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Wasserstein GAN
The Wasserstein Generative Adversarial Network (WGAN) is a variant of generative adversarial network (GAN) proposed in 2017 that aims to "improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches". Compared with the original GAN discriminator, the Wasserstein GAN discriminator provides a better learning signal to the generator. This allows the training to be more stable when generator is learning distributions in very high dimensional spaces. Motivation The GAN game The original GAN method is based on the GAN game, a zero-sum game with 2 players: generator and discriminator. The game is defined over a probability space (\Omega, \mathcal B, \mu_), The generator's strategy set is the set of all probability measures \mu_G on (\Omega, \mathcal B), and the discriminator's strategy set is the set of measurable functions D: \Omega \to , 1/math>. The objective of the game isL(\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generative Adversarial Network
A generative adversarial network (GAN) is a class of machine learning frameworks designed by Ian Goodfellow and his colleagues in June 2014. Two neural networks contest with each other in the form of a zero-sum game, where one agent's gain is another agent's loss. Given a training set, this technique learns to generate new data with the same statistics as the training set. For example, a GAN trained on photographs can generate new photographs that look at least superficially authentic to human observers, having many realistic characteristics. Though originally proposed as a form of generative model for unsupervised learning, GANs have also proved useful for semi-supervised learning, fully supervised learning, and reinforcement learning. The core idea of a GAN is based on the "indirect" training through the discriminator, another neural network that can tell how "realistic" the input seems, which itself is also being updated dynamically. This means that the generator is not trai ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Neural Network Architectures
In biology, the nervous system is the highly complex part of an animal that coordinates its actions and sensory information by transmitting signals to and from different parts of its body. The nervous system detects environmental changes that impact the body, then works in tandem with the endocrine system to respond to such events. Nervous tissue first arose in wormlike organisms about 550 to 600 million years ago. In vertebrates it consists of two main parts, the central nervous system (CNS) and the peripheral nervous system (PNS). The CNS consists of the brain and spinal cord. The PNS consists mainly of nerves, which are enclosed bundles of the long fibers or axons, that connect the CNS to every other part of the body. Nerves that transmit signals from the brain are called motor nerves or '' efferent'' nerves, while those nerves that transmit information from the body to the CNS are called sensory nerves or '' afferent''. Spinal nerves are mixed nerves that serve both func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transportation Theory (mathematics)
In mathematics and economics, transportation theory or transport theory is a name given to the study of optimal transportation and allocation of resources. The problem was formalized by the French mathematician Gaspard Monge in 1781.G. Monge. ''Mémoire sur la théorie des déblais et des remblais. Histoire de l’Académie Royale des Sciences de Paris, avec les Mémoires de Mathématique et de Physique pour la même année'', pages 666–704, 1781. In the 1920s A.N. Tolstoi was one of the first to study the transportation problem mathematically. In 1930, in the collection ''Transportation Planning Volume I'' for the National Commissariat of Transportation of the Soviet Union, he published a paper "Methods of Finding the Minimal Kilometrage in Cargo-transportation in space". Major advances were made in the field during World War II by the Soviet mathematician and economist Leonid Kantorovich.L. Kantorovich. ''On the translocation of masses.'' C.R. (Doklady) Acad. Sci. URSS (N.S. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Earth Mover's Distance
In statistics, the earth mover's distance (EMD) is a measure of the distance between two probability distributions over a region ''D''. In mathematics, this is known as the Wasserstein metric. Informally, if the distributions are interpreted as two different ways of piling up a certain amount of earth (dirt) over the region ''D'', the EMD is the minimum cost of turning one pile into the other; where the cost is assumed to be the amount of dirt moved times the distance by which it is moved. The above definition is valid only if the two distributions have the same integral (informally, if the two piles have the same amount of dirt), as in normalized histograms or probability density functions. In that case, the EMD is equivalent to the 1st Mallows distance or 1st Wasserstein distance between the two distributions. Theory Assume that we have a set of points in \mathbb^d (dimension d). Instead of assigning one distribution to the set of points, we can cluster them and represe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Memoization
In computing, memoization or memoisation is an optimization technique used primarily to speed up computer programs by storing the results of expensive function calls and returning the cached result when the same inputs occur again. Memoization has also been used in other contexts (and for purposes other than speed gains), such as in simple mutually recursive descent parsing. Although related to caching, memoization refers to a specific case of this optimization, distinguishing it from forms of caching such as buffering or page replacement. In the context of some logic programming languages, memoization is also known as tabling. Etymology The term "memoization" was coined by Donald Michie in 1968 and is derived from the Latin word "memorandum" ("to be remembered"), usually truncated as "memo" in American English, and thus carries the meaning of "turning he results ofa function into something to be remembered". While "memoization" might be confused with "memorization" (becaus ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Operators
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism. If a linear map is a bijection then it is called a . In the case where V = W, a linear map is called a (linear) ''endomorphism''. Sometimes the term refers to this case, but the term "linear operator" can have different meanings for different conventions: for example, it can be used to emphasize that V and W are real vector spaces (not necessarily with V = W), or it can be used to emphasize that V is a function space, which is a common convention in functional analysis. Sometimes the term ''linear function'' has the same meaning as ''linear map'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectral Radius
In mathematics, the spectral radius of a square matrix is the maximum of the absolute values of its eigenvalues. More generally, the spectral radius of a bounded linear operator is the supremum of the absolute values of the elements of its spectrum. The spectral radius is often denoted by . Definition Matrices Let be the eigenvalues of a matrix . The spectral radius of is defined as :\rho(A) = \max \left \. The spectral radius can be thought of as an infimum of all norms of a matrix. Indeed, on the one hand, \rho(A) \leqslant \, A\, for every natural matrix norm \, \cdot\, ; and on the other hand, Gelfand's formula states that \rho(A) = \lim_ \, A^k\, ^ . Both of these results are shown below. However, the spectral radius does not necessarily satisfy \, A\mathbf\, \leqslant \rho(A) \, \mathbf\, for arbitrary vectors \mathbf \in \mathbb^n . To see why, let r > 1 be arbitrary and consider the matrix : C_r = \begin 0 & r^ \\ r & 0 \end . The characteristic polynomial ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Value
In mathematics, in particular functional analysis, the singular values, or ''s''-numbers of a compact operator T: X \rightarrow Y acting between Hilbert spaces X and Y, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T^*T (where T^* denotes the adjoint of T). The singular values are non-negative real numbers, usually listed in decreasing order (''σ''1(''T''), ''σ''2(''T''), …). The largest singular value ''σ''1(''T'') is equal to the operator norm of ''T'' (see Min-max theorem). If ''T'' acts on Euclidean space \Reals ^n, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of T (the figure provides an example in \Reals^2). The singular values are the absolute values of the eigenvalues of a normal matrix ''A'', because the spectral theorem can be applied to obtain unitary diagonalization of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Operator Norm
In mathematics, the operator norm measures the "size" of certain linear operators by assigning each a real number called its . Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces. Introduction and definition Given two normed vector spaces V and W (over the same base field, either the real numbers \R or the complex numbers \Complex), a linear map A : V \to W is continuous if and only if there exists a real number c such that \, Av\, \leq c \, v\, \quad \mbox v\in V. The norm on the left is the one in W and the norm on the right is the one in V. Intuitively, the continuous operator A never increases the length of any vector by more than a factor of c. Thus the image of a bounded set under a continuous operator is also bounded. Because of this property, the continuous linear operators are also known as bounded operators. In order to "measure the size" of A, one can take the infimum of the numbers c such that the above i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chain Rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , then the chain rule is, in Lagrange's notation, :h'(x) = f'(g(x)) g'(x). or, equivalently, :h'=(f\circ g)'=(f'\circ g)\cdot g'. The chain rule may also be expressed in Leibniz's notation. If a variable depends on the variable , which itself depends on the variable (that is, and are dependent variables), then depends on as well, via the intermediate variable . In this case, the chain rule is expressed as :\frac = \frac \cdot \frac, and : \left.\frac\_ = \left.\frac\_ \cdot \left. \frac\_ , for indicating at which points the derivatives have to be evaluated. In integration, the counterpart to the chain rule is the substitution rule. Intuitive explanation Intuitively, the chain rule states that knowing the instantaneous rate of cha ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbolic Tangent Function
In mathematics, hyperbolic functions are analogues of the ordinary trigonometric functions, but defined using the hyperbola rather than the circle. Just as the points form a circle with a unit radius, the points form the right half of the unit hyperbola. Also, similarly to how the derivatives of and are and respectively, the derivatives of and are and respectively. Hyperbolic functions occur in the calculations of angles and distances in hyperbolic geometry. They also occur in the solutions of many linear differential equations (such as the equation defining a catenary), cubic equations, and Laplace's equation in Cartesian coordinates. Laplace's equations are important in many areas of physics, including electromagnetic theory, heat transfer, fluid dynamics, and special relativity. The basic hyperbolic functions are: * hyperbolic sine "" (), * hyperbolic cosine "" (),''Collins Concise Dictionary'', p. 328 from which are derived: * hyperbolic tangent "" (), * hy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
