A note on the complexity of L p minimization

Ge, Dongdong; Jiang, Xiaoye; Ye, Yinyu

doi:10.1007/s10107-011-0470-2

Cited by 239 publications

(207 citation statements)

References 21 publications

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“…However, standard interior-point algorithms for solving SDPs will always return the solution with the highest rank [21], which means that they are unlikely to deliver a feasible solution to the rank-constrained problem (8) in general. Thus, it is interesting to ask whether there are other efficient approaches for finding low-rank solutions to the SDP relaxation of (8).…”

Section: Non-convex Optimization Approaches To Network Localization Bmentioning

confidence: 99%

“…In a recent work, Ji et al [11] depart from the convex relaxation paradigm and develop a non-convex optimization approach for tackling Problem (8). Such an approach is motivated by ideas from low-rank matrix recovery-a topic that has received significant interest recently; see, e.g., the website [15] and the references therein.…”

Section: Non-convex Optimization Approaches To Network Localization Bmentioning

confidence: 99%

“…Indeed, the problem of minimizing the Schatten p-quasi-norm over a system of linear matrix inequalities is NP-hard for any fixed p 2 .0; 1/; cf. [8]. To circumvent this difficulty, Ji et al [11] design a potential reduction algorithm and show that it can approximate a first-order critical point of Problem (9) to any given accuracy in polynomial time.…”

Section: Non-convex Optimization Approaches To Network Localization Bmentioning

confidence: 99%

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Selected Open Problems in Discrete Geometry and Optimization

Bezdek

Deza

2013

Discrete Geometry and Optimization

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Section: Non-convex Optimization Approaches To Network Localization Bmentioning

confidence: 99%

Section: Non-convex Optimization Approaches To Network Localization Bmentioning

confidence: 99%

Section: Non-convex Optimization Approaches To Network Localization Bmentioning

confidence: 99%

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Selected Open Problems in Discrete Geometry and Optimization

Bezdek

Deza

2013

Discrete Geometry and Optimization

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“…Though finding the global optimal solution of p minimization is still NP-hard, computing a local minimizer can be done in polynomial time [11]. The global optimality of (3) has been studied and various conditions have been derived, for example, those based on restricted isometry property [7][8][9]12] and null space property [10,13].…”

Section: Introductionmentioning

confidence: 99%

On the Null Space Constant for <formula formulatype="inline"><tex Notation="TeX">${\ell _p}$</tex></formula> Minimization

Chen¹,

Gu²

2015

IEEE Signal Process. Lett.

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The literature on sparse recovery often adopts the p "norm" (p ∈ [0, 1]) as the penalty to induce sparsity of the signal satisfying an underdetermined linear system. The performance of the corresponding p minimization problem can be characterized by its null space constant. In spite of the NP-hardness of computing the constant, its properties can still help in illustrating the performance of p minimization. In this letter, we show the strict increase of the null space constant in the sparsity level k and its continuity in the exponent p. We also indicate that the constant is strictly increasing in p with probability 1 when the sensing matrix A is randomly generated. Finally, we show how these properties can help in demonstrating the performance of p minimization, mainly in the relationship between the the exponent p and the sparsity level k.

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“…In the most straightforward approach, optimizing the placement of sensors for a particular task with a well-defined cost function amounts to a combinatorial search. This search is exhaustive and NP-hard, and thus computationally intractable for all but the simplest examples [2,25]. Because of the brute-force nature of this optimization, prospects of scaling to larger systems do not improve with exponentially increasing computational resources.…”

mentioning

confidence: 99%

Sparse Sensor Placement Optimization for Classification

Brunton¹,

Brunton²,

Proctor³

et al. 2016

SIAM J. Appl. Math.

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Abstract. Choosing a limited set of sensor locations to characterize or classify a high-dimensional system is an important challenge in engineering design. Traditionally, optimizing the sensor locations involves a brute-force, combinatorial search, which is NP-hard and is computationally intractable for even moderately large problems. Using recent advances in sparsity-promoting techniques, we present a novel algorithm to solve this sparse sensor placement optimization for classification (SSPOC) that exploits low-dimensional structure exhibited by many high-dimensional systems. Our approach is inspired by compressed sensing, a framework that reconstructs data from few measurements. If only classification is required, reconstruction can be circumvented and the measurements needed are orders-of-magnitude fewer still. Our algorithm solves an 1 minimization to find the fewest nonzero entries of the full measurement vector that exactly reconstruct the discriminant vector in feature space; these entries represent sensor locations that best inform the decision task. We demonstrate the SSPOC algorithm on five classification tasks, using datasets from a diverse set of examples, including physical dynamical systems, image recognition, and microarray cancer identification. Once training identifies sensor locations, data taken at these locations forms a low-dimensional measurement space, and we perform computationally efficient classification with accuracy approaching that of classification using full-state data. The algorithm also works when trained on heavily subsampled data, eliminating the need for unrealistic full-state training data.

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A note on the complexity of L p minimization

Cited by 239 publications

References 21 publications

Selected Open Problems in Discrete Geometry and Optimization

Selected Open Problems in Discrete Geometry and Optimization

On the Null Space Constant for <formula formulatype="inline"><tex Notation="TeX">${\ell _p}$</tex></formula> Minimization

Sparse Sensor Placement Optimization for Classification

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