NP-hardness and inapproximability of sparse PCA

Magdon‐Ismail, Malik

doi:10.1016/j.ipl.2017.05.008

Cited by 36 publications

(25 citation statements)

References 18 publications

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“…The literature on the sparse principal component analysis problem informs us that finding a head projection for Σ is still an NP-hard problem[20, Theorem 2]. In our setting, though, there is room to relax the head projection to map into Σ[r ′ ] (s ′ ) with r ′ > r and s ′ > s.In this regard, Question 2 asks if one can actually compute a head projection for Σ with r ′ = Cr.…”

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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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confidence: 99%

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

et al. 2019

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“…For sparsity and low-rank models, this projection can be efficiently implemented by restricting to the largest coefficients or the largest principal components, respectively. For the joint low-rank and (bi-)sparse model, however, this projection is an instance of the Sparse Principal Component Analysis problem, which is known to be NP hard in general [25].…”

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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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“…Using Proposition 5.1 and the fact that rank-one SPCA is proven to be NP-hard and inapproximable in [15], we prove the following complexity result for SSVD (1.3).…”

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confidence: 99%

“…Exact and Approximation Algorithms of SSVD. We show that SSVD (1.3) is NP-hard with reduction to the rank-one SPCA problem that has been notoriously known to be NP-hard and inapproximable (see, e.g., [15]), which motivates us to develop efficient exact and approximation algorithms. Notably, the rank-one SPCA has been extensively studied in the literature and can be solved to optimality by the branch and bound algorithm [2].…”

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Beyond Symmetry: Best Submatrix Selection for the Sparse Truncated SVD

Li¹,

Xie²

2021

Preprint

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Truncated singular value decomposition (SVD), also known as the best low-rank matrix approximation, has been successfully applied to many domains such as biology, healthcare, and others, where high-dimensional datasets are prevalent. To enhance the interpretability of the truncated SVD, sparse SVD (SSVD) is introduced to select a few rows and columns of the original matrix along with the low rank approximation. Different from the literature, this paper presents a novel SSVD formulation that can select the best submatrix precisely up to a given size to maximize its truncated Ky Fan norm. The fact that the SSVD problem is NP-hard motivates us to study effective algorithms with provable performance guarantees. To do so, we first reformulate SSVD as a mixed-integer semidefinite program, which can be solved exactly for small-or medium-sized instances by a customized branch and cut algorithm with closed-form cuts, and is extremely useful to evaluate the quality of approximation algorithms. We next develop three selection algorithms based on different selection criteria and two searching algorithms-greedy and local search. We prove the approximation ratios for all the approximation algorithms and show that all the ratios are tight, i.e., we demonstrate that these approximation ratios are unimprovable. Finally, our numerical study demonstrates the high solution quality and computational efficiency of the proposed algorithms.

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NP-hardness and inapproximability of sparse PCA

Abstract: We give a reduction from clique to establish that sparse PCA is NP-hard. The reduction has a gap which we use to exclude an FPTAS for sparse PCA (unless P=NP). Under weaker complexity assumptions, we also exclude polynomial constant-factor approximation algorithms.

Cited by 36 publications

References 18 publications

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

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

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

Beyond Symmetry: Best Submatrix Selection for the Sparse Truncated SVD

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