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Babel Function
The Babel function (also known as cumulative coherence) measures the maximum total coherence between a fixed atom and a collection of other atoms in a dictionary. The Babel function was conceived of in the context of signals for which there exists a sparse representation consisting of atoms or columns of a redundant dictionary matrix, A. Definition and formulation The Babel function of a dictionary \boldsymbol with normalized columns is a real-valued function that is defined as ::\mu_1(p) = \max_ \ where \boldsymbol_k are the columns (atoms) of the dictionary \boldsymbol . Special case When p=1, the babel function is the mutual coherence. Practical Applications Li and Lin have used the Babel function to aid in creating effective dictionaries for Machine Learning applications. References {{reflist See also * Compressed sensing Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coherence (signal Processing)
In signal processing, the coherence is a statistic that can be used to examine the relation between two signals or data sets. It is commonly used to estimate the power transfer between input and output of a linear system. If the signals are ergodic, and the system function is linear, it can be used to estimate the causality between the input and output. Definition and formulation The coherence (sometimes called magnitude-squared coherence) between two signals x(t) and y(t) is a real-valued function that is defined as: ::C_(f) = \frac where Gxy(f) is the Cross-spectral density between x and y, and Gxx(f) and Gyy(f) the auto spectral density of x and y respectively. The magnitude of the spectral density is denoted as , G, . Given the restrictions noted above (ergodicity, linearity) the coherence function estimates the extent to which y(t) may be predicted from x(t) by an optimum linear least squares function. Values of coherence will always satisfy 0\le C_(f)\le 1. For an ''ideal' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atom (sparse Signal Analysis)
An atom is a particle that consists of a nucleus of protons and neutrons surrounded by a cloud of electrons. The atom is the basic particle of the chemical elements, and the chemical elements are distinguished from each other by the number of protons that are in their atoms. For example, any atom that contains 11 protons is sodium, and any atom that contains 29 protons is copper. The number of neutrons defines the isotope of the element. Atoms are extremely small, typically around 100 picometers across. A human hair is about a million carbon atoms wide. This is smaller than the shortest wavelength of visible light, which means humans cannot see atoms with conventional microscopes. Atoms are so small that accurately predicting their behavior using classical physics is not possible due to quantum effects. More than 99.94% of an atom's mass is in the nucleus. The protons have a positive electric charge, the electrons have a negative electric charge, and the neutrons have n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dictionary (sparse Signal Analysis)
A dictionary is a listing of lexemes from the lexicon of one or more specific languages, often arranged alphabetically (or by consonantal root for Semitic languages or radical and stroke for logographic languages), which may include information on definitions, usage, etymologies, pronunciations, translation, etc.Webster's New World College Dictionary, Fourth Edition, 2002 It is a lexicographical reference that shows inter-relationships among the data. A broad distinction is made between general and specialized dictionaries. Specialized dictionaries include words in specialist fields, rather than a complete range of words in the language. Lexical items that describe concepts in specific fields are usually called terms instead of words, although there is no consensus whether lexicology and terminology are two different fields of study. In theory, general dictionaries are supposed to be semasiological, mapping word to definition, while specialized dictionaries are suppose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real-valued Function
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain. Real-valued functions of a real variable (commonly called ''real functions'') and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions. Algebraic structure Let (X,) be the set of all functions from a set to real numbers \mathbb R. Because \mathbb R is a field, (X,) may be turned into a vector space and a commutative algebra over the reals with the following operations: *f+g: x \mapsto f(x) + g(x) – vector addition *\mathbf: x \mapsto 0 – additive identity *c f: x \mapsto c f(x),\quad c \in \mathbb R – scalar multiplication *f g: x \mapsto f(x)g(x) – pointwise multiplication These operations extend to partial functions from to \mathbb R, with the restricti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mutual Coherence (linear Algebra)
In linear algebra, the coherence or mutual coherence of a matrix ''A'' is defined as the maximum absolute value of the cross-correlations between the columns of ''A''. Formally, let a_1, \ldots, a_m\in ^d be the columns of the matrix ''A'', which are assumed to be normalized such that a_i^H a_i = 1. The mutual coherence of ''A'' is then defined as :M = \max_ \left, a_i^H a_j \. A lower bound is : M\ge \sqrt A deterministic matrix with the mutual coherence almost meeting the lower bound can be constructed by Weil's theorem. This concept was reintroduced by David Donoho and Michael Elad in the context of sparse representations. A special case of this definition for the two-ortho case appeared earlier in the paper by Donoho and Huo. The mutual coherence has since been used extensively in the field of sparse representations of signals. In particular, it is used as a measure of the ability of suboptimal algorithms such as matching pursuit and basis pursuit to correctly identif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Learning
Machine learning (ML) is a field of inquiry devoted to understanding and building methods that 'learn', that is, methods that leverage data to improve performance on some set of tasks. It is seen as a part of artificial intelligence. Machine learning algorithms build a model based on sample data, known as training data, in order to make predictions or decisions without being explicitly programmed to do so. Machine learning algorithms are used in a wide variety of applications, such as in medicine, email filtering, speech recognition, agriculture, and computer vision, where it is difficult or unfeasible to develop conventional algorithms to perform the needed tasks.Hu, J.; Niu, H.; Carrasco, J.; Lennox, B.; Arvin, F.,Voronoi-Based Multi-Robot Autonomous Exploration in Unknown Environments via Deep Reinforcement Learning IEEE Transactions on Vehicular Technology, 2020. A subset of machine learning is closely related to computational statistics, which focuses on making predicti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compressed Sensing
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a Signal (electronics), signal, by finding solutions to Underdetermined system, underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem. There are two conditions under which recovery is possible. The first one is sparsity, which requires the signal to be sparse in some domain. The second one is incoherence, which is applied through the isometric property, which is sufficient for sparse signals. Overview A common goal of the engineering field of signal processing is to reconstruct a signal from a series of sampling measurements. In general, this task is impossible because there is no way to reconstruct a signal during the times that the signa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
