remote sensing training data to the site. To solve this problem, a geographical stratification should be done during the preliminary stages of the project. All differences should be recorded (e.g. soil type, water turbidity, crop species, etc.). These differences should be recorded on the imagery and the selection training sites made based on the geographical stratification of this data. The final classification map would be a composite of the individual stratum classifications. After the data are organized in different training sites, a measurement vector is created. This vector would contain the brightness values for each pixel in each band in each training class. The mean, standard deviation, variance-covariance matrix, and correlation matrix are calculated from the measurement vectors. Once the statistics from each training site are determined, the most effective bands for each class should be selected. The objective of this discrimination is to eliminate the bands that can provide redundant information. Graphical and statistical methods can be used to achieve this objective. Some of the graphic methods are: Bar graph spectral plots Cospectral mean vector plots Feature space plots Cospectral parallelepiped or ellipse plots === Classification algorithm === The last step in supervised classification is selecting an appropriate algorithm. The choice of a specific algorithm depends on the input data and the desired output. Parametric algorithms are based on the fact that the data is normally distributed. If the data is not normally distributed, nonparametric algorithms should be used. The more common nonparametric algorithms are: One-dimensional density slicing Parallelipiped Minimum distance Nearest-neighbor Expert system analysis Convolutional neural network == Unsupervised classification == Unsupervised classification (also known as clustering) is a method of partitioning remote sensor image data in multispectral feature space and extracting land-cover information. Unsupervised classification require less input information from the analyst compared to supervised classification because clustering does not require training data. This process consists in a series of numerical operations to search for the spectral properties of pixels. From this process, a map with m spectral classes is obtained. Using the map, the analyst tries to assign or transform the spectral classes into thematic information of interest (i.e. forest, agriculture, urban). This process may not be easy because some spectral clusters represent mixed classes of surface materials and may not be useful. The analyst has to understand the spectral characteristics of the terrain to be able to label clusters as a specific information class. There are hundreds of clustering algorithms. Two of the most conceptually simple algorithms are the chain method and the ISODATA method. === Chain method === The algorithm used in this method operates in a two-pass mode (it passes through the multispectral dataset two times. In the first pass, the program reads through the dataset and sequentially builds clusters (groups of p

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  ImageNet

  The ImageNet project is a large visual database designed for use in visual object recognition software research. More than 14 million images have been hand-annotated by the project to indicate what objects are pictured and in at least one million of the images, bounding boxes are also provided. ImageNet contains more than 20,000 categories, with a typical category, such as "balloon" or "strawberry", consisting of several hundred images. The database of annotations of third-party image URLs is freely available directly from ImageNet, though the actual images are not owned by ImageNet. Since 2010, the ImageNet project runs an annual software contest, the ImageNet Large Scale Visual Recognition Challenge (ILSVRC), where software programs compete to correctly classify and detect objects and scenes. The challenge uses a "trimmed" list of one thousand non-overlapping classes. == History == AI researcher Fei-Fei Li began working on the idea for ImageNet in 2006. At a time when most AI research focused on models and algorithms, Li wanted to expand and improve the data available to train AI algorithms. In 2007, Li met with Princeton professor Christiane Fellbaum, one of the creators of WordNet, to discuss the project. As a result of this meeting, Li went on to build ImageNet starting from the roughly 22,000 nouns of WordNet and using many of its features. She was also inspired by a 1987 estimate that the average person recognizes roughly 30,000 different kinds of objects. As an assistant professor at Princeton, Li assembled a team of researchers to work on the ImageNet project. They used Amazon Mechanical Turk to help with the classification of images. Labeling started in July 2008 and ended in April 2010. It took 49K workers from 167 countries filtering and labeling over 160M candidate images. They had enough budget to have each of the 14 million images labelled three times. The original plan called for 10,000 images per category, for 40,000 categories at 400 million images, each verified 3 times. They found that humans can classify at most 2 images/sec. At this rate, it was estimated to take 19 human-years of labor (without rest). They presented their database for the first time as a poster at the 2009 Conference on Computer Vision and Pattern Recognition (CVPR) in Florida, titled "ImageNet: A Preview of a Large-scale Hierarchical Dataset". The poster was reused at Vision Sciences Society 2009. In 2009, Alex Berg suggested adding object localization as a task. Li approached PASCAL Visual Object Classes contest in 2009 for a collaboration. It resulted in the subsequent ImageNet Large Scale Visual Recognition Challenge starting in 2010, which has 1000 classes and object localization, as compared to PASCAL VOC which had just 20 classes and 19,737 images (in 2010). === Significance for deep learning === On 30 September 2012, a convolutional neural network (CNN) called AlexNet achieved a top-5 error of 15.3% in the ImageNet 2012 Challenge, more than 10.8 percentage points lower than that of the runner-up. Using convolutional neural networks was feasible due to the use of graphics processing units (GPUs) during training, an essential ingredient of the deep learning revolution. According to The Economist, "Suddenly people started to pay attention, not just within the AI community but across the technology industry as a whole." In 2015, AlexNet was outperformed by Microsoft's very deep CNN with over 100 layers, which won the ImageNet 2015 contest, having 3.57% error on the test set. Andrej Karpathy estimated in 2014 that with concentrated effort, he could reach 5.1% error rate, and ~10 people from his lab reached ~12-13% with less effort. It was estimated that with maximal effort, a human could reach 2.4%. == Dataset == ImageNet crowdsources its annotation process. Image-level annotations indicate the presence or absence of an object class in an image, such as "there are tigers in this image" or "there are no tigers in this image". Object-level annotations provide a bounding box around the (visible part of the) indicated object. ImageNet uses a variant of the broad WordNet schema to categorize objects, augmented with 120 categories of dog breeds to showcase fine-grained classification. In 2012, ImageNet was the world's largest academic user of Mechanical Turk. The average worker identified 50 images per minute. The original plan of the full ImageNet would have roughly 50M clean, diverse and full resolution images spread over approximately 50K synsets. This was not achieved. The summary statistics given on April 30, 2010: Total number of non-empty synsets: 21841 Total number of images: 14,197,122 Number of images with bounding box annotations: 1,034,908 Number of synsets with SIFT features: 1000 Number of images with SIFT features: 1.2 million === Categories === The categories of ImageNet were filtered from the WordNet concepts. Each concept, since it can contain multiple synonyms (for example, "kitty" and "young cat"), so each concept is called a "synonym set" or "synset". There were more than 100,000 synsets in WordNet 3.0, majority of them are nouns (80,000+). The ImageNet dataset filtered these to 21,841 synsets that are countable nouns that can be visually illustrated. Each synset in WordNet 3.0 has a "WordNet ID" (wnid), which is a concatenation of part of speech and an "offset" (a unique identifying number). Every wnid starts with "n" because ImageNet only includes nouns. For example, the wnid of synset "dog, domestic dog, Canis familiaris" is "n02084071". The categories in ImageNet fall into 9 levels, from level 1 (such as "mammal") to level 9 (such as "German shepherd"). === Image format === The images were scraped from online image search (Google, Picsearch, MSN, Yahoo, Flickr, etc) using synonyms in multiple languages. For example: German shepherd, German police dog, German shepherd dog, Alsatian, ovejero alemán, pastore tedesco, 德国牧羊犬. ImageNet consists of images in RGB format with varying resolutions. For example, in ImageNet 2012, "fish" category, the resolution ranges from 4288 x 2848 to 75 x 56. In machine learning, these are typically preprocessed into a standard constant resolution, and whitened, before further processing by neural networks. For example, in PyTorch, ImageNet images are by default normalized by dividing the pixel values so that they fall between 0 and 1, then subtracting by [0.485, 0.456, 0.406], then dividing by [0.229, 0.224, 0.225]. These are the mean and standard deviations for ImageNet, so this whitens the input data. === Labels and annotations === Each image is labelled with exactly one wnid. Dense SIFT features (raw SIFT descriptors, quantized codewords, and coordinates of each descriptor/codeword) for ImageNet-1K were available for download, designed for bag of visual words. The bounding boxes of objects were available for about 3000 popular synsets with on average 150 images in each synset. Furthermore, some images have attributes. They released 25 attributes for ~400 popular synsets: Color: black, blue, brown, gray, green, orange, pink, red, violet, white, yellow Pattern: spotted, striped Shape: long, round, rectangular, square Texture: furry, smooth, rough, shiny, metallic, vegetation, wooden, wet === ImageNet-21K === The full original dataset is referred to as ImageNet-21K. ImageNet-21k contains 14,197,122 images divided into 21,841 classes. Some papers round this up and name it ImageNet-22k. The full ImageNet-21k was released in Fall of 2011, as fall11_whole.tar. There is no official train-validation-test split for ImageNet-21k. Some classes contain only 1-10 samples, while others contain thousands. === ImageNet-1K === There are various subsets of the ImageNet dataset used in various context, sometimes referred to as "versions". One of the most highly used subsets of ImageNet is the "ImageNet Large Scale Visual Recognition Challenge (ILSVRC) 2012–2017 image classification and localization dataset". This is also referred to in the research literature as ImageNet-1K or ILSVRC2017, reflecting the original ILSVRC challenge that involved 1,000 classes. ImageNet-1K contains 1,281,167 training images, 50,000 validation images and 100,000 test images. Each category in ImageNet-1K is a leaf category, meaning that there are no child nodes below it, unlike ImageNet-21K. For example, in ImageNet-21K, there are some images categorized as simply "mammal", whereas in ImageNet-1K, there are only images categorized as things like "German shepherd", since there are no child-words below "German shepherd". === Later developments === In the WordNet they built ImageNet on, there were 2832 synsets in the "person" subtree. During 2018--2020 period, they removed the download of the ImageNet-21k as they went through extensive filtering in these person synsets. Out of these 2832 synsets, 1593 were deemed "potentially offensive". Out of the remaining 1239, 1081 were deemed not really "visual". The result was that only 158 syn

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  Occam learning

  In computational learning theory, Occam learning is a model of algorithmic learning where the objective of the learner is to output a succinct representation of received training data. This is closely related to probably approximately correct (PAC) learning, where the learner is evaluated on its predictive power of a test set. Occam learnability implies PAC learning, and for a wide variety of concept classes, the converse is also true: PAC learnability implies Occam learnability. == Introduction == Occam Learning is named after Occam's razor, which is a principle stating that, given all other things being equal, a shorter explanation for observed data should be favored over a lengthier explanation. The theory of Occam learning is a formal and mathematical justification for this principle. It was first shown by Blumer, et al. that Occam learning implies PAC learning, which is the standard model of learning in computational learning theory. In other words, parsimony (of the output hypothesis) implies predictive power. == Definition of Occam learning == The succinctness of a concept c {\displaystyle c} in concept class C {\displaystyle {\mathcal {C}}} can be expressed by the length s i z e ( c ) {\displaystyle size(c)} of the shortest bit string that can represent c {\displaystyle c} in C {\displaystyle {\mathcal {C}}} . Occam learning connects the succinctness of a learning algorithm's output to its predictive power on unseen data. Let C {\displaystyle {\mathcal {C}}} and H {\displaystyle {\mathcal {H}}} be concept classes containing target concepts and hypotheses respectively. Then, for constants α ≥ 0 {\displaystyle \alpha \geq 0} and 0 ≤ β < 1 {\displaystyle 0\leq \beta <1} , a learning algorithm L {\displaystyle L} is an ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} iff, given a set S = { x 1 , … , x m } {\displaystyle S=\{x_{1},\dots ,x_{m}\}} of m {\displaystyle m} samples labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} , L {\displaystyle L} outputs a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that h {\displaystyle h} is consistent with c {\displaystyle c} on S {\displaystyle S} (that is, h ( x ) = c ( x ) , ∀ x ∈ S {\displaystyle h(x)=c(x),\forall x\in S} ), and s i z e ( h ) ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle size(h)\leq (n\cdot size(c))^{\alpha }m^{\beta }} where n {\displaystyle n} is the maximum length of any sample x ∈ S {\displaystyle x\in S} . An Occam algorithm is called efficient if it runs in time polynomial in n {\displaystyle n} , m {\displaystyle m} , and s i z e ( c ) . {\displaystyle size(c).} We say a concept class C {\displaystyle {\mathcal {C}}} is Occam learnable with respect to a hypothesis class H {\displaystyle {\mathcal {H}}} if there exists an efficient Occam algorithm for C {\displaystyle {\mathcal {C}}} using H . {\displaystyle {\mathcal {H}}.} == The relation between Occam and PAC learning == Occam learnability implies PAC learnability, as the following theorem of Blumer, et al. shows: === Theorem (Occam learning implies PAC learning) === Let L {\displaystyle L} be an efficient ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm for C {\displaystyle {\mathcal {C}}} using H {\displaystyle {\mathcal {H}}} . Then there exists a constant a > 0 {\displaystyle a>0} such that for any 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} , for any distribution D {\displaystyle {\mathcal {D}}} , given m ≥ a ( 1 ϵ log ⁡ 1 δ + ( ( n ⋅ s i z e ( c ) ) α ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha }}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} samples drawn from D {\displaystyle {\mathcal {D}}} and labelled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, the algorithm L {\displaystyle L} will output a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } .Here, e r r o r ( h ) {\displaystyle error(h)} is with respect to the concept c {\displaystyle c} and distribution D {\displaystyle {\mathcal {D}}} . This implies that the algorithm L {\displaystyle L} is also a PAC learner for the concept class C {\displaystyle {\mathcal {C}}} using hypothesis class H {\displaystyle {\mathcal {H}}} . A slightly more general formulation is as follows: === Theorem (Occam learning implies PAC learning, cardinality version) === Let 0 < ϵ , δ < 1 {\displaystyle 0<\epsilon ,\delta <1} . Let L {\displaystyle L} be an algorithm such that, given m {\displaystyle m} samples drawn from a fixed but unknown distribution D {\displaystyle {\mathcal {D}}} and labeled according to a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} of length n {\displaystyle n} bits each, outputs a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} that is consistent with the labeled samples. Then, there exists a constant b {\displaystyle b} such that if log ⁡ | H n , m | ≤ b ϵ m − log ⁡ 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq b\epsilon m-\log {\frac {1}{\delta }}} , then L {\displaystyle L} is guaranteed to output a hypothesis h ∈ H n , m {\displaystyle h\in {\mathcal {H}}_{n,m}} such that e r r o r ( h ) ≤ ϵ {\displaystyle error(h)\leq \epsilon } with probability at least 1 − δ {\displaystyle 1-\delta } . While the above theorems show that Occam learning is sufficient for PAC learning, it doesn't say anything about necessity. Board and Pitt show that, for a wide variety of concept classes, Occam learning is in fact necessary for PAC learning. They proved that for any concept class that is polynomially closed under exception lists, PAC learnability implies the existence of an Occam algorithm for that concept class. Concept classes that are polynomially closed under exception lists include Boolean formulas, circuits, deterministic finite automata, decision-lists, decision-trees, and other geometrically defined concept classes. A concept class C {\displaystyle {\mathcal {C}}} is polynomially closed under exception lists if there exists a polynomial-time algorithm A {\displaystyle A} such that, when given the representation of a concept c ∈ C {\displaystyle c\in {\mathcal {C}}} and a finite list E {\displaystyle E} of exceptions, outputs a representation of a concept c ′ ∈ C {\displaystyle c'\in {\mathcal {C}}} such that the concepts c {\displaystyle c} and c ′ {\displaystyle c'} agree except on the set E {\displaystyle E} . == Proof that Occam learning implies PAC learning == We first prove the Cardinality version. Call a hypothesis h ∈ H {\displaystyle h\in {\mathcal {H}}} bad if e r r o r ( h ) ≥ ϵ {\displaystyle error(h)\geq \epsilon } , where again e r r o r ( h ) {\displaystyle error(h)} is with respect to the true concept c {\displaystyle c} and the underlying distribution D {\displaystyle {\mathcal {D}}} . The probability that a set of samples S {\displaystyle S} is consistent with h {\displaystyle h} is at most ( 1 − ϵ ) m {\displaystyle (1-\epsilon )^{m}} , by the independence of the samples. By the union bound, the probability that there exists a bad hypothesis in H n , m {\displaystyle {\mathcal {H}}_{n,m}} is at most | H n , m | ( 1 − ϵ ) m {\displaystyle |{\mathcal {H}}_{n,m}|(1-\epsilon )^{m}} , which is less than δ {\displaystyle \delta } if log ⁡ | H n , m | ≤ O ( ϵ m ) − log ⁡ 1 δ {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq O(\epsilon m)-\log {\frac {1}{\delta }}} . This concludes the proof of the second theorem above. Using the second theorem, we can prove the first theorem. Since we have a ( α , β ) {\displaystyle (\alpha ,\beta )} -Occam algorithm, this means that any hypothesis output by L {\displaystyle L} can be represented by at most ( n ⋅ s i z e ( c ) ) α m β {\displaystyle (n\cdot size(c))^{\alpha }m^{\beta }} bits, and thus log ⁡ | H n , m | ≤ ( n ⋅ s i z e ( c ) ) α m β {\displaystyle \log |{\mathcal {H}}_{n,m}|\leq (n\cdot size(c))^{\alpha }m^{\beta }} . This is less than O ( ϵ m ) − log ⁡ 1 δ {\displaystyle O(\epsilon m)-\log {\frac {1}{\delta }}} if we set m ≥ a ( 1 ϵ log ⁡ 1 δ + ( ( n ⋅ s i z e ( c ) ) α ) ϵ ) 1 1 − β ) {\displaystyle m\geq a\left({\frac {1}{\epsilon }}\log {\frac {1}{\delta }}+\left({\frac {(n\cdot size(c))^{\alpha })}{\epsilon }}\right)^{\frac {1}{1-\beta }}\right)} for some constant a > 0 {\displaystyle a>0} . Thus, by the Cardinality version Theorem, L {\displaystyle L} will output a consistent hypothesis h {\displaystyle h} with probability at least 1 − δ {\displaystyle 1-\delta } . This concludes the proof of the first theorem above. == Improving sample complexity for common problems == Though Occam and PAC learnability are equivalent, the Occam framework can be used to produce tighter bounds on the sample complexity of classical problems including conjunctions, co

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  Artificial development

  Artificial development, also known as artificial embryogeny or machine intelligence or computational development, is an area of computer science and engineering concerned with computational models motivated by genotype–phenotype mappings in biological systems. Artificial development is often considered a sub-field of evolutionary computation, although the principles of artificial development have also been used within stand-alone computational models. Within evolutionary computation, the need for artificial development techniques was motivated by the perceived lack of scalability and evolvability of direct solution encodings (Tufte, 2008). Artificial development entails indirect solution encoding. Rather than describing a solution directly, an indirect encoding describes (either explicitly or implicitly) the process by which a solution is constructed. Often, but not always, these indirect encodings are based upon biological principles of development such as morphogen gradients, cell division and cellular differentiation (e.g. Doursat 2008), gene regulatory networks (e.g. Guo et al., 2009), degeneracy (Whitacre et al., 2010), grammatical evolution (de Salabert et al., 2006), or analogous computational processes such as re-writing, iteration, and time. The influences of interaction with the environment, spatiality and physical constraints on differentiated multi-cellular development have been investigated more recently (e.g. Knabe et al. 2008). Artificial development approaches have been applied to a number of computational and design problems, including electronic circuit design (Miller and Banzhaf 2003), robotic controllers (e.g. Taylor 2004), and the design of physical structures (e.g. Hornby 2004).

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  Multi-surface method

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

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  Matrix regularization

  In the field of statistical learning theory, matrix regularization generalizes notions of vector regularization to cases where the object to be learned is a matrix. The purpose of regularization is to enforce conditions, for example sparsity or smoothness, that can produce stable predictive functions. For example, in the more common vector framework, Tikhonov regularization optimizes over min x ‖ A x − y ‖ 2 + λ ‖ x ‖ 2 {\displaystyle \min _{x}\left\|Ax-y\right\|^{2}+\lambda \left\|x\right\|^{2}} to find a vector x {\displaystyle x} that is a stable solution to the regression problem. When the system is described by a matrix rather than a vector, this problem can be written as min X ‖ A X − Y ‖ 2 + λ ‖ X ‖ 2 , {\displaystyle \min _{X}\left\|AX-Y\right\|^{2}+\lambda \left\|X\right\|^{2},} where the vector norm enforcing a regularization penalty on x {\displaystyle x} has been extended to a matrix norm on X {\displaystyle X} . Matrix regularization has applications in matrix completion, multivariate regression, and multi-task learning. Ideas of feature and group selection can also be extended to matrices, and these can be generalized to the nonparametric case of multiple kernel learning. == Basic definition == Consider a matrix W {\displaystyle W} to be learned from a set of examples, S = ( X i t , y i t ) {\displaystyle S=(X_{i}^{t},y_{i}^{t})} , where i {\displaystyle i} goes from 1 {\displaystyle 1} to n {\displaystyle n} , and t {\displaystyle t} goes from 1 {\displaystyle 1} to T {\displaystyle T} . Let each input matrix X i {\displaystyle X_{i}} be ∈ R D T {\displaystyle \in \mathbb {R} ^{DT}} , and let W {\displaystyle W} be of size D × T {\displaystyle D\times T} . A general model for the output y {\displaystyle y} can be posed as y i t = ⟨ W , X i t ⟩ F , {\displaystyle y_{i}^{t}=\left\langle W,X_{i}^{t}\right\rangle _{F},} where the inner product is the Frobenius inner product. For different applications the matrices X i {\displaystyle X_{i}} will have different forms, but for each of these the optimization problem to infer W {\displaystyle W} can be written as min W ∈ H E ( W ) + R ( W ) , {\displaystyle \min _{W\in {\mathcal {H}}}E(W)+R(W),} where E {\displaystyle E} defines the empirical error for a given W {\displaystyle W} , and R ( W ) {\displaystyle R(W)} is a matrix regularization penalty. The function R ( W ) {\displaystyle R(W)} is typically chosen to be convex and is often selected to enforce sparsity (using ℓ 1 {\displaystyle \ell ^{1}} -norms) and/or smoothness (using ℓ 2 {\displaystyle \ell ^{2}} -norms). Finally, W {\displaystyle W} is in the space of matrices H {\displaystyle {\mathcal {H}}} with Frobenius inner product ⟨ … ⟩ F {\displaystyle \langle \dots \rangle _{F}} . == General applications == === Matrix completion === In the problem of matrix completion, the matrix X i t {\displaystyle X_{i}^{t}} takes the form X i t = e t ⊗ e i ′ , {\displaystyle X_{i}^{t}=e_{t}\otimes e_{i}',} where ( e t ) t {\displaystyle (e_{t})_{t}} and ( e i ′ ) i {\displaystyle (e_{i}')_{i}} are the canonical basis in R T {\displaystyle \mathbb {R} ^{T}} and R D {\displaystyle \mathbb {R} ^{D}} . In this case the role of the Frobenius inner product is to select individual elements w i t {\displaystyle w_{i}^{t}} from the matrix W {\displaystyle W} . Thus, the output y {\displaystyle y} is a sampling of entries from the matrix W {\displaystyle W} . The problem of reconstructing W {\displaystyle W} from a small set of sampled entries is possible only under certain restrictions on the matrix, and these restrictions can be enforced by a regularization function. For example, it might be assumed that W {\displaystyle W} is low-rank, in which case the regularization penalty can take the form of a nuclear norm. R ( W ) = λ ‖ W ‖ ∗ = λ ∑ i | σ i | , {\displaystyle R(W)=\lambda \left\|W\right\|_{}=\lambda \sum _{i}\left|\sigma _{i}\right|,} where σ i {\displaystyle \sigma _{i}} , with i {\displaystyle i} from 1 {\displaystyle 1} to min D , T {\displaystyle \min D,T} , are the singular values of W {\displaystyle W} . === Multivariate regression === Models used in multivariate regression are parameterized by a matrix of coefficients. In the Frobenius inner product above, each matrix X {\displaystyle X} is X i t = e t ⊗ x i {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}} such that the output of the inner product is the dot product of one row of the input with one column of the coefficient matrix. The familiar form of such models is Y = X W + b {\displaystyle Y=XW+b} Many of the vector norms used in single variable regression can be extended to the multivariate case. One example is the squared Frobenius norm, which can be viewed as an ℓ 2 {\displaystyle \ell ^{2}} -norm acting either entrywise, or on the singular values of the matrix: R ( W ) = λ ‖ W ‖ F 2 = λ ∑ i ∑ j | w i j | 2 = λ Tr ⁡ ( W ∗ W ) = λ ∑ i σ i 2 . {\displaystyle R(W)=\lambda \left\|W\right\|_{F}^{2}=\lambda \sum _{i}\sum _{j}\left|w_{ij}\right|^{2}=\lambda \operatorname {Tr} \left(W^{}W\right)=\lambda \sum _{i}\sigma _{i}^{2}.} In the multivariate case the effect of regularizing with the Frobenius norm is the same as the vector case; very complex models will have larger norms, and, thus, will be penalized more. === Multi-task learning === The setup for multi-task learning is almost the same as the setup for multivariate regression. The primary difference is that the input variables are also indexed by task (columns of Y {\displaystyle Y} ). The representation with the Frobenius inner product is then X i t = e t ⊗ x i t . {\displaystyle X_{i}^{t}=e_{t}\otimes x_{i}^{t}.} The role of matrix regularization in this setting can be the same as in multivariate regression, but matrix norms can also be used to couple learning problems across tasks. In particular, note that for the optimization problem min W ‖ X W − Y ‖ 2 2 + λ ‖ W ‖ 2 2 {\displaystyle \min _{W}\left\|XW-Y\right\|_{2}^{2}+\lambda \left\|W\right\|_{2}^{2}} the solutions corresponding to each column of Y {\displaystyle Y} are decoupled. That is, the same solution can be found by solving the joint problem, or by solving an isolated regression problem for each column. The problems can be coupled by adding an additional regularization penalty on the covariance of solutions min W , Ω ‖ X W − Y ‖ 2 2 + λ 1 ‖ W ‖ 2 2 + λ 2 Tr ⁡ ( W T Ω − 1 W ) {\displaystyle \min _{W,\Omega }\left\|XW-Y\right\|_{2}^{2}+\lambda _{1}\left\|W\right\|_{2}^{2}+\lambda _{2}\operatorname {Tr} \left(W^{T}\Omega ^{-1}W\right)} where Ω {\displaystyle \Omega } models the relationship between tasks. This scheme can be used to both enforce similarity of solutions across tasks, and to learn the specific structure of task similarity by alternating between optimizations of W {\displaystyle W} and Ω {\displaystyle \Omega } . When the relationship between tasks is known to lie on a graph, the Laplacian matrix of the graph can be used to couple the learning problems. == Spectral regularization == Regularization by spectral filtering has been used to find stable solutions to problems such as those discussed above by addressing ill-posed matrix inversions (see for example Filter function for Tikhonov regularization). In many cases the regularization function acts on the input (or kernel) to ensure a bounded inverse by eliminating small singular values, but it can also be useful to have spectral norms that act on the matrix that is to be learned. There are a number of matrix norms that act on the singular values of the matrix. Frequently used examples include the Schatten p-norms, with p = 1 or 2. For example, matrix regularization with a Schatten 1-norm, also called the nuclear norm, can be used to enforce sparsity in the spectrum of a matrix. This has been used in the context of matrix completion when the matrix in question is believed to have a restricted rank. In this case the optimization problem becomes: min ‖ W ‖ ∗ subject to W i , j = Y i j . {\displaystyle \min \left\|W\right\|_{}~~{\text{ subject to }}~~W_{i,j}=Y_{ij}.} Spectral Regularization is also used to enforce a reduced rank coefficient matrix in multivariate regression. In this setting, a reduced rank coefficient matrix can be found by keeping just the top n {\displaystyle n} singular values, but this can be extended to keep any reduced set of singular values and vectors. == Structured sparsity == Sparse optimization has become the focus of much research interest as a way to find solutions that depend on a small number of variables (see e.g. the Lasso method). In principle, entry-wise sparsity can be enforced by penalizing the entry-wise ℓ 0 {\displaystyle \ell ^{0}} -norm of the matrix, but the ℓ 0 {\displaystyle \ell ^{0}} -norm is not convex. In practice this can be implemented by convex relaxation to the ℓ 1 {\displaystyle \ell ^{1}} -norm. While entry-wise regularization with an ℓ 1 {\displaystyle \ell ^{1}} -norm will find solutions with a small number of nonzero elements, applying an ℓ 1 {

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  Facial recognition system

  A facial recognition system is a technology potentially capable of matching a human face from a digital image or a video frame against a database of faces. Such a system is typically employed to authenticate users through ID verification services, and works by pinpointing and measuring facial features from a given image. Development on similar systems began in the 1960s as a form of computer application. Since their inception, facial recognition systems have seen wider uses in recent times on smartphones and in other forms of technology, such as robotics. Because computerized facial recognition involves the measurement of a human's physiological characteristics, facial recognition systems are categorized as biometrics. Although the accuracy of facial recognition systems as a biometric technology is lower than iris recognition, fingerprint image acquisition, palm recognition or voice recognition, it is widely adopted due to its contactless process. Facial recognition systems have been deployed in advanced human–computer interaction, video surveillance, law enforcement, passenger screening, decisions on employment and housing, and automatic indexing of images. Facial recognition systems are employed throughout the world today by governments and private companies. Their effectiveness varies, and some systems have previously been scrapped because of their ineffectiveness. The use of facial recognition systems has also raised controversy, with claims that the systems violate citizens' privacy, commonly make incorrect identifications, encourage gender norms and racial profiling, and do not protect important biometric data. The appearance of synthetic media such as deepfakes has also raised concerns about its security. These claims have led to the ban of facial recognition systems in several cities in the United States. Growing societal concerns led social networking company Meta Platforms to shut down its Facebook facial recognition system in 2021, deleting the face-scan data of more than one billion users. The change represented one of the largest shifts in facial recognition usage in the technology's history. IBM also stopped offering facial recognition technology due to similar concerns. == History of facial recognition technology == Automated facial recognition was pioneered in the 1960s by Woody Bledsoe, Helen Chan Wolf, and Charles Bisson, whose work focused on teaching computers to recognize human faces. Their early facial recognition project was dubbed "man-machine" because a human first needed to establish the coordinates of facial features in a photograph before they could be used by a computer for recognition. Using a graphics tablet, a human would pinpoint facial features coordinates, such as the pupil centers, the inside and outside corners of eyes, and the widows peak in the hairline. The coordinates were used to calculate 20 individual distances, including the width of the mouth and of the eyes. A human could process about 40 pictures an hour, building a database of these computed distances. A computer would then automatically compare the distances for each photograph, calculate the difference between the distances, and return the closed records as a possible match. In 1970, Takeo Kanade publicly demonstrated a face-matching system that located anatomical features such as the chin and calculated the distance ratio between facial features without human intervention. Later tests revealed that the system could not always reliably identify facial features. Nonetheless, interest in the subject grew and in 1977 Kanade published the first detailed book on facial recognition technology. In 1993, the Defense Advanced Research Project Agency (DARPA) and the Army Research Laboratory (ARL) established the face recognition technology program FERET to develop "automatic face recognition capabilities" that could be employed in a productive real life environment "to assist security, intelligence, and law enforcement personnel in the performance of their duties." Face recognition systems that had been trialled in research labs were evaluated. The FERET tests found that while the performance of existing automated facial recognition systems varied, a handful of existing methods could viably be used to recognize faces in still images taken in a controlled environment. The FERET tests spawned three US companies that sold automated facial recognition systems. Vision Corporation and Miros Inc were founded in 1994, by researchers who used the results of the FERET tests as a selling point. Viisage Technology was established by an identification card defense contractor in 1996 to commercially exploit the rights to the facial recognition algorithm developed by Alex Pentland at MIT. Following the 1993 FERET face-recognition vendor test, the Department of Motor Vehicles (DMV) offices in West Virginia and New Mexico became the first DMV offices to use automated facial recognition systems to prevent people from obtaining multiple driving licenses using different names. Driver's licenses in the United States were at that point a commonly accepted form of photo identification. DMV offices across the United States were undergoing a technological upgrade and were in the process of establishing databases of digital ID photographs. This enabled DMV offices to deploy the facial recognition systems on the market to search photographs for new driving licenses against the existing DMV database. DMV offices became one of the first major markets for automated facial recognition technology and introduced US citizens to facial recognition as a standard method of identification. The increase of the US prison population in the 1990s prompted U.S. states to established connected and automated identification systems that incorporated digital biometric databases, in some instances this included facial recognition. In 1999, Minnesota incorporated the facial recognition system FaceIT by Visionics into a mug shot booking system that allowed police, judges and court officers to track criminals across the state. Until the 1990s, facial recognition systems were developed primarily by using photographic portraits of human faces. Research on face recognition to reliably locate a face in an image that contains other objects gained traction in the early 1990s with the principal component analysis (PCA). The PCA method of face detection is also known as Eigenface and was developed by Matthew Turk and Alex Pentland. Turk and Pentland combined the conceptual approach of the Karhunen–Loève theorem and factor analysis, to develop a linear model. Eigenfaces are determined based on global and orthogonal features in human faces. A human face is calculated as a weighted combination of a number of Eigenfaces. Because few Eigenfaces were used to encode human faces of a given population, Turk and Pentland's PCA face detection method greatly reduced the amount of data that had to be processed to detect a face. Pentland in 1994 defined Eigenface features, including eigen eyes, eigen mouths and eigen noses, to advance the use of PCA in facial recognition. In 1997, the PCA Eigenface method of face recognition was improved upon using linear discriminant analysis (LDA) to produce Fisherfaces. LDA Fisherfaces became dominantly used in PCA feature based face recognition. While Eigenfaces were also used for face reconstruction. In these approaches no global structure of the face is calculated which links the facial features or parts. Purely feature based approaches to facial recognition were overtaken in the late 1990s by the Bochum system, which used Gabor filter to record the face features and computed a grid of the face structure to link the features. Christoph von der Malsburg and his research team at the University of Bochum developed Elastic Bunch Graph Matching in the mid-1990s to extract a face out of an image using skin segmentation. By 1997, the face detection method developed by Malsburg outperformed most other facial detection systems on the market. The so-called "Bochum system" of face detection was sold commercially on the market as ZN-Face to operators of airports and other busy locations. The software was "robust enough to make identifications from less-than-perfect face views. It can also often see through such impediments to identification as mustaches, beards, changed hairstyles and glasses—even sunglasses". Real-time face detection in video footage became possible in 2001 with the Viola–Jones object detection framework for faces. Paul Viola and Michael Jones combined their face detection method with the Haar-like feature approach to object recognition in digital images to launch AdaBoost, the first real-time frontal-view face detector. By 2015, the Viola–Jones algorithm had been implemented using small low power detectors on handheld devices and embedded systems. Therefore, the Viola–Jones algorithm has not only broadened the practical application of face recognition systems but

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  Stress majorization

  Stress majorization is an optimization strategy used in multidimensional scaling (MDS) where, for a set of n {\displaystyle n} m {\displaystyle m} -dimensional data items, a configuration X {\displaystyle X} of n {\displaystyle n} points in r {\displaystyle r} ( ≪ m ) {\displaystyle (\ll m)} -dimensional space is sought that minimizes the so-called stress function σ ( X ) {\displaystyle \sigma (X)} . Usually r {\displaystyle r} is 2 {\displaystyle 2} or 3 {\displaystyle 3} , i.e. the ( n × r ) {\displaystyle (n\times r)} matrix X {\displaystyle X} lists points in 2 − {\displaystyle 2-} or 3 − {\displaystyle 3-} dimensional Euclidean space so that the result may be visualised (i.e. an MDS plot). The function σ {\displaystyle \sigma } is a cost or loss function that measures the squared differences between ideal ( m {\displaystyle m} -dimensional) distances and actual distances in r-dimensional space. It is defined as: σ ( X ) = ∑ i < j ≤ n w i j ( d i j ( X ) − δ i j ) 2 {\displaystyle \sigma (X)=\sum _{i Read more →