This paper considers the problem of recovering an unknown sparse p⇥ p matrix X from an m ⇥ m matrix Y = AXB T ,w hereA and B are known m ⇥ p matrices with m ⌧ p. The main result shows that there exist constructions of the “sketching” matrices A and B so that even if X has O(p) non-zeros, it can be recovered exactly and eciently using a convex program as long as these non-zeros are not concentrated in any single row column of X. Furthermore, it suces for the size of Y (the sketch dimension) to scale as m = O p # nonzeros in X ⇥ logp . The results also show that the recovery is robust and stable in the sense that if X is equal to a sparse matrix plus a perturbation, then the convex program we propose produces an approximation with accuracy proportional to the size of the perturbation. Unlike traditional results on sparse recovery, where the sensing matrix produces independent measurements, our sensing operator is highly constrained (it assumes a tensor product structure). Therefore, proving recovery guarantees require non-standard techniques. Indeed our approach relies on a novel result concerning tensor products of bipartite graphs, which may be of independent interest. This problem is motivated by the following application, among others. Consider a p ⇥ n data matrix D, consisting of n observations of p variables. Assume that the correlation matrix X := DD T is (approximately) sparse in the sense that each of the p variables is significantly correlated with only a few others. Our results show that these significant correlations can be detected even if we have access to only a sketch of the data S = AD with A 2 R m⇥p .
@cite_37: Given a dictionary D = dk of vectors dk, we seek to represent a signal S as a linear combination S = ∑k γ(k)dk, with scalar coefficients γ(k). In particular, we aim for the sparsest representation possible. In general, this requires a combinatorial optimization process. Previous work considered the special case where D is an overcomplete system consisting of exactly two orthobases and has shown that, under a condition of mutual incoherence of the two bases, and assuming that S has a sufficiently sparse representation, this representation is unique and can be found by solving a convex optimization problem: specifically, minimizing the l1 norm of the coefficients γ. In this article, we obtain parallel results in a more general setting, where the dictionary D can arise from two or several bases, frames, or even less structured systems. We sketch three applications: separating linear features from planar ones in 3D data, noncooperative multiuser encoding, and identification of over-complete independent component models.
@cite_22: This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries. It provides a sufficient condition under which both OMP and Donoho's basis pursuit (BP) paradigm can recover the optimal representation of an exactly sparse signal. It leverages this theory to show that both OMP and BP succeed for every sparse input signal from a wide class of dictionaries. These quasi-incoherent dictionaries offer a natural generalization of incoherent dictionaries, and the cumulative coherence function is introduced to quantify the level of incoherence. This analysis unifies all the recent results on BP and extends them to OMP. Furthermore, the paper develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal. From there, it argues that OMP is an approximation algorithm for the sparse problem over a quasi-incoherent dictionary. That is, for every input signal, OMP calculates a sparse approximant whose error is only a small factor worse than the minimal error that can be attained with the same number of terms.
@cite_8: In this note will consider sufficient conditions for sparse recovery such as Spark, coherence, restricted isometry property (RIP) and null space property (NSP). Then we will discuss the solution of underdetermined linear equations when the matrix is the Kronecker product of matrices. Specially we will explain how NSP behave in the case where the matrix is the Kronecker product of matrices.
@cite_41: In applications throughout science and engineering one is often faced with the challenge of solving an ill-posed inverse problem, where the number of available measurements is smaller than the dimension of the model to be estimated. However in many practical situations of interest, models are constrained structurally so that they only have a few degrees of freedom relative to their ambient dimension. This paper provides a general framework to convert notions of simplicity into convex penalty functions, resulting in convex optimization solutions to linear, underdetermined inverse problems. The class of simple models considered includes those formed as the sum of a few atoms from some (possibly infinite) elementary atomic set; examples include well-studied cases from many technical fields such as sparse vectors (signal processing, statistics) and low-rank matrices (control, statistics), as well as several others including sums of a few permutation matrices (ranked elections, multiobject tracking), low-rank tensors (computer vision, neuroscience), orthogonal matrices (machine learning), and atomic measures (system identification). The convex programming formulation is based on minimizing the norm induced by the convex hull of the atomic set; this norm is referred to as the atomic norm. The facial structure of the atomic norm ball carries a number of favorable properties that are useful for recovering simple models, and an analysis of the underlying convex geometry provides sharp estimates of the number of generic measurements required for exact and robust recovery of models from partial information. These estimates are based on computing the Gaussian widths of tangent cones to the atomic norm ball. When the atomic set has algebraic structure the resulting optimization problems can be solved or approximated via semidefinite programming. The quality of these approximations affects the number of measurements required for recovery, and this tradeoff is characterized via some examples. Thus this work extends the catalog of simple models (beyond sparse vectors and low-rank matrices) that can be recovered from limited linear information via tractable convex programming.
@cite_32: This paper considers a natural error correcting problem with real valued input output. We wish to recover an input vector f spl isin R sup n from corrupted measurements y=Af+e. Here, A is an m by n (coding) matrix and e is an arbitrary and unknown vector of errors. Is it possible to recover f exactly from the data y? We prove that under suitable conditions on the coding matrix A, the input f is the unique solution to the spl lscr sub 1 -minimization problem ( spl par x spl par sub spl lscr 1 := spl Sigma sub i |x sub i |) min(g spl isin R sup n ) spl par y - Ag spl par sub spl lscr 1 provided that the support of the vector of errors is not too large, spl par e spl par sub spl lscr 0 :=| i:e sub i spl ne 0 | spl les spl rho spl middot m for some spl rho >0. In short, f can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program). In addition, numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted. This work is related to the problem of finding sparse solutions to vastly underdetermined systems of linear equations. There are also significant connections with the problem of recovering signals from highly incomplete measurements. In fact, the results introduced in this paper improve on our earlier work. Finally, underlying the success of spl lscr sub 1 is a crucial property we call the uniform uncertainty principle that we shall describe in detail.
@cite_6: The purpose of this correspondence is to generalize a result by Donoho and Huo and Elad and Bruckstein on sparse representations of signals in a union of two orthonormal bases for R sup N . We consider general (redundant) dictionaries for R sup N , and derive sufficient conditions for having unique sparse representations of signals in such dictionaries. The special case where the dictionary is given by the union of L spl ges 2 orthonormal bases for R sup N is studied in more detail. In particular, it is proved that the result of Donoho and Huo, concerning the replacement of the spl lscr sup 0 optimization problem with a linear programming problem when searching for sparse representations, has an analog for dictionaries that may be highly redundant.
@cite_3: A machine for measuring and cutting predetermined lengths of electric cable such as armored (BX) or insulated cable which is supplied from a coil or reel to a "caterpillar" feed assembly and then to a cutting assembly, the feeding and cutting being synchronized and controlled to effect the cutting, automatically, of a predetermined number of pieces cut to predetermined uniform lengths, the controls including means for measuring the fed cable and for counting the number of pieces.
@cite_44: Suppose a discrete-time signal S(t), 0 spl les t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1 sub t= spl tau and sinusoids exp 2 spl pi iwt N spl radic N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l sup 1 norm of the coefficients among all decompositions. Here "highly sparse" means that N sub t +N sub w < spl radic N 2 where N sub t is the number of time atoms, N sub w is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l sup 1 uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l sup 1 norm atomic decomposition. The condition "sufficiently sparse" becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2 sup j 2 .
@cite_23: Compressive sensing (CS) is an emerging approach for acquisition of signals having a sparse or compressible representation in some basis. While CS literature has mostly focused on problems involving 1-D and 2-D signals, many important applications involve signals that are multidimensional. We propose the use of Kronecker product matrices in CS for two purposes. First, we can use such matrices as sparsifying bases that jointly model the different types of structure present in the signal. Second, the measurement matrices used in distributed measurement settings can be easily expressed as Kronecker products. This new formulation enables the derivation of analytical bounds for sparse approximation and CS recovery of multidimensional signals.
One interesting aspect of our work is that we show that it is possible to use highly constrained sensing matrices (i.e. those with tensor product structure) to recover the signal of interest. Many standard techniques fail in this setting. Restricted isometry based approaches @cite_32 and coherence based approaches @cite_37 @cite_6 @cite_22 fail due to a lack of independence structure in the sensing matrix. Indeed, the restricted isometry constants as well as the coherence constants are known to be weak for tensor product sensing operators @cite_23 @cite_8 . Gaussian width based analysis approaches @cite_41 fail because the kernel of the sensing matrix is not a uniformly random subspace and hence not amenable to a similar application of Gordon's ( escape through the mesh'') theorem. We overcome these technical difficulties by working directly with combinatorial properties of the tensor product of a random bipartite graph, and exploiting those to prove the so-called nullspace property @cite_44 @cite_3 .