Basis Pursuit Denoising

	Basis Pursuit Denoising In applied mathematics and statistics, basis pursuit denoising (BPDN) refers to a mathematical optimization problem of the form : \min_x \left(\frac \, y - Ax\, ^2_2 + \lambda \, x\, _1\right), where \lambda is a parameter that controls the trade-off between sparsity and reconstruction fidelity, x is an N \times 1 solution vector, y is an M \times 1 vector of observations, A is an M \times N transform matrix and M < N. This is an instance of convex optimization . Some authors refer to basis pursuit denoising as the following closely related problem: : $\min_x \, x\, _1 \text \, y - Ax\, ^2_2 \le \delta,$ which, for any given $\lambda$ , is equivalent to the unconstrained formulation for some (usually unknown ''a priori'') value of $\delta$ . The two problems are quite similar. In practice, the unconstrained fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Applied Mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the profession, professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of Mathematical analysis, applied analysis, most notably differential equations; approximation theory (broadly construed, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Basis Pursuit Basis pursuit is the mathematical optimization problem of the form : \min_x \, x\, _1 \quad \text \quad y = Ax, where ''x'' is a ''N''-dimensional solution vector (signal), ''y'' is a ''M''-dimensional vector of observations (measurements), ''A'' is a ''M'' × ''N'' transform matrix (usually measurement matrix) and ''M'' < ''N''. The version of basis pursuit that seeks to minimize the ''L''₀ norm is NP-hard. It is usually applied in cases where there is an underdetermined system of linear equations ''y'' = ''Ax'' that must be exactly satisfied, and the sparsest solution in the ''L''₁ sense is desired. When it is desirable to trade off exact equality of ''Ax'' and ''y'' in exchange for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Fixed-point Continuation Fixed point may refer to: * Fixed point (mathematics), a value that does not change under a given transformation * Fixed-point arithmetic, a manner of doing arithmetic on computers * Fixed point, a benchmark (surveying) used by geodesists * Fixed point join, also called a recursive join * Fixed point, in quantum field theory, a coupling where the beta function vanishes – see * Temperature reference point, usually defined by a phase change or triple point In thermodynamics, the triple point of a substance is the temperature and pressure at which the three Phase (matter), phases (gas, liquid, and solid) of that substance coexist in thermodynamic equilibrium.. It is that temperature and pressure at ... . {{disambiguation, math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Homotopy Continuation Numerical algebraic geometry is a field of computational mathematics, particularly computational algebraic geometry, which uses methods from numerical analysis to study and manipulate the solutions of systems of polynomial equations. Homotopy continuation The primary computational method used in numerical algebraic geometry is homotopy continuation, in which a homotopy is formed between two polynomial systems, and the isolated solutions (points) of one are continued to the other. This is a specialization of the more general method of numerical continuation. Let z represent the variables of the system. By abuse of notation, and to facilitate the spectrum of ambient spaces over which one can solve the system, we do not use vector notation for z. Similarly for the polynomial systems f and g. Current canonical notation calls the start system g, and the target system, i.e., the system to solve, f. A very common homotopy, the straight-line homotopy, between f and g is H(z,t) = (1- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	MATLAB MATLAB (an abbreviation of "MATrix LABoratory") is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks. MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages. Although MATLAB is intended primarily for numeric computing, an optional toolbox uses the MuPAD symbolic engine allowing access to symbolic computing abilities. An additional package, Simulink, adds graphical multi-domain simulation and model-based design for dynamic and embedded systems. , MATLAB has more than four million users worldwide. They come from various backgrounds of engineering, science, and economics. , more than 5000 global colleges and universities use MATLAB to support instruction and research. History Origins MATLAB was invented by mathematician and computer programmer Cleve Moler. The idea for MATLAB was base ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	IEEE Transactions On Signal Processing The ''IEEE Transactions on Signal Processing'' is a biweekly peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers covering research on signal processing. It was established in 1953 as the ''IRE Transactions on Audio'', renamed to ''IEEE Transactions on Audio and Electroacoustics'' in 1966 and to ''IEEE Transactions on Acoustics, Speech, and Signal Processing'' in 1974, before obtaining its current name in 1992. The journal is abstracted and indexed in MEDLINE/PubMed and the Science Citation Index Expanded. According to the ''Journal Citation Reports'', the journal has a 2022 impact factor of 5.4. The editor-in-chief is Wing-Kin (Ken) Ma (Chinese University of Hong Kong The Chinese University of Hong Kong (CUHK) is a public university, public research university in Sha Tin, New Territories, Hong Kong. Established in 1963 as a federation of three university college, collegesChung Chi College, New Asia Coll ...). See also '' IEE ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	In-crowd Algorithm The in-crowd algorithm is a numerical method for solving basis pursuit denoising quickly; faster than any other algorithm for large, sparse problems.See ''The In-Crowd Algorithm for Fast Basis Pursuit Denoising'', IEEE Trans Sig Proc 59 (10), Oct 1 2011, pp. 4595 - 4605 demo MATLAB code availabl/ref> This algorithm is an active set method, which minimizes iteratively sub-problems of the global basis pursuit denoising: \min_x \frac\, y-Ax\, ^2_2+\lambda\, x\, _1. where y is the observed signal, x is the sparse signal to be recovered, Ax is the expected signal under x, and \lambda is the regularization parameter trading off signal fidelity and simplicity. The simplicity is here measured using the sparsity of the solution x, measure through its \ell_1-norm. The active set strategies are very efficient in this context as only few coefficient are expected to be non-zero. Thus, if they can be identified, solving the problem restricted to these coefficients yield the solution. Here, the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Interior-point Method Interior-point methods (also referred to as barrier methods or IPMs) are algorithms for solving linear and non-linear convex optimization problems. IPMs combine two advantages of previously-known algorithms: * Theoretically, their run-time is polynomial—in contrast to the simplex method, which has exponential run-time in the worst case. * Practically, they run as fast as the simplex method—in contrast to the ellipsoid method, which has polynomial run-time in theory but is very slow in practice. In contrast to the simplex method which traverses the ''boundary'' of the feasible region, and the ellipsoid method which bounds the feasible region from ''outside'', an IPM reaches a best solution by traversing the ''interior'' of the feasible region—hence the name. History An interior point method was discovered by Soviet mathematician I. I. Dikin in 1967. The method was reinvented in the U.S. in the mid-1980s. In 1984, Narendra Karmarkar developed a method for linear programming ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Robert Tibshirani Robert Tibshirani (born July 10, 1956) is a professor in the Departments of Statistics and Biomedical Data Science at Stanford University. He was a professor at the University of Toronto from 1985 to 1998. In his work, he develops statistical tools for the analysis of complex datasets, most recently in genomics and proteomics. His most well-known contributions are the Lasso method, which proposed the use of L1 penalization in regression and related problems, and Significance Analysis of Microarrays. Education and early life Tibshirani was born on 10 July 1956 in Niagara Falls, Ontario, Canada. He received his B. Math. in statistics and computer science from the University of Waterloo in 1979 and a Master's degree in Statistics from the University of Toronto in 1980. Tibshirani joined the doctoral program at Stanford University in 1981 and received his Ph.D. in 1984 under the supervision of Bradley Efron. His dissertation was entitled "Local likelihood estimation". Honors ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	David Donoho David Leigh Donoho (born March 5, 1957) is an American statistician. He is a professor of statistics at Stanford University, where he is also the Anne T. and Robert M. Bass Professor in the Humanities and Sciences. His work includes the development of effective methods for the construction of low-dimensional representations for high-dimensional data problems ( multiscale geometric analysis), development of wavelets for denoising and compressed sensing. He was elected a Member of the American Philosophical Society in 2019. Academic biography Donoho did his undergraduate studies at Princeton University, graduating in 1978. His undergraduate thesis advisor was John W. Tukey. Donoho obtained his Ph.D. from Harvard University in 1983, under the supervision of Peter J. Huber.. He was on the faculty of the University of California, Berkeley, from 1984 to 1990 before moving to Stanford. He has been the Ph.D. advisor of at least 20 doctoral students, including Jianqing Fan and Emmanu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	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. Compressed sensing has applications in, for example, magnetic resonance imaging (MRI) where the incoherence condition is typically satisfied. Overview A common goal of the engineering field of signal processing is to reconstruct a signal from a series ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Statistics Statistics (from German language, German: ', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments. When census data (comprising every member of the target population) cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling assures that inferences and conclusions can reasonably extend from the sample ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]