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Small-bias Sample Space
In theoretical computer science, a small-bias sample space (also known as \epsilon-biased sample space, \epsilon-biased generator, or small-bias probability space) is a probability distribution that fools parity functions. In other words, no parity function can distinguish between a small-bias sample space and the uniform distribution with high probability, and hence, small-bias sample spaces naturally give rise to pseudorandom generators for parity functions. The main useful property of small-bias sample spaces is that they need far fewer truly random bits than the uniform distribution to fool parities. Efficient constructions of small-bias sample spaces have found many applications in computer science, some of which are derandomization, error-correcting codes, and probabilistically checkable proofs. The connection with error-correcting codes is in fact very strong since \epsilon-biased sample spaces are ''equivalent'' to \epsilon-balanced error-correcting codes. Definition Bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theoretical Computer Science
Theoretical computer science (TCS) is a subset of general computer science and mathematics that focuses on mathematical aspects of computer science such as the theory of computation, lambda calculus, and type theory. It is difficult to circumscribe the theoretical areas precisely. The Association for Computing Machinery, ACM's ACM SIGACT, Special Interest Group on Algorithms and Computation Theory (SIGACT) provides the following description: History While logical inference and mathematical proof had existed previously, in 1931 Kurt Gödel proved with his incompleteness theorem that there are fundamental limitations on what statements could be proved or disproved. Information theory was added to the field with a 1948 mathematical theory of communication by Claude Shannon. In the same decade, Donald Hebb introduced a mathematical model of Hebbian learning, learning in the brain. With mounting biological data supporting this hypothesis with some modification, the fields of n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Justesen Code
In coding theory, Justesen codes form a class of error-correcting codes that have a constant rate, constant relative distance, and a constant alphabet size. Before the Justesen error correction code was discovered, no error correction code was known that had all of these three parameters as a constant. Subsequently, other ECC codes with this property have been discovered, for example expander codes. These codes have important applications in computer science such as in the construction of small-bias sample spaces. Justesen codes are derived as the code concatenation of a Reed–Solomon code and the Wozencraft ensemble. The Reed–Solomon codes used achieve constant rate and constant relative distance at the expense of an alphabet size that is ''linear'' in the message length. The Wozencraft ensemble is a family of codes that achieve constant rate and constant alphabet size, but the relative distance is only constant for most of the codes in the family. The concatenation o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Mapping
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] |
P-norm
In mathematics, the spaces are function spaces defined using a natural generalization of the -norm for finite-dimensional vector spaces. They are sometimes called Lebesgue spaces, named after Henri Lebesgue , although according to the Bourbaki group they were first introduced by Frigyes Riesz . spaces form an important class of Banach spaces in functional analysis, and of topological vector spaces. Because of their key role in the mathematical analysis of measure and probability spaces, Lebesgue spaces are used also in the theoretical discussion of problems in physics, statistics, economics, finance, engineering, and other disciplines. Applications Statistics In statistics, measures of central tendency and statistical dispersion, such as the mean, median, and standard deviation, are defined in terms of metrics, and measures of central tendency can be characterized as solutions to variational problems. In penalized regression, "L1 penalty" and "L2 penalty" refer to penaliz ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform Distribution (discrete)
In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution wherein a finite number of values are equally likely to be observed; every one of ''n'' values has equal probability 1/''n''. Another way of saying "discrete uniform distribution" would be "a known, finite number of outcomes equally likely to happen". A simple example of the discrete uniform distribution is throwing a fair dice. The possible values are 1, 2, 3, 4, 5, 6, and each time the die is thrown the probability of a given score is 1/6. If two dice are thrown and their values added, the resulting distribution is no longer uniform because not all sums have equal probability. Although it is convenient to describe discrete uniform distributions over integers, such as this, one can also consider discrete uniform distributions over any finite set. For instance, a random permutation is a permutation generated uniformly from the permutations of a given length, and a unif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Marginal Distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables in the subset without reference to the values of the other variables. This contrasts with a conditional distribution, which gives the probabilities contingent upon the values of the other variables. Marginal variables are those variables in the subset of variables being retained. These concepts are "marginal" because they can be found by summing values in a table along rows or columns, and writing the sum in the margins of the table. The distribution of the marginal variables (the marginal distribution) is obtained by marginalizing (that is, focusing on the sums in the margin) over the distribution of the variables being discarded, and the discarded variables are said to have been marginalized out. The context here is that the theoretical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebraic Geometric Code
In mathematics, an algebraic geometric code (AG-code), otherwise known as a Goppa code, is a general type of linear code constructed by using an algebraic curve X over a finite field \mathbb_q. Such codes were introduced by Valerii Denisovich Goppa. In particular cases, they can have interesting extremal property, extremal properties, making them useful for a variety of error detection and correction problems. They should not be confused with binary Goppa codes that are used, for instance, in the McEliece cryptosystem. Construction Traditionally, an AG-code is constructed from a non-singular projective curve X over a finite field \mathbb_q by using a number of fixed distinct \mathbb_q-rational points on \mathbf: :\mathcal:= \ \subset \mathbf (\mathbb_q). Let G be a divisor (algebraic geometry), divisor on X, with a Support (mathematics), support that consists of only rational points and that is disjoint from the P_i (i.e., \mathcal \cap \operatorname(G) = \varnothing). By the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hadamard Code
The Hadamard code is an error-correcting code named after Jacques Hadamard that is used for error detection and correction when transmitting messages over very noisy or unreliable channels. In 1971, the code was used to transmit photos of Mars back to Earth from the NASA space probe Mariner 9. Because of its unique mathematical properties, the Hadamard code is not only used by engineers, but also intensely studied in coding theory, mathematics, and theoretical computer science. The Hadamard code is also known under the names Walsh code, Walsh family, and Walsh–Hadamard code in recognition of the American mathematician Joseph Leonard Walsh. The Hadamard code is an example of a linear code of length 2^m over a binary alphabet. Unfortunately, this term is somewhat ambiguous as some references assume a message length k = m while others assume a message length of k = m+1. In this article, the first case is called the Hadamard code while the second is called the augmented Hadamar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expander Walk Sampling
In the mathematical discipline of graph theory, the expander walk sampling theorem intuitively states that sampling vertices in an expander graph by doing relatively short random walk can simulate sampling the vertices independently from a uniform distribution. The earliest version of this theorem is due to , and the more general version is typically attributed to . Statement Let G=(V,E) be an n-vertex expander graph with positively weighted edges, and let A\subset V. Let P denote the stochastic matrix of the graph, and let \lambda_2 be the second largest eigenvalue of P. Let y_0, y_1, \ldots, y_ denote the vertices encountered in a (k-1)-step random walk on G starting at vertex y_0, and let \pi (A):= \lim_ \frac \sum_^ \mathbf_A(y_i). Where \mathbf_A(y)\begin 1, & \texty \in A \\ 0, & \text\end (It is well known that almost all trajectories y_0, y_1, \ldots, y_ converges to some limiting point, \pi (A), as k \rightarrow \infty .) The theorem states that for a weighted graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wozencraft Ensemble
In coding theory, the Wozencraft ensemble is a set of linear codes in which most of codes satisfy the Gilbert-Varshamov bound. It is named after John Wozencraft, who proved its existence. The ensemble is described by , who attributes it to Wozencraft. used the Wozencraft ensemble as the inner codes in his construction of strongly explicit asymptotically good code. Existence theorem :Theorem: Let \varepsilon > 0. For a large enough k, there exists an ensemble of inner codes C_^1,\cdots,C_^N of rate \tfrac, where N = q^k - 1, such that for at least (1 - \varepsilon)N values of i, C_^i has relative distance \geqslant H_q^ \left(\tfrac - \varepsilon \right ). Here relative distance is the ratio of minimum distance to block length. And H_q is the q-ary entropy function defined as follows: :H_q(x) = x\log_q(q-1)-x\log_qx-(1-x)\log_q(1-x). In fact, to show the existence of this set of linear codes, we will specify this ensemble explicitly as follows: for \alpha \in \mathbb_ - \, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generator Matrix
In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. The codewords are all of the linear combinations of the rows of this matrix, that is, the linear code is the row space of its generator matrix. Terminology If G is a matrix, it generates the codewords of a linear code ''C'' by : w=sG where w is a codeword of the linear code ''C'', and s is any input vector. Both w and s are assumed to be row vectors. A generator matrix for a linear , k, dq-code has format k \times n, where ''n'' is the length of a codeword, ''k'' is the number of information bits (the dimension of ''C'' as a vector subspace), ''d'' is the minimum distance of the code, and ''q'' is size of the finite field, that is, the number of symbols in the alphabet (thus, ''q'' = 2 indicates a binary code, etc.). The number of redundant bits is denoted by r = n - k. The ''standard'' form for a generator matrix is, : G = \begin I_k , P \end, where I_k is the k \times k identity ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
