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Asymptotic Geometric Analysis
Asymptotic geometry, also known as asymptotic geometric analysis or high-dimensional geometry, is a field of mathematics that investigates the geometric properties of finite-dimensional objects, such as convex bodies and normed spaces, as the dimension tends to infinity. It is at the intersection of convex geometry and functional analysis. The primary objects of study are typically finite-dimensional normed spaces, which can be represented as \mathbb^n equipped with a norm \, \cdot\, , or equivalently, a unit ball K_X = \, which is a centrally symmetric, compact, convex set with a non-empty interior. History One early approach to Banach space theory was the "local theory of normed spaces", which aimed to understand infinite-dimensional Banach spaces by examining their finite-dimensional subspaces and quotient spaces. John von Neumann in 1942 studied the asymptotic behavior of E^n (n-dimensional Euclidean space) and M_n (the space of n×n matrices) for finite n as n \to \infty, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Convex Body
In mathematics, a convex body in n-dimensional Euclidean space \R^n is a compact convex set with non- empty interior. Some authors do not require a non-empty interior, merely that the set is non-empty. A convex body K is called symmetric if it is centrally symmetric with respect to the origin; that is to say, a point x lies in K if and only if its antipode, - x also lies in K. Symmetric convex bodies are in a one-to-one correspondence In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivale ... with the unit balls of Norm (mathematics), norms on \R^n. Some commonly known examples of convex bodies are the Euclidean ball, the hypercube and the cross-polytope. Metric space structure Write \mathcal K^n for the set of convex bodies in \mathbb R^n. Then \mathcal K^n is a complete metric spac ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Lebesgue Measure
In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of higher dimensional Euclidean '-spaces. For lower dimensions or , it coincides with the standard measure of length, area, or volume. In general, it is also called '-dimensional volume, '-volume, hypervolume, or simply volume. It is used throughout real analysis, in particular to define Lebesgue integration. Sets that can be assigned a Lebesgue measure are called Lebesgue-measurable; the measure of the Lebesgue-measurable set A is here denoted by \lambda(A). Henri Lebesgue described this measure in the year 1901 which, a year after, was followed up by his description of the Lebesgue integral. Both were published as part of his dissertation in 1902. Definition For any interval I = ,b/math>, or I = (a, b), in the set \mathbb of real numbers, let \ell(I)= b - a denote its length. For any subset E\subseteq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Functional Analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (for example, Inner product space#Definition, inner product, Norm (mathematics)#Definition, norm, or Topological space#Definitions, topology) and the linear transformation, linear functions defined on these spaces and suitably respecting these structures. The historical roots of functional analysis lie in the study of function space, spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining, for example, continuous function, continuous or unitary operator, unitary operators between function spaces. This point of view turned out to be particularly useful for the study of differential equations, differential and integral equations. The usage of the word ''functional (mathematics), functional'' as a noun goes back to the calculus of v ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Dvoretzky's Theorem
In mathematics, Dvoretzky's theorem is an important structural theorem about normed vector spaces proved by Aryeh Dvoretzky in the early 1960s, answering a question of Alexander Grothendieck. In essence, it says that every sufficiently high-dimensional normed vector space will have low-dimensional subspaces that are approximately Euclidean. Equivalently, every high-dimensional bounded symmetric convex set has low-dimensional sections that are approximately ellipsoids. A new proof found by Vitali Milman in the 1970s was one of the starting points for the development of asymptotic geometric analysis (also called ''asymptotic functional analysis'' or the ''local theory of Banach spaces''). Original formulations For every natural number ''k'' ∈ N and every ''ε'' > 0 there exists a natural number ''N''(''k'', ''ε'') ∈ N such that if (''X'', ‖·‖) is any normed space of dimension ''N''(''k'', ''ε''), th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Vapnik–Chervonenkis Dimension
In Vapnik–Chervonenkis theory, the Vapnik–Chervonenkis (VC) dimension is a measure of the size (capacity, complexity, expressive power, richness, or flexibility) of a class of sets. The notion can be extended to classes of binary functions. It is defined as the cardinality of the largest set of points that the algorithm can shatter, which means the algorithm can always learn a perfect classifier for any labeling of at least one configuration of those data points. It was originally defined by Vladimir Vapnik and Alexey Chervonenkis. Informally, the capacity of a classification model is related to how complicated it can be. For example, consider the thresholding of a high- degree polynomial: if the polynomial evaluates above zero, that point is classified as positive, otherwise as negative. A high-degree polynomial can be wiggly, so that it can fit a given set of training points well. But one can expect that the classifier will make errors on other points, because it is too wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Rademacher Complexity
In computational learning theory (machine learning and theory of computation), Rademacher complexity, named after Hans Rademacher, measures richness of a class of sets with respect to a probability distribution. The concept can also be extended to real valued functions. Definitions Rademacher complexity of a set Given a set A\subseteq \mathbb^m, the Rademacher complexity of ''A'' is defined as follows:Chapter 26 in : \operatorname(A) := \frac \mathbb_\sigma \left \sup_ \sum_^m \sigma_i a_i \right where \sigma_1, \sigma_2, \dots, \sigma_m are independent random variables drawn from the Rademacher distribution i.e. \Pr(\sigma_i = +1) = \Pr(\sigma_i = -1) = 1/2 for i=1,2,\dots,m, and a=(a_1, \ldots, a_m). Some authors take the absolute value of the sum before taking the supremum, but if A is symmetric this makes no difference. Rademacher complexity of a function class Let S=\ \subset Z be a sample of points and consider a function class \mathcal of real-valued ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Covering Number
In mathematics, a covering number is the number of balls of a given size needed to completely cover a given space, with possible overlaps between the balls. The covering number quantifies the size of a set and can be applied to general metric spaces. Two related concepts are the ''packing number'', the number of disjoint balls that fit in a space, and the ''metric entropy'', the number of points that fit in a space when constrained to lie at some fixed minimum distance apart. Definition Let (''M'', ''d'') be a metric space, let ''K'' be a subset of ''M'', and let ''r'' be a positive real number. Let ''B''''r''(''x'') denote the ball of radius ''r'' centered at ''x''. A subset ''C'' of ''M'' is an ''r-external covering'' of ''K'' if: :K \subseteq \bigcup_ B_r(x). In other words, for every y\in K there exists x\in C such that d(x,y)\leq r. If furthermore ''C'' is a subset of ''K'', then it is an ''r-internal covering''. The external covering number of ''K'', denoted N^_r(K), is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaussian Measure
In mathematics, Gaussian measure is a Borel measure on finite-dimensional Euclidean space \mathbb^n, closely related to the normal distribution in statistics. There is also a generalization to infinite-dimensional spaces. Gaussian measures are named after the German mathematician Carl Friedrich Gauss. One reason why Gaussian measures are so ubiquitous in probability theory is the central limit theorem. Loosely speaking, it states that if a random variable X is obtained by summing a large number N of independent random variables with variance 1, then X has variance N and its law is approximately Gaussian. Definitions Let n \in N and let B_0(\mathbb^n) denote the completion of the Borel \sigma-algebra on \mathbb^n. Let \lambda^n : B_0(\mathbb^n) \to , +\infty/math> denote the usual n-dimensional Lebesgue measure. Then the standard Gaussian measure \gamma^n : B_0(\mathbb^n) \to , 1/math> is defined by \gamma^ (A) = \frac \int_ \exp \left( - \frac \left\, x \right\, _^ \right) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Gaussian Process
In probability theory and statistics, a Gaussian process is a stochastic process (a collection of random variables indexed by time or space), such that every finite collection of those random variables has a multivariate normal distribution. The distribution of a Gaussian process is the joint distribution of all those (infinitely many) random variables, and as such, it is a distribution over functions with a continuous domain, e.g. time or space. The concept of Gaussian processes is named after Carl Friedrich Gauss because it is based on the notion of the Gaussian distribution (normal distribution). Gaussian processes can be seen as an infinite-dimensional generalization of multivariate normal distributions. Gaussian processes are useful in statistical modelling, benefiting from properties inherited from the normal distribution. For example, if a random process is modelled as a Gaussian process, the distributions of various derived quantities can be obtained explicitly. Such quanti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Mahler Volume
In convex geometry, the Mahler volume of a centrally symmetric convex body is a dimensionless quantity that is associated with the body and is invariant under linear transformations. It is named after German-English mathematician Kurt Mahler. It is known that the shapes with the largest possible Mahler volume are the balls and solid ellipsoids; this is now known as the Blaschke–Santaló inequality. The still-unsolved Mahler conjecture states that the minimum possible Mahler volume is attained by a hypercube. Definition A convex body in Euclidean space is defined as a compact convex set with non-empty interior. If B is a centrally symmetric convex body in n-dimensional Euclidean space, the polar body B^\circ is another centrally symmetric body in the same space, defined as the set \left\. The Mahler volume of B is the product of the volumes of B and B^\circ.. If T is an invertible linear transformation, then (TB)^\circ = (T^)^\ast B^\circ. Applying T to B multiplies its volu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
