By David Donoho, and Jared Tanner
Proceedings of the National Academy of Sciences of the United States of America (2005) Abstract Paper
Coders:

David Donoho

Stanford University

United States

Jared Tanner

University of Oxford

United Kingdom

81 Visits

4 Runs

Ranking 17

This code outputs a graphic that presents the fraction of successes in convex (LP) optimization recovering the combinatorial (NP) optimization outcomes. The equivalence phase transition is depicted as a function of the aspect ratio gamma and the sparsity through the phase transition rho. The user must specify the number of points generated, the length of the solution (n), the number of points for δ and ρ, as well as the number of trials performed to compute the fraction of equivalence. The larger the parameters specified (in particular the number of trials), the longer the code takes to run! For more information, please visit the SparseLab (Seeking Sparse Solutions to Linear Systems of Equations) website (http://sparselab.stanford.edu/).
Created June 27, 2012
Last update October 08, 2012
Software Matlab 2009b
Abstract
Close
Consider an underdetermined system of linear equations y = Ax with known d*n matrix A and known y. We seek the sparsest nonnegative solution, i.e. the nonnegative x with fewest nonzeros satisfying y = Ax. In general this problem is NP-hard. However, for many matrices A there is a threshold phenomenon: if the sparsest solution is sufficiently sparse, it can be found by linear programming. In classical convex polytope theory, a polytope P is called k-neighborly if every set of k vertices of P span a face of P. Let aj denote the j-th column of A, 1<=_j<=_n, let a0 = 0 and let P denote the convex hull of the aj . We say P is outwardly k-neighborly if every subset of k vertices not including 0 spans a face of P. We show that outward k-neighborliness is completely equivalent to the statement that, whenever y = Ax has a nonnegative solution with at most k nonzeros, it is the nonnegative solution to y = Ax having minimal sum. Using this and classical results on polytope neighborliness we obtain two types of corollaries. First, because many [d/2]-neighborly polytopes are known, there are many systems where the sparsest solution is available by convex optimization rather than combinatorial optimization - provided the answer has fewer nonzeros than half the number of equations. We mention examples involving incompletely-observed Fourier transforms and Laplace transforms. Second, results on classical neighborliness of high-dimensional randomly-projected simplices imply that, if A is a typical uniformly-distributed random orthoprojector with n = 2d and n large, the sparsest nonnegative solution to y = Ax can be found by linear programming provided it has fewer nonzeros than 1/8 the number of equations. We also consider a notion of weak neighborliness, in which the overwhelming majority of k-sets of aj's not containing 0 span a face. This implies that most nonnegative vectors x with k nonzeros are uniquely determined by y = Ax. As a corollary of recent work counting faces of random simplices, it is known that most polytopes P generated by large n by 2n uniformly-distributed orthoprojectors A are weakly k-neighborly with k _~.558n. We infer that for most n by 2n underdetermined systems having a sparse solution with fewer nonzeros than roughly half the number of equations, the sparsest solution can be found by linear programming.
Donoho, D., and J. Tanner, "Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming ", Proceedings of the National Academy of Sciences of the United States of America , 102, 9446–9451.
2. RunMyCode
No. Points
No. Points
Solution's length
Solution's length
No. points ρ
No. points ρ
No. points δ
No. points δ
No. tests
No. tests
Waiting time

Please cite the publication as :

Donoho, D., and J. Tanner, "Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming ", Proceedings of the National Academy of Sciences of the United States of America , 102, 9446–9451.

Please cite the companion website as :

Donoho, D., and J. Tanner, "Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming ", RunMyCode companion website, http://www.runmycode.org/CompanionSite/Site129

Inputs and Inputs description

No. Points
The number of points between 0 and 1 to be generated.
Solution's length
The length of the solution.
No. points ρ
Number of points for the phase transition ρ.
No. points δ
Number of points for the aspect ratio δ.
No. tests
The number of tests performed to calculate the fraction of equivalence between LP and NP.

Inputs and inputs description

Variable/Parameters Description Visualisation
No. Points 20 points are used.
Solution's length n=100 for the demo.
No. points ρ 20 points in the demo exercise.
No. points δ 20 points are considered in the exercise.
No. tests 10 Tests are performed.

Results

Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming
D. Donoho, and J. Tanner (2012)

Computing queue

Computing Date Status Actions
Coders:
• David Donoho

Stanford University

United States

• Jared Tanner

University of Oxford

United Kingdom

David Donoho also created these companion sites

Article Authors Coders Last update Ranking Visits Runs
High-Dimensional Centrally-Symmetric Polytopes With Neighborliness Proportional to Dimension
Abstract
Close
Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in Rn. It has recently been shown that properties of the centrosymmetric polytope P = AC are of interest for finding sparse solutions to the underdetermined system of equations y = Ax; [9]. In particular, it is valuable to know that P is centrally k-neighborly. We study the face numbers of randomly-projected cross-polytopes in the proportional dimensional case where dn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rn. We derive ?N(d) > 0 with the property that, for any ? < ?N(d), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 k d. This implies that P is centrally bdc-neighborly, and its skeleton Skel[? d](P) is combinatorially equivalent to Skel[? d]©. We display graphs of ?N. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: facial neighborliness and sectional neighborliness [9]; we study both. The weakest, (k, e)-facial neighborliness, asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P) >= fk(C)(1 - e). We characterize and compute the critical proportion ?F (d) > 0 at which phase transition occurs in k/d. The other, (k, e)- sectional neighborliness, asks whether all, except for a small fraction epsilon, of the k-dimensional intrinsic sections of P are k-dimensional cross-polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ?S(d) > 0 guaranteeing this property for k/d ~ ? < ?S(d). We display graphs of ?S and ?F.
Donoho, D., "High-Dimensional Centrally-Symmetric Polytopes With Neighborliness Proportional to Dimension", Discrete & Computational Geometry, 35, 617-652.
Donoho
Donoho
10/08/2012 16 33 3
Neighborliness of Randomly-Projected Simplices in High Dimensions
Abstract
Close
Let A be a d by n matrix, d < n. Let T = T^n-1 be the standard regular simplex in R^n. We count the faces of the projected simplex AT in the case where the projection is random, the dimension d is large and n and d are comparable: d ~ dn, d in (0, 1). The projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R^n. We derive ?N( d) > 0 with the property that, for any ? < ? N( deter), with overwhelming probability for large d, the number of k-dimensional faces of P = AT is exactly the same as for T, for 0<=k<= ?d. This implies that P is [?d]-neighborly, and its skeleton Skel[? d] ( P) is combinatorially equivalent to Skel[?d] (T). We display graphs of ?N. We also study a weaker notion of neighborliness it asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P) >= fk(T)(1-e). This was already considered by Vershik and Sporyshev, who obtained qualitative results about the existence of a threshold ? VS(d) > 0 at which phase transition occurs in k/d. We compute and display ?VS and compare to ?N. Our results imply that the convex hull of n Gaussian samples in R^d, with n large and proportional to d, ‘looks like a simplex’ in the following sense. In a typical realization of such a high-dimensional Gaussian point cloud d~ dn, all points are on the boundary of the convex hull, and all pairwise line segments, triangles, quadrangles, …, [?d]-angles are on the boundary, for ?<? N(d/n). Our results also quantify a precise phase transition in the ability of linear programming to find the sparsest nonnegative solution to typical systems of underdetermined linear equations; when there is a solution with fewer than ?VS(d/n)d nonzeros, linear programming will find that solution.
Donoho, D., and J. Tanner, "Neighborliness of Randomly-Projected Simplices in High Dimensions ", Proceedings of the National Academy of Sciences of the United States of America , 102, 9452-9457.
Donoho
Tanner
Donoho
Tanner
10/08/2012 19 52 3
Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise
Abstract
Close
Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimal-sparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
Donoho, D., M. Elad, and V. Temlyakov, "Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise", Transactions on Information Theory, 52.
Donoho
Temlyakov
Donoho
Temlyakov
10/08/2012 14 89 10
On the Stability of the Basis Pursuit in the Presence of Noise
Abstract
Close
Given a signal S ( R^N and a full-rank matrix D ( R^NL with N<L, we define the signal’s over-complete representation as a ( R^L satisfying S=Da. Among the infinitely many solutions of this under-determined linear system of equations, we have special interest in the sparsest representation, i.e., the one minimizing ||a||0. This problem has a combinatorial flavor to it, and its direct solution is impossible even for moderate L. Approximation algorithms are thus required, and one such appealing technique is the basis pursuit (BP) algorithm. This algorithm has been the focus of recent theoretical research effort. It was found that if indeed the representation is sparse enough, BP finds it accurately. When an error is permitted in the composition of the signal, we no longer require exact equality S=Da. The BP has been extended to treat this case, leading to a denoizing algorithm. The natural question to pose is how the abovementioned theoretical results generalize to this more practical mode of operation. In this paper we propose such a generalization. The behavior of the basis pursuit in the presence of noise has been the subject of two independent very wide contributions released for publication very recently. This paper is another contribution in this direction, but as opposed to the others mentioned, this paper aims to present a somewhat simplified picture of the topic, and thus could be referred to as a primer to this field. Specifically, we establish here the stability of the BP in the presence of noise for sparse enough representations. We study both the case of a general dictionary D, and a special case where D is built as a union of orthonormal bases. This work is a direct generalization of noiseless BP study, and indeed, when the noise power is reduced to zero, we obtain the known results of the noiseless BP.
Donoho, D., and M. Elad, "On the Stability of the Basis Pursuit in the Presence of Noise ", Signal Processing , 86 , 511-532.
Donoho
Donoho
10/08/2012 29 63 N.A.
Deterministic Matrices Matching the Compressed Sensing Phase Transitions of Gaussian Random Matrices
Abstract
Close
In compressed sensing, one takes n < N samples of an N -dimensional vector x0 using an n × N matrix A, obtaining un-dersampled measurements y = Ax0 . For random matrices with Gaussian i.i.d entries, it is known that, when x0 is k-sparse, there is a precisely determined phase transition: for a certain region in the (k/n, n/N )-phase diagram, convex optimization min ||x||_1 subject to y = Ax, x ∈ X^N typically ﬁnds the sparsest solution, while outside that region, it typically fails. It has been shown empirically that the same property – with the same phase transition location – holds for a wide range of non-Gaussian random matrix ensembles. We consider speciﬁc deterministic matrices including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Extensive experiments show that for a typical k-sparse object, convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian matrices. In our experiments, we considered coefﬁcients constrained to X^N for four different sets X ∈ {[0, 1], R_+ , R, C}. We establish this ﬁnding for each of the associated four phase transitions.
Monajemi, H., D. Donoho, "Deterministic Matrices Matching the Compressed Sensing Phase Transitions of Gaussian Random Matrices", Stanford University.
Monajemi
Jafarpour
Gavish
Donoho
Monajemi
Donoho
01/04/2013 9999 86 13

Jared Tanner also created these companion sites

Article Authors Coders Last update Ranking Visits Runs
Neighborliness of Randomly-Projected Simplices in High Dimensions
Abstract
Close
Let A be a d by n matrix, d < n. Let T = T^n-1 be the standard regular simplex in R^n. We count the faces of the projected simplex AT in the case where the projection is random, the dimension d is large and n and d are comparable: d ~ dn, d in (0, 1). The projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R^n. We derive ?N( d) > 0 with the property that, for any ? < ? N( deter), with overwhelming probability for large d, the number of k-dimensional faces of P = AT is exactly the same as for T, for 0<=k<= ?d. This implies that P is [?d]-neighborly, and its skeleton Skel[? d] ( P) is combinatorially equivalent to Skel[?d] (T). We display graphs of ?N. We also study a weaker notion of neighborliness it asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P) >= fk(T)(1-e). This was already considered by Vershik and Sporyshev, who obtained qualitative results about the existence of a threshold ? VS(d) > 0 at which phase transition occurs in k/d. We compute and display ?VS and compare to ?N. Our results imply that the convex hull of n Gaussian samples in R^d, with n large and proportional to d, ‘looks like a simplex’ in the following sense. In a typical realization of such a high-dimensional Gaussian point cloud d~ dn, all points are on the boundary of the convex hull, and all pairwise line segments, triangles, quadrangles, …, [?d]-angles are on the boundary, for ?<? N(d/n). Our results also quantify a precise phase transition in the ability of linear programming to find the sparsest nonnegative solution to typical systems of underdetermined linear equations; when there is a solution with fewer than ?VS(d/n)d nonzeros, linear programming will find that solution.
Donoho, D., and J. Tanner, "Neighborliness of Randomly-Projected Simplices in High Dimensions ", Proceedings of the National Academy of Sciences of the United States of America , 102, 9452-9457.
Donoho
Tanner
Donoho
Tanner
10/08/2012 19 52 3

Other Companion Sites on same paper

Sparse Nonnegative Solution of Underdetermined Linear Equations by Linear Programming
Article Authors Coders Last update Ranking Visits Runs

Other Companion Sites relative to similar papers

Article Authors Coders Last update Ranking Visits Runs
High-Dimensional Centrally-Symmetric Polytopes With Neighborliness Proportional to Dimension
Abstract
Close
Let A be a d by n matrix, d < n. Let C be the regular cross polytope (octahedron) in Rn. It has recently been shown that properties of the centrosymmetric polytope P = AC are of interest for finding sparse solutions to the underdetermined system of equations y = Ax; [9]. In particular, it is valuable to know that P is centrally k-neighborly. We study the face numbers of randomly-projected cross-polytopes in the proportional dimensional case where dn, where the projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of Rn. We derive ?N(d) > 0 with the property that, for any ? < ?N(d), with overwhelming probability for large d, the number of k-dimensional faces of P = AC is the same as for C, for 0 k d. This implies that P is centrally bdc-neighborly, and its skeleton Skel[? d](P) is combinatorially equivalent to Skel[? d]©. We display graphs of ?N. Two weaker notions of neighborliness are also important for understanding sparse solutions of linear equations: facial neighborliness and sectional neighborliness [9]; we study both. The weakest, (k, e)-facial neighborliness, asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P) >= fk(C)(1 - e). We characterize and compute the critical proportion ?F (d) > 0 at which phase transition occurs in k/d. The other, (k, e)- sectional neighborliness, asks whether all, except for a small fraction epsilon, of the k-dimensional intrinsic sections of P are k-dimensional cross-polytopes. (Intrinsic sections intersect P with k-dimensional subspaces spanned by vertices of P.) We characterize and compute a proportion ?S(d) > 0 guaranteeing this property for k/d ~ ? < ?S(d). We display graphs of ?S and ?F.
Donoho, D., "High-Dimensional Centrally-Symmetric Polytopes With Neighborliness Proportional to Dimension", Discrete & Computational Geometry, 35, 617-652.
Donoho
Donoho
10/08/2012 16 33 3
Neighborliness of Randomly-Projected Simplices in High Dimensions
Abstract
Close
Let A be a d by n matrix, d < n. Let T = T^n-1 be the standard regular simplex in R^n. We count the faces of the projected simplex AT in the case where the projection is random, the dimension d is large and n and d are comparable: d ~ dn, d in (0, 1). The projector A is chosen uniformly at random from the Grassmann manifold of d-dimensional orthoprojectors of R^n. We derive ?N( d) > 0 with the property that, for any ? < ? N( deter), with overwhelming probability for large d, the number of k-dimensional faces of P = AT is exactly the same as for T, for 0<=k<= ?d. This implies that P is [?d]-neighborly, and its skeleton Skel[? d] ( P) is combinatorially equivalent to Skel[?d] (T). We display graphs of ?N. We also study a weaker notion of neighborliness it asks if the k-faces are all simplicial and if the numbers of k-dimensional faces fk(P) >= fk(T)(1-e). This was already considered by Vershik and Sporyshev, who obtained qualitative results about the existence of a threshold ? VS(d) > 0 at which phase transition occurs in k/d. We compute and display ?VS and compare to ?N. Our results imply that the convex hull of n Gaussian samples in R^d, with n large and proportional to d, ‘looks like a simplex’ in the following sense. In a typical realization of such a high-dimensional Gaussian point cloud d~ dn, all points are on the boundary of the convex hull, and all pairwise line segments, triangles, quadrangles, …, [?d]-angles are on the boundary, for ?<? N(d/n). Our results also quantify a precise phase transition in the ability of linear programming to find the sparsest nonnegative solution to typical systems of underdetermined linear equations; when there is a solution with fewer than ?VS(d/n)d nonzeros, linear programming will find that solution.
Donoho, D., and J. Tanner, "Neighborliness of Randomly-Projected Simplices in High Dimensions ", Proceedings of the National Academy of Sciences of the United States of America , 102, 9452-9457.
Donoho
Tanner
Donoho
Tanner
10/08/2012 19 52 3
The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising
Abstract
Close
Let X_0 be an unknown M by N matrix. In matrix recovery, one takes n < MN linear measurements y_1, ... , y_n of X_0, where y_i = Trace(a_i' X_0) and each a_i is a M by N matrix. For measurement matrices with Gaussian i.i.d entries, it known that if X_0 is of low rank, it is recoverable from just a few measurements. A popular approach for matrix recovery is Nuclear Norm Minimization (NNM): solving the convex optimization problem min ||X||_* subject to y_i=Trace(a_i' X) for all 1<= i<= n, where || . ||_* denotes the nuclear norm, namely, the sum of singular values. Empirical work reveals a phase transition curve, stated in terms of the undersampling fraction \delta(n,M,N) = n/(MN), rank fraction \rho=r/N and aspect ratio \beta=M/N. Specifically, a curve \delta^* = \delta^*(\rho;\beta) exists such that, if \delta > \delta^*(\rho;\beta), NNM typically succeeds, while if \delta < \delta^*(\rho;\beta), it typically fails. An apparently quite different problem is matrix denoising in Gaussian noise, where an unknown M by N matrix X_0 is to be estimated based on direct noisy measurements Y = X_0 + Z, where the matrix Z has iid Gaussian entries. It has been empirically observed that, if X_0 has low rank, it may be recovered quite accurately from the noisy measurement Y. A popular matrix denoising scheme solves the unconstrained optimization problem min || Y - X ||_F^2/2 + \lambda ||X||_*. When optimally tuned, this scheme achieves the asymptotic minimax MSE, M( \rho ) = \lim_{N-> \infty} \inf_\lambda \sup_{\rank(X) <= \rho * N} MSE(X,\hat{X}_\lambda). We report extensive experiments showing that the phase transition \delta^*(\rho) in the first problem (Matrix Recovery from Gaussian Measurements) coincides with the minimax risk curve M(\rho) in the second problem (Matrix Denoising in Gaussian Noise): \delta^*(\rho) = M(\rho), for any rank fraction 0 < \rho < 1. Our experiments considered matrices belonging to two constraint classes: real M by N matrices, of various ranks and aspect ratios, and real symmetric positive semidefinite N by N matrices, of various ranks. Different predictions M(\rho) of the phase transition location were used in the two different cases, and were validated by the experimental data.
Gavish, M., "The Phase Transition of Matrix Recovery from Gaussian Measurements Matches the Minimax MSE of Matrix Denoising", Stanford University.
Donoho
Gavish
Montanari
Gavish
02/15/2013 9999 N.A. 8
Deterministic Matrices Matching the Compressed Sensing Phase Transitions of Gaussian Random Matrices
Abstract
Close
In compressed sensing, one takes n < N samples of an N -dimensional vector x0 using an n × N matrix A, obtaining un-dersampled measurements y = Ax0 . For random matrices with Gaussian i.i.d entries, it is known that, when x0 is k-sparse, there is a precisely determined phase transition: for a certain region in the (k/n, n/N )-phase diagram, convex optimization min ||x||_1 subject to y = Ax, x ∈ X^N typically ﬁnds the sparsest solution, while outside that region, it typically fails. It has been shown empirically that the same property – with the same phase transition location – holds for a wide range of non-Gaussian random matrix ensembles. We consider speciﬁc deterministic matrices including Spikes and Sines, Spikes and Noiselets, Paley Frames, Delsarte-Goethals Frames, Chirp Sensing Matrices, and Grassmannian Frames. Extensive experiments show that for a typical k-sparse object, convex optimization is successful over a region of the phase diagram that coincides with the region known for Gaussian matrices. In our experiments, we considered coefﬁcients constrained to X^N for four different sets X ∈ {[0, 1], R_+ , R, C}. We establish this ﬁnding for each of the associated four phase transitions.
Monajemi, H., D. Donoho, "Deterministic Matrices Matching the Compressed Sensing Phase Transitions of Gaussian Random Matrices", Stanford University.
Monajemi
Jafarpour
Gavish
Donoho
Monajemi
Donoho
01/04/2013 9999 86 13
Adaptive Estimation of Vector Autoregressive Models with Time-Varying Variance: Application to Testing Linear Causality in Mean
Abstract
Close
Linear Vector AutoRegressive (VAR) models where the innovations could be unconditionally heteroscedastic and serially dependent are considered. The volatility structure is deterministic and quite general, including breaks or trending variances as special cases. In this framework we propose Ordinary Least Squares (OLS), Generalized Least Squares (GLS) and Adaptive Least Squares (ALS) procedures. The GLS estimator requires the knowledge of the time-varying variance structure while in the ALS approach the unknown variance is estimated by kernel smoothing with the outer product of the OLS residuals vectors. Different bandwidths for the different cells of the time-varying variance matrix are also allowed. We derive the asymptotic distribution of the proposed estimators for the VAR model coefficients and compare their properties. In particular we show that the ALS estimator is asymptotically equivalent to the infeasible GLS estimator. This asymptotic equivalence is obtained uniformly with respect to the bandwidth(s) in a given range and hence justifies data-driven bandwidth rules. Using these results we build Wald tests for the linear Granger causality in mean which are adapted to VAR processes driven by errors with a non stationary volatility. It is also shown that the commonly used standard Wald test for the linear Granger causality in mean is potentially unreliable in our framework (incorrect level and lower asymptotic power). Monte Carlo and real-data experiments illustrate the use of the different estimation approaches for the analysis of VAR models with time-varying variance innovations.
Raïssi, H., "Adaptive Estimation of Vector Autoregressive Models with Time-Varying Variance: Application to Testing Linear Causality in Mean", IRMAR-INSA and CREST ENSAI.
Patilea
Raïssi
10/08/2012 58 169 9
Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise
Abstract
Close
Overcomplete representations are attracting interest in signal processing theory, particularly due to their potential to generate sparse representations of signals. However, in general, the problem of finding sparse representations must be unstable in the presence of noise. This paper establishes the possibility of stable recovery under a combination of sufficient sparsity and favorable structure of the overcomplete system. Considering an ideal underlying signal that has a sufficiently sparse representation, it is assumed that only a noisy version of it can be observed. Assuming further that the overcomplete system is incoherent, it is shown that the optimally sparse approximation to the noisy data differs from the optimally sparse decomposition of the ideal noiseless signal by at most a constant multiple of the noise level. As this optimal-sparsity method requires heavy (combinatorial) computational effort, approximation algorithms are considered. It is shown that similar stability is also available using the basis and the matching pursuit algorithms. Furthermore, it is shown that these methods result in sparse approximation of the noisy data that contains only terms also appearing in the unique sparsest representation of the ideal noiseless sparse signal.
Donoho, D., M. Elad, and V. Temlyakov, "Stable Recovery of Sparse Overcomplete Representations in the Presence of Noise", Transactions on Information Theory, 52.
Donoho
Temlyakov
Donoho
Temlyakov
10/08/2012 14 89 10
On the Stability of the Basis Pursuit in the Presence of Noise
Abstract
Close
Given a signal S ( R^N and a full-rank matrix D ( R^NL with N<L, we define the signal’s over-complete representation as a ( R^L satisfying S=Da. Among the infinitely many solutions of this under-determined linear system of equations, we have special interest in the sparsest representation, i.e., the one minimizing ||a||0. This problem has a combinatorial flavor to it, and its direct solution is impossible even for moderate L. Approximation algorithms are thus required, and one such appealing technique is the basis pursuit (BP) algorithm. This algorithm has been the focus of recent theoretical research effort. It was found that if indeed the representation is sparse enough, BP finds it accurately. When an error is permitted in the composition of the signal, we no longer require exact equality S=Da. The BP has been extended to treat this case, leading to a denoizing algorithm. The natural question to pose is how the abovementioned theoretical results generalize to this more practical mode of operation. In this paper we propose such a generalization. The behavior of the basis pursuit in the presence of noise has been the subject of two independent very wide contributions released for publication very recently. This paper is another contribution in this direction, but as opposed to the others mentioned, this paper aims to present a somewhat simplified picture of the topic, and thus could be referred to as a primer to this field. Specifically, we establish here the stability of the BP in the presence of noise for sparse enough representations. We study both the case of a general dictionary D, and a special case where D is built as a union of orthonormal bases. This work is a direct generalization of noiseless BP study, and indeed, when the noise power is reduced to zero, we obtain the known results of the noiseless BP.
Donoho, D., and M. Elad, "On the Stability of the Basis Pursuit in the Presence of Noise ", Signal Processing , 86 , 511-532.
Donoho