Sparse power factorization: balancing peakiness and sample complexity

Geppert, Jakob Alexander; Krahmer, Felix; Stöger, Dominik

doi:10.1007/s10444-019-09698-6

Cited by 12 publications

(9 citation statements)

References 39 publications

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“…Other locally convergent methods applied to the recovery of row-sparse (or column-sparse) and low-rank matrices are the sparse power factorization (SPF) and its subspaceconcatenated variant (SCSPF), see [19]. While the latter work assumes a high peak-to-average power ratio on the singular vectors of the observed matrix, [13] recently enlarged the class of recoverable matrices by relaxing this constraint.…”

Section: Jointly Low-rank and Bisparse Recoverymentioning

confidence: 99%

Jointly low-rank and bisparse recovery: Questions and partial answers

et al. 2019

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We investigate the problem of recovering jointly r-rank and s-bisparse matrices from as few linear measurements as possible, considering arbitrary measurements as well as rank-one measurements. In both cases, we show that m ≍ rs ln(en/s) measurements make the recovery possible in theory, meaning via a nonpractical algorithm.In case of arbitrary measurements, we investigate the possibility of achieving practical recovery via an iterative-hard-thresholding algorithm when m ≍ rs γ ln(en/s) for some exponent γ > 0. We show that this is feasible for γ = 2, and that the proposed analysis cannot cover the case γ ≤ 1. The precise value of the optimal exponent γ ∈ [1, 2] is the object of a question, raised but unresolved in this paper, about head projections for the jointly low-rank and bisparse structure.Some related questions are partially answered in passing. For rank-one measurements, we suggest on arcane grounds an iterative-hard-thresholding algorithm modified to exploit the nonstandard restricted isometry property obeyed by this type of measurements.

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Section: Jointly Low-rank and Bisparse Recoverymentioning

confidence: 99%

Jointly low-rank and bisparse recovery: Questions and partial answers

et al. 2019

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“…A class of structured signals that is important in many applications are matrices which are simultaneously sparse and of low rank. Such matrices occur in sparse phase retrieval 1 [5,17,18], dictionary learning and sparse encoding [19], sparse matrix approximation [20], sparse PCA [21], bilinear compressed sensing problems like sparse blind deconvolution [22][23][24][25][26][27] or, more general, sparse self-calibration [28]. For example, upcoming challenges in communication engineering and signal processing require efficient algorithms for such problems with theoretical guarantees [29][30][31].…”

Section: Simultaneously Sparse and Low-rank Matricesmentioning

confidence: 99%

“…For a suboptimal but tractable initialization recovery can only be guaranteed for a considerably restricted set of very peaky signals. Relaxed conditions have been worked out recently [27] with the added benefit that the intrinsic balance between additivity and multiplicativity in sparsity is more explicitly established. Further alternating algorithms like [33] have been proposed with guaranteed local convergence and which have better empirical performance.…”

Section: Some More Details On Related Workmentioning

confidence: 99%

Simultaneous Structures in Convex Signal Recovery—Revisiting the Convex Combination of Norms

Kliesch

Szarek

Jung

2019

Front. Appl. Math. Stat.

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In compressed sensing one uses known structures of otherwise unknown signals to recover them from as few linear observations as possible. The structure comes in form of some compressibility including different notions of sparsity and low rankness. In many cases convex relaxations allow to efficiently solve the inverse problems using standard convex solvers at almost-optimal sampling rates. A standard practice to account for multiple simultaneous structures in convex optimization is to add further regularizers or constraints. From the compressed sensing perspective there is then the hope to also improve the sampling rate. Unfortunately, when taking simple combinations of regularizers, this seems not to be automatically the case as it has been shown for several examples in recent works. Here, we give an overview over ideas of combining multiple structures in convex programs by taking weighted sums and weighted maximums. We discuss explicitly cases where optimal weights are used reflecting an optimal tuning of the reconstruction. In particular, we extend known lower bounds on the number of required measurements to the optimally weighted maximum by using geometric arguments. As examples, we discuss simultaneously low rank and sparse matrices and notions of matrix norms (in the "square deal" sense) for regularizing for tensor products. We state an SDP formulation for numerically estimating the statistical dimensions and find a tensor case where the lower bound is roughly met up to a factor of two.

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“…For unstructured Gaussian measurements, local guarantees are available for the alternating algorithms Sparse Power Factorization [20] and Alternating Tikhonov regularization and Lasso [11]. Suitable initialization procedures to complement these methods by constructing a starting point in a small enough neighborhood of the solution, however, are known only for certain special classes of signals such as signals with few dominant entries [20,14]. Despite the recent progress, it remains largely an open problem whether and how joint (bi-)sparse and low rank signals can be efficiently recovered from a near-minimal number of measurements when no such initialization is provided.…”

Section: Introductionmentioning

confidence: 99%

Riemannian thresholding methods for row-sparse and low-rank matrix recovery

Eisenmann¹,

Krahmer²,

Pfeffer³

et al. 2021

Preprint

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In this paper, we present modifications of the iterative hard thresholding (IHT) method for recovery of jointly row-sparse and low-rank matrices. In particular a Riemannian version of IHT is considered which significantly reduces computational cost of the gradient projection in the case of rank-one measurement operators, which have concrete applications in blind deconvolution. Experimental results are reported that show nearoptimal recovery for Gaussian and rank-one measurements, and that adaptive stepsizes give crucial improvement. A Riemannian proximal gradient method is derived for the special case of unknown sparsity.

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Sparse power factorization: balancing peakiness and sample complexity

Cited by 12 publications

References 39 publications

Jointly low-rank and bisparse recovery: Questions and partial answers

Jointly low-rank and bisparse recovery: Questions and partial answers

Simultaneous Structures in Convex Signal Recovery—Revisiting the Convex Combination of Norms

Riemannian thresholding methods for row-sparse and low-rank matrix recovery

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