AI Apps Like Chat Gpt

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  Salience (neuroscience)

  Salience (also called saliency, from Latin saliō meaning "leap, spring") is the property by which some thing stands out. Salient events are an attentional mechanism by which organisms learn and survive; those organisms can focus their limited perceptual and cognitive resources on the pertinent (that is, salient) subset of the sensory data available to them. Saliency typically arises from contrasts between items and their neighborhood. They might be represented, for example, by a red dot surrounded by white dots, or by a flickering message indicator of an answering machine, or a loud noise in an otherwise quiet environment. Saliency detection is often studied in the context of the visual system, but similar mechanisms operate in other sensory systems. Just what is salient can be influenced by training: for example, for human subjects particular letters can become salient by training. There can be a sequence of necessary events, each of which has to be salient, in turn, in order for successful training in the sequence; the alternative is a failure, as in an illustrated sequence when tying a bowline; in the list of illustrations, even the first illustration is a salient: the rope in the list must cross over, and not under the bitter end of the rope (which can remain fixed, and not free to move); failure to notice that the first salient has not been satisfied means the knot will fail to hold, even when the remaining salient events have been satisfied. When attention deployment is driven by salient stimuli, it is considered to be bottom-up, memory-free, and reactive. Conversely, attention can also be guided by top-down, memory-dependent, or anticipatory mechanisms, such as when looking ahead of moving objects or sideways before crossing streets. Humans and other animals have difficulty paying attention to more than one item simultaneously, so they are faced with the challenge of continuously integrating and prioritizing different bottom-up and top-down influences. == Neuroanatomy == The brain component named the hippocampus helps with the assessment of salience and context by using past memories to filter new incoming stimuli, and placing those that are most important into long term memory. The entorhinal cortex is the pathway into and out of the hippocampus, and is an important part of the brain's memory network; research shows that it is a brain region that suffers damage early on in Alzheimer's disease, one of the effects of which is altered (diminished) salience. The pulvinar nuclei (in the thalamus) modulate physical/perceptual salience in attentional selection. One group of neurons (i.e., D1-type medium spiny neurons) within the nucleus accumbens shell (NAcc shell) assigns appetitive motivational salience ("want" and "desire", which includes a motivational component), aka incentive salience, to rewarding stimuli, while another group of neurons (i.e., D2-type medium spiny neurons) within the NAcc shell assigns aversive motivational salience to aversive stimuli. The primary visual cortex (V1) generates a bottom-up saliency map from visual inputs to guide reflexive attentional shifts or gaze shifts. According to V1 Saliency Hypothesis, the saliency of a location is higher when V1 neurons give higher responses to that location relative to V1 neurons' responses to other visual locations. For example, a unique red item among green items, or a unique vertical bar among horizontal bars, is salient since it evokes higher V1 responses and attracts attention or gaze. The V1 neural responses are sent to the superior colliculus to guide gaze shifts to the salient locations. A fingerprint of the saliency map in V1 is that attention or gaze can be captured by the location of an eye-of-origin singleton in visual inputs, e.g., a bar uniquely shown to the left eye in a background of many other bars shown to the right eye, even when observers cannot tell the difference between the singleton and the background bars. == In psychology == The term is widely used in the study of perception and cognition to refer to any aspect of a stimulus that, for any of many reasons, stands out from the rest. Salience may be the result of emotional, motivational or cognitive factors and is not necessarily associated with physical factors such as intensity, clarity or size. Although salience is thought to determine attentional selection, salience associated with physical factors does not necessarily influence selection of a stimulus. === Salience bias === Salience bias (also referred to as perceptual salience) is a cognitive bias that predisposes individuals to focus on or attend to items, information, or stimuli that are more prominent, visible, or emotionally striking. This is as opposed to stimuli that are unremarkable, or less salient, even though this difference is often irrelevant by objective standards. The American Psychological Association (APA) defines the salience hypothesis as a theory regarding perception where "motivationally significant" information is more readily perceived than information with little or less significant motivational importance. Perceptual salience (salience bias) is linked to the vividness effect, whereby a more pronounced response is produced by a more vivid perception of a stimulus than the mere knowledge of the stimulus. Salience bias assumes that more dynamic, conspicuous, or distinctive stimuli engage attention more than less prominent stimuli, disproportionately impacting decision making, it is a bias which favors more salient information. ==== Application ==== ===== Cognitive Psychology ===== Salience bias, like all other cognitive biases, is an applicable concept to various disciplines. For example, cognitive psychology investigates cognitive functions and processes, such as perception, attention, memory, problem solving, and decision making, all of which could be influenced by salience bias. Salience bias acts to combat cognitive overload by focusing attention on prominent stimuli, which affects how individuals perceive the world as other, less vivid stimuli that could add to or change this perception, are ignored. Human attention gravitates towards novel and relevant stimuli and unconsciously filters out less prominent information, demonstrating salience bias, which influences behavior as human behavior is affected by what is attended to. Behavioral economists Tversky and Kahneman also suggest that the retrieval of instances is influenced by their salience, such as how witnessing or experiencing an event first-hand has a greater impact than when it is less salient, like if it were read about, implying that memory is affected by salience. ===== Language ===== It is also relevant in language understanding and acquisition. Focusing on more salient phenomena allows people to detect language patterns and dialect variations more easily, making dialect categorization more efficient. ===== Social Behavior ===== Furthermore, social behaviors and interactions can also be influenced by perceptual salience. Changes in the perceptual salience of an individual heavily influences their social behavior and subjective experience of their social interactions, confirming a "social salience effect". Social salience relates to how individuals perceive and respond to other people. ===== Behavioral Science ===== The connection between salience bias and other heuristics, like availability and representativeness, links it to the fields of behavioral science and behavioral economics. Salience bias is closely related to the availability heuristic in behavioral economics, based on the influence of information vividness and visibility, such as recency or frequency, on judgements, for example:Accessibility and salience are closely related to availability, and they are important as well. If you have personally experienced a serious earthquake, you're more likely to believe that an earthquake is likely than if you read about it in a weekly magazine. Thus, vivid and easily imagined causes of death (for example, tornadoes) often receive inflated estimates of probability, and less-vivid causes (for example, asthma attacks) receive low estimates, even if they occur with a far greater frequency (here, by a factor of twenty). Timing counts too: more recent events have a greater impact on our behavior, and on our fears, than earlier ones.Humans have bounded rationality, which refers to their limited ability to be rational in decision making, due to a limited capacity to process information and cognitive ability. Heuristics, such as availability, are employed to reduce the complexity of cognitive and social tasks or judgements, in order to decrease the cognitive load that result from bounded rationality. Despite the effectiveness of heuristics in doing so, they are limited by systematic errors that occur, often the result of influencing biases, such as salience. This can lead to misdirected or misinformed judgements, based on an overemphasis or overweighting of

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  Causal Markov condition

  The Causal Markov (CM) condition states that, conditional on the set of all its direct causes, a node is independent of all variables which are not effects or direct causes of that node. In the event that the structure of a Bayesian network accurately depicts causality, the two conditions are equivalent. This is related to the Markov condition, an assumption made in Bayesian probability theory, that every node in a Bayesian network is conditionally independent of its nondescendants, given its parents. Stated loosely, it is assumed that a node has no bearing on nodes which do not descend from it. In a DAG, this local Markov condition is equivalent to the global Markov condition, which states that d-separations in the graph also correspond to conditional independence relations. This also means that a node is conditionally independent of the entire network, given its Markov blanket. A network may accurately embody the Markov condition without depicting causality, in which case it should not be assumed to embody the causal Markov condition. == Motivation == Statisticians are enormously interested in the ways in which certain events and variables are connected. The precise notion of what constitutes a cause and effect is necessary to understand the connections between them. The central idea behind the philosophical study of probabilistic causation is that causes raise the probabilities of their effects, all else being equal. A deterministic interpretation of causation means that if A causes B, then A must always be followed by B. In this sense, smoking does not cause cancer because some smokers never develop cancer. On the other hand, a probabilistic interpretation simply means that causes raise the probability of their effects. In this sense, changes in meteorological readings associated with a storm do cause that storm, since they raise its probability. (However, simply looking at a barometer does not change the probability of the storm, for a more detailed analysis, see:). == Examples == In a simple view, releasing one's hand from a hammer causes the hammer to fall. However, doing so in outer space does not produce the same outcome, calling into question if releasing one's fingers from a hammer always causes it to fall. A causal graph could be created to acknowledge that both the presence of gravity and the release of the hammer contribute to its falling. However, it would be very surprising if the surface underneath the hammer affected its falling. This essentially states the Causal Markov Condition, that given the existence of gravity the release of the hammer, it will fall regardless of what is beneath it. == Implications == === Dependence and Causation === It follows from the definition that if X and Y are in V and are probabilistically dependent, then either X causes Y, Y causes X, or X and Y are both effects of some common cause Z in V. This definition was seminally introduced by Hans Reichenbach as the Common Cause Principle (CCP). === Screening === It once again follows from the definition that the parents of X screen X from other "indirect causes" of X (parents of Parents(X)) and other effects of Parents(X) which are not also effects of X.

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  Autoencoder

  An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg ⁡ min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-

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  Autoencoder

  An autoencoder is a type of artificial neural network used to learn efficient codings of unlabeled data (unsupervised learning). An autoencoder learns two functions: an encoding function that transforms the input data, and a decoding function that recreates the input data from the encoded representation. The autoencoder learns an efficient representation (encoding) for a set of data, typically for dimensionality reduction, to generate lower-dimensional embeddings for subsequent use by other machine learning algorithms. Variants exist which aim to make the learned representations assume useful properties. Examples are regularized autoencoders (sparse, denoising and contractive autoencoders), which are effective in learning representations for subsequent classification tasks, and variational autoencoders, which can be used as generative models. Autoencoders are applied to many problems, including facial recognition, feature detection, anomaly detection, and learning the meaning of words. In terms of data synthesis, autoencoders can also be used to randomly generate new data that is similar to the input (training) data. == Mathematical principles == === Definition === An autoencoder is defined by the following components: Two sets: the space of encoded messages Z {\displaystyle {\mathcal {Z}}} ; the space of decoded messages X {\displaystyle {\mathcal {X}}} . Typically X {\displaystyle {\mathcal {X}}} and Z {\displaystyle {\mathcal {Z}}} are Euclidean spaces, that is, X = R m , Z = R n {\displaystyle {\mathcal {X}}=\mathbb {R} ^{m},{\mathcal {Z}}=\mathbb {R} ^{n}} with m > n . {\displaystyle m>n.} Two parametrized families of functions: the encoder family E ϕ : X → Z {\displaystyle E_{\phi }:{\mathcal {X}}\rightarrow {\mathcal {Z}}} , parametrized by ϕ {\displaystyle \phi } ; the decoder family D θ : Z → X {\displaystyle D_{\theta }:{\mathcal {Z}}\rightarrow {\mathcal {X}}} , parametrized by θ {\displaystyle \theta } .For any x ∈ X {\displaystyle x\in {\mathcal {X}}} , we usually write z = E ϕ ( x ) {\displaystyle z=E_{\phi }(x)} , and refer to it as the code, the latent variable, latent representation, latent vector, etc. Conversely, for any z ∈ Z {\displaystyle z\in {\mathcal {Z}}} , we usually write x ′ = D θ ( z ) {\displaystyle x'=D_{\theta }(z)} , and refer to it as the (decoded) message. Usually, both the encoder and the decoder are defined as multilayer perceptrons (MLPs). For example, a one-layer-MLP encoder E ϕ {\displaystyle E_{\phi }} is: E ϕ ( x ) = σ ( W x + b ) {\displaystyle E_{\phi }(\mathbf {x} )=\sigma (Wx+b)} where σ {\displaystyle \sigma } is an element-wise activation function, W {\displaystyle W} is a "weight" matrix, and b {\displaystyle b} is a "bias" vector. === Training an autoencoder === An autoencoder, by itself, is simply a tuple of two functions. To judge its quality, we need a task. A task is defined by a reference probability distribution μ r e f {\displaystyle \mu _{ref}} over X {\displaystyle {\mathcal {X}}} , and a "reconstruction quality" function d : X × X → [ 0 , ∞ ] {\displaystyle d:{\mathcal {X}}\times {\mathcal {X}}\to [0,\infty ]} , such that d ( x , x ′ ) {\displaystyle d(x,x')} measures how much x ′ {\displaystyle x'} differs from x {\displaystyle x} . With those, we can define the loss function for the autoencoder as L ( θ , ϕ ) := E x ∼ μ r e f [ d ( x , D θ ( E ϕ ( x ) ) ) ] {\displaystyle L(\theta ,\phi ):=\mathbb {\mathbb {E} } _{x\sim \mu _{ref}}[d(x,D_{\theta }(E_{\phi }(x)))]} The optimal autoencoder for the given task ( μ r e f , d ) {\displaystyle (\mu _{ref},d)} is then arg ⁡ min θ , ϕ L ( θ , ϕ ) {\displaystyle \arg \min _{\theta ,\phi }L(\theta ,\phi )} . The search for the optimal autoencoder can be accomplished by any mathematical optimization technique, but usually by gradient descent. This search process is referred to as "training the autoencoder". In most situations, the reference distribution is just the empirical distribution given by a dataset { x 1 , . . . , x N } ⊂ X {\displaystyle \{x_{1},...,x_{N}\}\subset {\mathcal {X}}} , so that μ r e f = 1 N ∑ i = 1 N δ x i {\displaystyle \mu _{ref}={\frac {1}{N}}\sum _{i=1}^{N}\delta _{x_{i}}} where δ x i {\displaystyle \delta _{x_{i}}} is the Dirac measure, the quality function is just L 2 {\displaystyle L^{2}} loss: d ( x , x ′ ) = ‖ x − x ′ ‖ 2 2 {\displaystyle d(x,x')=\|x-x'\|_{2}^{2}} , and ‖ ⋅ ‖ 2 {\displaystyle \|\cdot \|_{2}} is the Euclidean norm. Then the problem of searching for the optimal autoencoder is just a least-squares optimization: min θ , ϕ L ( θ , ϕ ) , where L ( θ , ϕ ) = 1 N ∑ i = 1 N ‖ x i − D θ ( E ϕ ( x i ) ) ‖ 2 2 {\displaystyle \min _{\theta ,\phi }L(\theta ,\phi ),\qquad {\text{where }}L(\theta ,\phi )={\frac {1}{N}}\sum _{i=1}^{N}\|x_{i}-D_{\theta }(E_{\phi }(x_{i}))\|_{2}^{2}} === Interpretation === An autoencoder has two main parts: an encoder that maps the message to a code, and a decoder that reconstructs the message from the code. An optimal autoencoder would perform as close to perfect reconstruction as possible, with "close to perfect" defined by the reconstruction quality function d {\displaystyle d} . The simplest way to perform the copying task perfectly would be to duplicate the signal. To suppress this behavior, the code space Z {\displaystyle {\mathcal {Z}}} usually has fewer dimensions than the message space X {\displaystyle {\mathcal {X}}} . Such an autoencoder is called undercomplete. It can be interpreted as compressing the message, or reducing its dimensionality. At the limit of an ideal undercomplete autoencoder, every possible code z {\displaystyle z} in the code space is used to encode a message x {\displaystyle x} that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} , and the decoder is also perfect: D θ ( E ϕ ( x ) ) = x {\displaystyle D_{\theta }(E_{\phi }(x))=x} . This ideal autoencoder can then be used to generate messages indistinguishable from real messages, by feeding its decoder arbitrary code z {\displaystyle z} and obtaining D θ ( z ) {\displaystyle D_{\theta }(z)} , which is a message that really appears in the distribution μ r e f {\displaystyle \mu _{ref}} . If the code space Z {\displaystyle {\mathcal {Z}}} has dimension larger than (overcomplete), or equal to, the message space X {\displaystyle {\mathcal {X}}} , or the hidden units are given enough capacity, an autoencoder can learn the identity function and become useless. However, experimental results found that overcomplete autoencoders might still learn useful features. In the ideal setting, the code dimension and the model capacity could be set on the basis of the complexity of the data distribution to be modeled. A standard way to do so is to add modifications to the basic autoencoder, to be detailed below. == Variations == === Variational autoencoder (VAE) === Variational autoencoders (VAEs) belong to the families of variational Bayesian methods. Despite the architectural similarities with basic autoencoders, VAEs are architected with different goals and have a different mathematical formulation. The latent space is, in this case, composed of a mixture of distributions instead of fixed vectors. Given an input dataset x {\displaystyle x} characterized by an unknown probability function P ( x ) {\displaystyle P(x)} and a multivariate latent encoding vector z {\displaystyle z} , the objective is to model the data as a distribution p θ ( x ) {\displaystyle p_{\theta }(x)} , with θ {\displaystyle \theta } defined as the set of the network parameters so that p θ ( x ) = ∫ z p θ ( x , z ) d z {\displaystyle p_{\theta }(x)=\int _{z}p_{\theta }(x,z)dz} . === Sparse autoencoder (SAE) === Inspired by the sparse coding hypothesis in neuroscience, sparse autoencoders (SAE) are variants of autoencoders, such that the codes E ϕ ( x ) {\displaystyle E_{\phi }(x)} for messages tend to be sparse codes, that is, E ϕ ( x ) {\displaystyle E_{\phi }(x)} is close to zero in most entries. Sparse autoencoders may include more (rather than fewer) hidden units than inputs, but only a small number of the hidden units are allowed to be active at the same time. Encouraging sparsity improves performance on classification tasks. There are two main ways to enforce sparsity. One way is to simply clamp all but the highest-k activations of the latent code to zero. This is the k-sparse autoencoder. The k-sparse autoencoder inserts the following "k-sparse function" in the latent layer of a standard autoencoder: f k ( x 1 , . . . , x n ) = ( x 1 b 1 , . . . , x n b n ) {\displaystyle f_{k}(x_{1},...,x_{n})=(x_{1}b_{1},...,x_{n}b_{n})} where b i = 1 {\displaystyle b_{i}=1} if | x i | {\displaystyle |x_{i}|} ranks in the top k, and 0 otherwise. Backpropagating through f k {\displaystyle f_{k}} is simple: set gradient to 0 for b i = 0 {\displaystyle b_{i}=0} entries, and keep gradient for b i = 1 {\displaystyle b_{i}=1} entries. This is essentially a generalized ReLU function. The other way is a relaxed version of the k-

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  Digital image correlation and tracking

  Digital image correlation and tracking is an optical method that employs tracking and image registration techniques for accurate 2D and 3D measurements of changes in 2D images or 3D volumes. This method is often used to measure full-field displacement and strains, and it is widely applied in many areas of science and engineering. Compared to strain gauges and extensometers, digital image correlation methods provide finer details about deformation, due to the ability to provide both local and average data. == Overview == Digital image correlation (DIC) techniques have been increasing in popularity, especially in micro- and nano-scale mechanical testing applications due to their relative ease of implementation and use. Advances in computer technology and digital cameras have been the enabling technologies for this method and while white-light optics has been the predominant approach, DIC can be and has been extended to almost any imaging technology. The concept of using cross-correlation to measure shifts in datasets has been known for a long time, and it has been applied to digital images since at least the early 1970s. The present-day applications are almost innumerable, including image analysis, image compression, velocimetry, and strain estimation. Much early work in DIC in the field of mechanics was led by researchers at the University of South Carolina in the early 1980s and has been optimized and improved in recent years. Commonly, DIC relies on finding the maximum of the correlation array between pixel intensity array subsets on two or more corresponding images, which gives the integer translational shift between them. It is also possible to estimate shifts to a finer resolution than the resolution of the original images, which is often called "sub-pixel" registration because the measured shift is smaller than an integer pixel unit. For sub-pixel interpolation of the shift, other methods do not simply maximize the correlation coefficient. An iterative approach can also be used to maximize the interpolated correlation coefficient by using non-linear optimization techniques. The non-linear optimization approach tends to be conceptually simpler and can handle large deformations more accurately, but as with most nonlinear optimization techniques, it is slower. The two-dimensional discrete cross correlation r i j {\displaystyle r_{ij}} can be defined in several ways, one possibility being: r i j = ∑ m ∑ n [ f ( m + i , n + j ) − f ¯ ] [ g ( m , n ) − g ¯ ] ∑ m ∑ n [ f ( m , n ) − f ¯ ] 2 ∑ m ∑ n [ g ( m , n ) − g ¯ ] 2 . {\displaystyle r_{ij}={\frac {\sum _{m}\sum _{n}[f(m+i,n+j)-{\bar {f}}][g(m,n)-{\bar {g}}]}{\sqrt {\sum _{m}\sum _{n}{[f(m,n)-{\bar {f}}]^{2}}\sum _{m}\sum _{n}{[g(m,n)-{\bar {g}}]^{2}}}}}.} Here f(m, n) is the pixel intensity or the gray-scale value at a point (m, n) in the original image, g(m, n) is the gray-scale value at a point (m, n) in the translated image, f ¯ {\displaystyle {\bar {f}}} and g ¯ {\displaystyle {\bar {g}}} are mean values of the intensity matrices f and g respectively. However, in practical applications, the correlation array is usually computed using Fourier-transform methods, since the fast Fourier transform is a much faster method than directly computing the correlation. F = F { f } , G = F { g } . {\displaystyle \mathbf {F} ={\mathcal {F}}\{f\},\quad \mathbf {G} ={\mathcal {F}}\{g\}.} Then taking the complex conjugate of the second result and multiplying the Fourier transforms together elementwise, we obtain the Fourier transform of the correlogram, R {\displaystyle \ R} : R = F ∘ G ∗ , {\displaystyle R=\mathbf {F} \circ \mathbf {G} ^{},} where ∘ {\displaystyle \circ } is the Hadamard product (entry-wise product). It is also fairly common to normalize the magnitudes to unity at this point, which results in a variation called phase correlation. Then the cross-correlation is obtained by applying the inverse Fourier transform: r = F − 1 { R } . {\displaystyle \ r={\mathcal {F}}^{-1}\{R\}.} At this point, the coordinates of the maximum of r i j {\displaystyle r_{ij}} give the integer shift: ( Δ x , Δ y ) = arg ⁡ max ( i , j ) { r } . {\displaystyle (\Delta x,\Delta y)=\arg \max _{(i,j)}\{r\}.} == Deformation mapping == For deformation mapping, the mapping function that relates the images can be derived from comparing a set of subwindow pairs over the whole images. (Figure 1). The coordinates or grid points (xi, yj) and (xi, yj) are related by the translations that occur between the two images. If the deformation is small and perpendicular to the optical axis of the camera, then the relation between (xi, yj) and (xi, yj) can be approximated by a 2D affine transformation such as: x ∗ = x + u + ∂ u ∂ x Δ x + ∂ u ∂ y Δ y , {\displaystyle x^{}=x+u+{\frac {\partial u}{\partial x}}\Delta x+{\frac {\partial u}{\partial y}}\Delta y,} y ∗ = y + v + ∂ v ∂ x Δ x + ∂ v ∂ y Δ y . {\displaystyle y^{}=y+v+{\frac {\partial v}{\partial x}}\Delta x+{\frac {\partial v}{\partial y}}\Delta y.} Here u and v are translations of the center of the sub-image in the X and Y directions respectively. The distances from the center of the sub-image to the point (x, y) are denoted by Δ x {\displaystyle \Delta x} and Δ y {\displaystyle \Delta y} . Thus, the correlation coefficient rij is a function of displacement components (u, v) and displacement gradients ∂ u ∂ x , ∂ u ∂ y , ∂ v ∂ x , ∂ v ∂ y . {\displaystyle {\frac {\partial u}{\partial x}},{\frac {\partial u}{\partial y}},{\frac {\partial v}{\partial x}},{\frac {\partial v}{\partial y}}.} DIC has proven to be very effective at mapping deformation in macroscopic mechanical testing, where the application of specular markers (e.g. paint, toner powder) or surface finishes from machining and polishing provide the needed contrast to correlate images well. However, these methods for applying surface contrast do not extend to the application of free-standing thin films for several reasons. First, vapor deposition at normal temperatures on semiconductor grade substrates results in mirror-finish quality films with RMS roughnesses that are typically on the order of several nanometers. No subsequent polishing or finishing steps are required, and unless electron imaging techniques are employed that can resolve microstructural features, the films do not possess enough useful surface contrast to adequately correlate images. Typically this challenge can be circumvented by applying paint that results in a random speckle pattern on the surface, although the large and turbulent forces resulting from either spraying or applying paint to the surface of a free-standing thin film are too high and would break the specimens. In addition, the sizes of individual paint particles are on the order of μms, while the film thickness is only several hundred nanometers, which would be analogous to supporting a large boulder on a thin sheet of paper. == Digital volume correlation == Digital Volume Correlation (DVC, and sometimes called Volumetric-DIC) extends the 2D-DIC algorithms into three dimensions to calculate the full-field 3D deformation from a pair of 3D images. This technique is distinct from 3D-DIC, which only calculates the 3D deformation of an exterior surface using conventional optical images. The DVC algorithm is able to track full-field displacement information in the form of voxels instead of pixels. The theory is similar to above except that another dimension is added: the z-dimension. The displacement is calculated from the correlation of 3D subsets of the reference and deformed volumetric images, which is analogous to the correlation of 2D subsets described above. DVC can be performed using volumetric image datasets. These images can be obtained using confocal microscopy, X-ray computed tomography, Magnetic Resonance Imaging or other techniques. Similar to the other DIC techniques, the images must exhibit a distinct, high-contrast 3D "speckle pattern" to ensure accurate displacement measurement. DVC was first developed in 1999 to study the deformation of trabecular bone using X-ray computed tomography images. Since then, applications of DVC have grown to include granular materials, metals, foams, composites and biological materials. To date it has been used with images acquired by MRI imaging, Computer Tomography (CT), micro-CT, confocal microscopy, and lightsheet microscopy. DVC is currently considered to be ideal in the research world for 3D quantification of local displacements, strains, and stress in biological specimens. It is preferred because of the non-invasiveness of the method over traditional experimental methods. Two of the key challenges are improving the speed and reliability of the DVC measurement. The 3D imaging techniques produce noisier images than conventional 2D optical images, which reduces the quality of the displacement measurement. Computational speed is restricted by the file sizes of 3D images, which are significantly larger than 2D images. For example, an

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  Tensor sketch

  In statistics, machine learning and algorithms, a tensor sketch is a type of dimensionality reduction that is particularly efficient when applied to vectors that have tensor structure. Such a sketch can be used to speed up explicit kernel methods, bilinear pooling in neural networks and is a cornerstone in many numerical linear algebra algorithms. == Mathematical definition == Mathematically, a dimensionality reduction or sketching matrix is a matrix M ∈ R k × d {\displaystyle M\in \mathbb {R} ^{k\times d}} , where k < d {\displaystyle k Read more →

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  Multiple discriminant analysis

  Multiple Discriminant Analysis (MDA) is a multivariate dimensionality reduction technique. It has been used to predict signals as diverse as neural memory traces and corporate failure. MDA is not directly used to perform classification. It merely supports classification by yielding a compressed signal amenable to classification. The method described in Duda et al. (2001) §3.8.3 projects the multivariate signal down to an M−1 dimensional space where M is the number of categories. MDA is useful because most classifiers are strongly affected by the curse of dimensionality. In other words, when signals are represented in very-high-dimensional spaces, the classifier's performance is catastrophically impaired by the overfitting problem. This problem is reduced by compressing the signal down to a lower-dimensional space as MDA does. MDA has been used to reveal neural codes.

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  Arabic Speech Corpus

  The Arabic Speech Corpus is a Modern Standard Arabic (MSA) speech corpus for speech synthesis. The corpus contains phonetic and orthographic transcriptions of more than 3.7 hours of MSA speech aligned with recorded speech on the phoneme level. The annotations include word stress marks on the individual phonemes. The Arabic Speech Corpus was built as part of a doctoral project by Nawar Halabi at the University of Southampton funded by MicroLinkPC who own an exclusive license to commercialise the corpus, but the corpus is available for strictly non-commercial purposes through the official Arabic Speech Corpus website. It is distributed under the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License. == Purpose == The corpus was mainly built for speech synthesis purposes, specifically Speech Synthesis, but the corpus has been used for building HMM based voices in Arabic. It was also used to automatically align other speech corpora with their phonetic transcript and could be used as part of a larger corpus for training speech recognition systems. == Contents == The package contains the following: 1813 .wav files containing spoken utterances. 1813 .lab files containing text utterances. 1813 .TextGrid files containing the phoneme labels with time stamps of the boundaries where these occur in the .wav files. phonetic-transcript.txt which has the form "[wav_filename]" "[Phoneme Sequence]" in every line. orthographic-transcript.txt which has the form "[wav_filename]" "[Orthographic Transcript]" in every line. Orthography is in Buckwalter Format which is friendlier where there is software that does not read Arabic script. It can be easily converted back to Arabic. There is an extra 18 minutes of fully annotated corpus (separate from above but with the same structure as above) which was used to evaluated the corpus (see PhD thesis). The corpus was also used to prove that using automatically extracted, orthography-based stress marks improve the quality of speech synthesis in MSA.

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  Google Tasks

  Google Tasks is a task management application developed by Google and included with Google Workspace. Included initially as a feature in Gmail and Google Calendar, Google Tasks launched as a core product with a standalone app in 2018. It is available for Android and iOS, as well as in the right-hand side panel on Google Workspace apps on the web and in Google Calendar. == History and development == Google Tasks began as an integration within other apps in G Suite (now Google Workspace), allowing to-do items to be created in Calendar and Gmail. Upon graduating to a core service on June 28, 2018, Google Tasks launched as a dedicated mobile app in which tasks can be sorted into lists, managed, and completed. Google Tasks launched the ability to create tasks from Google Chat messages in 2022.

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  Representer theorem

  For computer science, in statistical learning theory, a representer theorem is any of several related results stating that a minimizer f ∗ {\displaystyle f^{}} of a regularized empirical risk functional defined over a reproducing kernel Hilbert space can be represented as a finite linear combination of kernel products evaluated on the input points in the training set data. == Formal statement == The following Representer Theorem and its proof are due to Schölkopf, Herbrich, and Smola: Theorem: Consider a positive-definite real-valued kernel k : X × X → R {\displaystyle k:{\mathcal {X}}\times {\mathcal {X}}\to \mathbb {R} } on a non-empty set X {\displaystyle {\mathcal {X}}} with a corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} . Let there be given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\dotsc ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } , a strictly increasing real-valued function g : [ 0 , ∞ ) → R {\displaystyle g\colon [0,\infty )\to \mathbb {R} } , and an arbitrary error function E : ( X × R 2 ) n → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{n}\to \mathbb {R} \cup \lbrace \infty \rbrace } , which together define the following regularized empirical risk functional on H k {\displaystyle H_{k}} : f ↦ E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) . {\displaystyle f\mapsto E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right).} Then, any minimizer of the empirical risk f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + g ( ‖ f ‖ ) } , ( ∗ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+g\left(\lVert f\rVert \right)\right\rbrace ,\quad ()} admits a representation of the form: f ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} where α i ∈ R {\displaystyle \alpha _{i}\in \mathbb {R} } for all 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . Proof: Define a mapping φ : X → H k φ ( x ) = k ( ⋅ , x ) {\displaystyle {\begin{aligned}\varphi \colon {\mathcal {X}}&\to H_{k}\\\varphi (x)&=k(\cdot ,x)\end{aligned}}} (so that φ ( x ) = k ( ⋅ , x ) {\displaystyle \varphi (x)=k(\cdot ,x)} is itself a map X → R {\displaystyle {\mathcal {X}}\to \mathbb {R} } ). Since k {\displaystyle k} is a reproducing kernel, then φ ( x ) ( x ′ ) = k ( x ′ , x ) = ⟨ φ ( x ′ ) , φ ( x ) ⟩ , {\displaystyle \varphi (x)(x')=k(x',x)=\langle \varphi (x'),\varphi (x)\rangle ,} where ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the inner product on H k {\displaystyle H_{k}} . Given any x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} , one can use orthogonal projection to decompose any f ∈ H k {\displaystyle f\in H_{k}} into a sum of two functions, one lying in span ⁡ { φ ( x 1 ) , … , φ ( x n ) } {\displaystyle \operatorname {span} \left\lbrace \varphi (x_{1}),\ldots ,\varphi (x_{n})\right\rbrace } , and the other lying in the orthogonal complement: f = ∑ i = 1 n α i φ ( x i ) + v , {\displaystyle f=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,} where ⟨ v , φ ( x i ) ⟩ = 0 {\displaystyle \langle v,\varphi (x_{i})\rangle =0} for all i {\displaystyle i} . The above orthogonal decomposition and the reproducing property together show that applying f {\displaystyle f} to any training point x j {\displaystyle x_{j}} produces f ( x j ) = ⟨ ∑ i = 1 n α i φ ( x i ) + v , φ ( x j ) ⟩ = ∑ i = 1 n α i ⟨ φ ( x i ) , φ ( x j ) ⟩ , {\displaystyle f(x_{j})=\left\langle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v,\varphi (x_{j})\right\rangle =\sum _{i=1}^{n}\alpha _{i}\langle \varphi (x_{i}),\varphi (x_{j})\rangle ,} which we observe is independent of v {\displaystyle v} . Consequently, the value of the error function E {\displaystyle E} in () is likewise independent of v {\displaystyle v} . For the second term (the regularization term), since v {\displaystyle v} is orthogonal to ∑ i = 1 n α i φ ( x i ) {\displaystyle \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})} and g {\displaystyle g} is strictly monotonic, we have g ( ‖ f ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) + v ‖ ) = g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ 2 + ‖ v ‖ 2 ) ≥ g ( ‖ ∑ i = 1 n α i φ ( x i ) ‖ ) . {\displaystyle {\begin{aligned}g\left(\lVert f\rVert \right)&=g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})+v\rVert \right)\\&=g\left({\sqrt {\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert ^{2}+\lVert v\rVert ^{2}}}\right)\\&\geq g\left(\lVert \sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})\rVert \right).\end{aligned}}} Therefore, setting v = 0 {\displaystyle v=0} does not affect the first term of (), while it strictly decreases the second term. Consequently, any minimizer f ∗ {\displaystyle f^{}} in () must have v = 0 {\displaystyle v=0} , i.e., it must be of the form f ∗ ( ⋅ ) = ∑ i = 1 n α i φ ( x i ) = ∑ i = 1 n α i k ( ⋅ , x i ) , {\displaystyle f^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}\varphi (x_{i})=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i}),} which is the desired result. == Generalizations == The Theorem stated above is a particular example of a family of results that are collectively referred to as "representer theorems"; here we describe several such. The first statement of a representer theorem was due to Kimeldorf and Wahba for the special case in which E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) = 1 n ∑ i = 1 n ( f ( x i ) − y i ) 2 , g ( ‖ f ‖ ) = λ ‖ f ‖ 2 {\displaystyle {\begin{aligned}E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)&={\frac {1}{n}}\sum _{i=1}^{n}(f(x_{i})-y_{i})^{2},\\g(\lVert f\rVert )&=\lambda \lVert f\rVert ^{2}\end{aligned}}} for λ > 0 {\displaystyle \lambda >0} . Schölkopf, Herbrich, and Smola generalized this result by relaxing the assumption of the squared-loss cost and allowing the regularizer to be any strictly monotonically increasing function g ( ⋅ ) {\displaystyle g(\cdot )} of the Hilbert space norm. It is possible to generalize further by augmenting the regularized empirical risk functional through the addition of unpenalized offset terms. For example, Schölkopf, Herbrich, and Smola also consider the minimization f ~ ∗ = argmin ⁡ { E ( ( x 1 , y 1 , f ~ ( x 1 ) ) , … , ( x n , y n , f ~ ( x n ) ) ) + g ( ‖ f ‖ ) ∣ f ~ = f + h ∈ H k ⊕ span ⁡ { ψ p ∣ 1 ≤ p ≤ M } } , ( † ) {\displaystyle {\tilde {f}}^{}=\operatorname {argmin} \left\lbrace E\left((x_{1},y_{1},{\tilde {f}}(x_{1})),\ldots ,(x_{n},y_{n},{\tilde {f}}(x_{n}))\right)+g\left(\lVert f\rVert \right)\mid {\tilde {f}}=f+h\in H_{k}\oplus \operatorname {span} \lbrace \psi _{p}\mid 1\leq p\leq M\rbrace \right\rbrace ,\quad (\dagger )} i.e., we consider functions of the form f ~ = f + h {\displaystyle {\tilde {f}}=f+h} , where f ∈ H k {\displaystyle f\in H_{k}} and h {\displaystyle h} is an unpenalized function lying in the span of a finite set of real-valued functions { ψ p : X → R ∣ 1 ≤ p ≤ M } {\displaystyle \lbrace \psi _{p}\colon {\mathcal {X}}\to \mathbb {R} \mid 1\leq p\leq M\rbrace } . Under the assumption that the n × M {\displaystyle n\times M} matrix ( ψ p ( x i ) ) i p {\displaystyle \left(\psi _{p}(x_{i})\right)_{ip}} has rank M {\displaystyle M} , they show that the minimizer f ~ ∗ {\displaystyle {\tilde {f}}^{}} in ( † ) {\displaystyle (\dagger )} admits a representation of the form f ~ ∗ ( ⋅ ) = ∑ i = 1 n α i k ( ⋅ , x i ) + ∑ p = 1 M β p ψ p ( ⋅ ) {\displaystyle {\tilde {f}}^{}(\cdot )=\sum _{i=1}^{n}\alpha _{i}k(\cdot ,x_{i})+\sum _{p=1}^{M}\beta _{p}\psi _{p}(\cdot )} where α i , β p ∈ R {\displaystyle \alpha _{i},\beta _{p}\in \mathbb {R} } and the β p {\displaystyle \beta _{p}} are all uniquely determined. The conditions under which a representer theorem exists were investigated by Argyriou, Micchelli, and Pontil, who proved the following: Theorem: Let X {\displaystyle {\mathcal {X}}} be a nonempty set, k {\displaystyle k} a positive-definite real-valued kernel on X × X {\displaystyle {\mathcal {X}}\times {\mathcal {X}}} with corresponding reproducing kernel Hilbert space H k {\displaystyle H_{k}} , and let R : H k → R {\displaystyle R\colon H_{k}\to \mathbb {R} } be a differentiable regularization function. Then given a training sample ( x 1 , y 1 ) , … , ( x n , y n ) ∈ X × R {\displaystyle (x_{1},y_{1}),\ldots ,(x_{n},y_{n})\in {\mathcal {X}}\times \mathbb {R} } and an arbitrary error function E : ( X × R 2 ) m → R ∪ { ∞ } {\displaystyle E\colon ({\mathcal {X}}\times \mathbb {R} ^{2})^{m}\to \mathbb {R} \cup \lbrace \infty \rbrace } , a minimizer f ∗ = argmin f ∈ H k { E ( ( x 1 , y 1 , f ( x 1 ) ) , … , ( x n , y n , f ( x n ) ) ) + R ( f ) } ( ‡ ) {\displaystyle f^{}={\underset {f\in H_{k}}{\operatorname {argmin} }}\left\lbrace E\left((x_{1},y_{1},f(x_{1})),\ldots ,(x_{n},y_{n},f(x_{n}))\right)+R(f)\right\rbrace \quad (\ddagger )} of the regularized empirical risk admits a repr

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  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.

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  Transkribus

  Transkribus is a platform for the text recognition, image analysis and structure recognition of historical documents. The platform was created in the context of the two EU projects "tranScriptorium" (2013–2015) and "READ" (Recognition and Enrichment of Archival Documents – 2016–2019). It was developed by the University of Innsbruck. Since July 1, 2019 the platform has been directed and further developed by the READ-COOP, a non-profit cooperative. The platform integrates tools developed by research groups throughout Europe, including the Pattern Recognition and Human Language Technology (PRHLT) group of the Technical University of Valencia and the Computational Intelligence Technology Lab (CITlab) group of University of Rostock. Comparable programs that offer similar functions are eScriptorium and OCR4All.

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  Hard sigmoid

  In artificial intelligence, especially computer vision and artificial neural networks, a hard sigmoid is non-smooth function used in place of a sigmoid function. These retain the basic shape of a sigmoid, rising from 0 to 1, but using simpler functions, especially piecewise linear functions or piecewise constant functions. These are preferred where speed of computation is more important than precision. == Examples == The most extreme examples are the sign function or Heaviside step function, which go from −1 to 1 or 0 to 1 (which to use depends on normalization) at 0. Other examples include the Theano library, which provides two approximations: ultra_fast_sigmoid, which is a multi-part piecewise approximation and hard_sigmoid, which is a 3-part piecewise linear approximation (output 0, line with slope 0.2, output 1).

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  C4.5 algorithm

  C4.5 is an algorithm used to generate a decision tree developed by Ross Quinlan. C4.5 is an extension of Quinlan's earlier ID3 algorithm. The decision trees generated by C4.5 can be used for classification, and for this reason, C4.5 is often referred to as a statistical classifier. In 2011, authors of the Weka machine learning software described the C4.5 algorithm as "a landmark decision tree program that is probably the machine learning workhorse most widely used in practice to date". It became quite popular after ranking #1 in the Top 10 Algorithms in Data Mining pre-eminent paper published by Springer LNCS in 2008. == Algorithm == C4.5 builds decision trees from a set of training data in the same way as ID3, using the concept of information entropy. The training data is a set S = s 1 , s 2 , . . . {\displaystyle S={s_{1},s_{2},...}} of already classified samples. Each sample s i {\displaystyle s_{i}} consists of a p-dimensional vector ( x 1 , i , x 2 , i , . . . , x p , i ) {\displaystyle (x_{1,i},x_{2,i},...,x_{p,i})} , where the x j {\displaystyle x_{j}} represent attribute values or features of the sample, as well as the class in which s i {\displaystyle s_{i}} falls. At each node of the tree, C4.5 chooses the attribute of the data that most effectively splits its set of samples into subsets enriched in one class or the other. The splitting criterion is the normalized information gain (difference in entropy). The attribute with the highest normalized information gain is chosen to make the decision. The C4.5 algorithm then recurses on the partitioned sublists. This algorithm has a few base cases. All the samples in the list belong to the same class. When this happens, it simply creates a leaf node for the decision tree saying to choose that class. None of the features provide any information gain. In this case, C4.5 creates a decision node higher up the tree using the expected value of the class. Instance of previously unseen class encountered. Again, C4.5 creates a decision node higher up the tree using the expected value. === Pseudocode === In pseudocode, the general algorithm for building decision trees is: Check for the above base cases. For each attribute a, find the normalized information gain ratio from splitting on a. Let a_best be the attribute with the highest normalized information gain. Create a decision node that splits on a_best. Recurse on the sublists obtained by splitting on a_best, and add those nodes as children of node. == Improvements from ID3 algorithm == C4.5 made a number of improvements to ID3. Some of these are: Handling both continuous and discrete attributes: In order to handle continuous attributes, C4.5 creates a threshold and then splits the list into those whose attribute value is above the threshold and those that are less than or equal to it. Handling training data with missing attribute values: C4.5 allows attribute values to be marked as missing. Missing attribute values are simply not used in gain and entropy calculations. Handling attributes with differing costs. Pruning trees after creation: C4.5 goes back through the tree once it's been created and attempts to remove branches that do not help by replacing them with leaf nodes. == Improvements in C5.0/See5 algorithm == Quinlan went on to create C5.0 and See5 (C5.0 for Unix/Linux, See5 for Windows) which he markets commercially. C5.0 offers a number of improvements on C4.5. Some of these are: Speed - C5.0 is significantly faster than C4.5 (several orders of magnitude) Memory usage - C5.0 is more memory efficient than C4.5 Smaller decision trees - C5.0 gets similar results to C4.5 with considerably smaller decision trees. Support for boosting - Boosting improves the trees and gives them more accuracy. Weighting - C5.0 allows you to weight different cases and misclassification types. Winnowing - a C5.0 option automatically winnows the attributes to remove those that may be unhelpful. Source for a single-threaded Linux version of C5.0 is available under the GNU General Public License (GPL).

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  Correlation clustering

  Clustering is the problem of partitioning data points into groups based on similarity or dissimilarity. Correlation clustering is a clustering framework in which a set of objects is partitioned into clusters based on pairwise similarity and dissimilarity information, without requiring the number of clusters to be specified in advance. == Description of the problem == In machine learning, correlation clustering (also known as cluster editing) considers settings in which pairwise similarity or dissimilarity relationships between objects are known. A standard formulation models the input as an unweighted complete graph G = ( V , E ) {\displaystyle G=(V,E)} , where each edge is labeled either + {\displaystyle +} or − {\displaystyle -} (that is, the graph is a signed graph), indicating whether the corresponding endpoints are similar or dissimilar. The goal is to find a clustering (that is, a partition of V {\displaystyle V} ) that either maximizes the number of agreements—the sum of positive edges whose endpoints lie in the same cluster and negative edges whose endpoints lie in different clusters—or minimizes the number of disagreements—the sum of positive edges whose endpoints are separated and negative edges whose endpoints lie in the same cluster. Unlike other clustering methods such as k-means, correlation clustering does not require choosing the number of clusters k {\displaystyle k} in advance. It is not always possible to find a clustering with zero disagreements. For example, consider a triangle graph containing two positive edges and one negative edge. In this case, every clustering incurs at least one disagreement. Such configurations are referred to in the literature as bad triangles. From a computational perspective, optimizing the correlation clustering objective is challenging. The (decision version of the) problem is NP-complete. A large body of subsequent work has developed approximation algorithms for correlation clustering under various assumptions, including complete or general graphs and unweighted or weighted graphs, for both minimization and maximization objectives. This problem is considered one of the fundamental combinatorial optimization problems, and many algorithmic techniques have been developed to address it. The problem has also been studied extensively across multiple disciplines. A comprehensive literature review of early correlation clustering research is provided by Wahid and Hassini. == Formal Definitions == Let G = ( V , E ) {\displaystyle G=(V,E)} be a graph with nodes V {\displaystyle V} and edges E {\displaystyle E} . A clustering of G {\displaystyle G} is a partition of its node set Π = { π 1 , … , π k } {\displaystyle \Pi =\{\pi _{1},\dots ,\pi _{k}\}} with V = π 1 ∪ ⋯ ∪ π k {\displaystyle V=\pi _{1}\cup \dots \cup \pi _{k}} and π i ∩ π j = ∅ {\displaystyle \pi _{i}\cap \pi _{j}=\emptyset } for i ≠ j {\displaystyle i\neq j} . For a given clustering Π {\displaystyle \Pi } , let δ ( Π ) = { { u , v } ∈ E ∣ { u , v } ⊈ π ∀ π ∈ Π } {\displaystyle \delta (\Pi )=\{\{u,v\}\in E\mid \{u,v\}\not \subseteq \pi \;\forall \pi \in \Pi \}} denote the subset of edges of G {\displaystyle G} whose endpoints are in different subsets of the clustering Π {\displaystyle \Pi } . Now, let w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} be a function that assigns a non-negative weight to each edge of the graph and let E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} be a partition of the edges into attractive ( E + {\displaystyle E^{+}} ) and repulsive ( E − {\displaystyle E^{-}} ) edges; that is, the edges are signed. The minimum disagreement correlation clustering problem is the following optimization problem: minimize Π ∑ e ∈ E + ∩ δ ( Π ) w e + ∑ e ∈ E − ∖ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in E^{+}\cap \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\setminus \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∩ δ ( Π ) {\displaystyle E^{+}\cap \delta (\Pi )} contains the attractive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } and the set E − ∖ δ ( Π ) {\displaystyle E^{-}\setminus \delta (\Pi )} contains the repulsive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that disagree with the clustering Π {\displaystyle \Pi } . Similarly to the minimum disagreement correlation clustering problem, the maximum agreement correlation clustering problem is defined as maximize Π ∑ e ∈ E + ∖ δ ( Π ) w e + ∑ e ∈ E − ∩ δ ( Π ) w e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E^{+}\setminus \delta (\Pi )}w_{e}+\sum _{e\in E^{-}\cap \delta (\Pi )}w_{e}\;.\end{aligned}}} Here, the set E + ∖ δ ( Π ) {\displaystyle E^{+}\setminus \delta (\Pi )} contains the attractive edges whose endpoints are in the same component with respect to the clustering Π {\displaystyle \Pi } and the set E − ∩ δ ( Π ) {\displaystyle E^{-}\cap \delta (\Pi )} contains the repulsive edges whose endpoints are in different components with respect to the clustering Π {\displaystyle \Pi } . Together these two sets contain all edges that agree with the clustering Π {\displaystyle \Pi } . Instead of formulating the correlation clustering problem in terms of non-negative edge weights and a partition of the edges into attractive and repulsive edges the problem is also formulated in terms of positive and negative edge costs without partitioning the set of edges explicitly. For given weights w : E → R ≥ 0 {\displaystyle w\colon E\to \mathbb {R} _{\geq 0}} and a given partition E = E + ∪ E − {\displaystyle E=E^{+}\cup E^{-}} of the edges into attractive and repulsive edges, the edge costs can be defined by c e = { w e if e ∈ E + − w e if e ∈ E − {\displaystyle {\begin{aligned}c_{e}={\begin{cases}\;\;w_{e}&{\text{if }}e\in E^{+}\\-w_{e}&{\text{if }}e\in E^{-}\end{cases}}\end{aligned}}} for all e ∈ E {\displaystyle e\in E} . An edge whose endpoints are in different clusters is said to be cut. The set δ ( Π ) {\displaystyle \delta (\Pi )} of all edges that are cut is often called a multicut of G {\displaystyle G} . The minimum cost multicut problem is the problem of finding a clustering Π {\displaystyle \Pi } of G {\displaystyle G} such that the sum of the costs of the edges whose endpoints are in different clusters is minimal: minimize Π ∑ e ∈ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {minimize} }}&&\sum _{e\in \delta (\Pi )}c_{e}\;.\end{aligned}}} Similar to the minimum cost multicut problem, coalition structure generation in weighted graph games is the problem of finding a clustering such that the sum of the costs of the edges that are not cut is maximal: maximize Π ∑ e ∈ E ∖ δ ( Π ) c e . {\displaystyle {\begin{aligned}&{\underset {\Pi }{\operatorname {maximize} }}&&\sum _{e\in E\setminus \delta (\Pi )}c_{e}\;.\end{aligned}}} This formulation is also known as the clique partitioning problem. It can be shown that all four problems that are formulated above are equivalent. This means that a clustering that is optimal with respect to any of the four objectives is optimal for all of the four objectives. == Algorithms == If the graph admits a clustering with zero disagreements, then deleting all negative edges and computing the connected components of the remaining graph yields an optimal clustering. A necessary and sufficient condition for the existence of such a clustering was given by Davis: no cycle in the graph may contain exactly one negative edge. Bansal et al. discuss the NP-completeness proof and also present both a constant factor approximation algorithm and polynomial-time approximation scheme to find the clusters in this setting. Ailon et al. propose a randomized 3-approximation algorithm for the same problem. CC-Pivot(G=(V,E+,E−)) Pick random pivot i ∈ V Set C = { i } {\displaystyle C=\{i\}} , V'=Ø For all j ∈ V, j ≠ i; If (i,j) ∈ E+ then Add j to C Else (If (i,j) ∈ E−) Add j to V' Let G' be the subgraph induced by V' Return clustering C,CC-Pivot(G') The authors show that the above algorithm is a 3-approximation algorithm for correlation clustering. The best polynomial-time approximation algorithm known at the moment for this problem achieves a ~2.06 approximation by rounding a linear program, as shown by Chawla, Makarychev, Schramm, and Yaroslavtsev. Karpinski and Schudy proved existence of a polynomial time approximation scheme (PTAS) for that problem on complete graphs and fixed number of clusters. == Optimal number of clusters == In 2011, it was shown by Bagon and Galun that the optimization of the correlation clustering functional is closely related to well known discrete optimization methods. In their work they proposed a probabilistic analysis of the underlying implicit model that allows the correlation clustering functional to estimate the

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