fast algorithm for computing minimal-norm solutions

. In contrast, heuristic attempts to sparsely solve such systems – greedy algorithms and thresholding – perform poorly in this challenging setting. The techniques include the use of random proportional embeddings and almost-spherical sections in Banach space theory, and deviation bounds for the eigenvalues

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yk = 〈x0, φk〉, k = 1,...,M or y = Φx0 Fewer measurements than degrees of freedom, M N y! x0= Treat acquisition as a linear inverse problem Compressive Sampling: for sparse x0, we can “invert ” incoherent Φ Sparse recovery Given M linear measurements of an S-sparse signal y = Φx0 + noise when can we recover x0? Sparse recovery Given M linear measurements of an S-sparse signal y = Φx0 + noise when can we recover x0? Key condition: matrix Φ is a restricted isometry: (1 − δ)‖x‖22 ≤ ‖Φx‖22 ≤ (1 + δ)‖x‖22 for all 2S-sparse x [Candes and Tao ’06] Random matrices Example: Φi,j ∼ ±1 w / prob 1/2, iid Can recover S-sparse x0 from M & S · log(N/S) measurements using convex programming, greedy algorithms,... Random matrices Example: Φi,j ∼ ±1 w / prob 1/2, iid Can recover S-sparse x0 from M & S · log(N/S) measurements using `1 minimization: min ‖x‖1 subject to Φx = y Random matrices Example: Φi,j ∼ ±1 w / prob 1/2, iid Can recover S-sparse (in basis Ψ) x0 = Ψα0 from M & S · log(N/S) measurements using `1 minimization: min ‖α‖1 subject to ΦΨα = y Random matrices Example: Φi,j ∼ ±1 w / prob 1/2, iid Can stably recover ≈ S-sparse (in basis Ψ) x0 = Ψα0 from M & S · log(N/S) noisy measurements using `1 minimization: min λ‖α‖1 + 1 2 ‖ΦΨα − y‖22 for appropriate λ. Sparsity Decompose signal/image x(t) in orthobasis {ψi(t)}i x(t) = i αiψi(t) wavelet transform zoom x0 {αi}i Wavelet approximation Take 1 % of largest coefficients, set the rest to zero (adaptive) original approximated rel. error = 0.031 Integrating compression and sensing Goal: a dynamical framework for sparse recovery Given y and Φ, solve min x λ‖x‖1 + 1

The minimal 2-norm solution to an underdetermined system Ax = b of full rank can be computed using a QR factorization of A T in two di erent ways. One requires storage and re-use of the orthogo-nal matrix Q while the method of semi-normal equations does not. Existing error analyses show that both

Abstract—Sparse signal priors help in a variety of modern signal processing tasks. In many cases, a sparse signal needs to be recovered from an underdetermined system of equations. For instance, sparse approximation of a signal with an overcomplete dictionary or reconstruction of a sparse signal

methods in the usual case of an equal number of equations and unknowns. This in turn gives a local convergence analysis for augmented Jacobian algorithms which use least-change secant updates. In conclusion, the results of some numerical experiments are given. Key words, underdetermined systems, least

An underdetermined linear system of equations Ax = b with nonnegativity constraint x 0 is considered. It is shown that for matrices A with a row-span intersecting the positive orthant, if this problem admits a sufficiently sparse solution, it is necessarily unique. The bound on the required

This paper focuses on defining a measure, appropriate for obtaining optimally sparse solutions to underdetermined systems of linear equations.1 The general idea is the extension of metrics in n-dimensional spaces via the Cartesian product of metric spaces. 1

This article gives an introduction to the main ideas of numerical path following and presents some advances in the subject regarding adaptations, applications, analysis of efficiency, and complexity. Both theoretical and implementing aspects of the predictor corrector path following methods are thoroughly discussed. Piecewise linear methods are also studied. At the end, a list for some available software related to path following is provided.