Stable Rank-One Matrix Completion is Solved by the Level 2 Lasserre Relaxation

Cosse, Augustin; Demanet, Laurent

doi:10.1007/s10208-020-09471-y

Cited by 4 publications

(4 citation statements)

References 34 publications

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“…One can study the stability of Algorithm 3.1 under the noisy measurement case. Also, the application of Algorithm 2.1 under the noisy case might suffer from instability issues, as the propagation scheme to complete missing entries with the rank-one constraint is unstable [4], so it is necessary to adapt Algorithm 2.1 in order to allow for stable recovery of exact matrix decomposition with fixed-support.…”

Section: Discussionmentioning

confidence: 99%

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Efficient Identification of Butterfly Sparse Matrix Factorizations

Zheng,

Riccietti,

Gribonval

2021

Preprint

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Many well-known matrices Z are associated to fast transforms corresponding to factorizations of the form Z = X (L) . . . X (1) , where each factor X ( ) is sparse. Based on general result for the case with two factors, established in a companion paper, we investigate essential uniqueness of such factorizations. We show some identifiability results for the sparse factorization into two factors of the discrete Fourier Transform, discrete cosine transform or discrete sine transform matrices of size N = 2 L , when enforcing N 2 -sparsity by column on the left factor, and 2-sparsity by row on the right factor. We also show that the analysis with two factors can be extended to the multilayer case, based on a hierarchical factorization method. We prove that any matrix which is the product of L factors whose supports are exactly the so-called butterfly supports, admits a unique sparse factorization into L factors. This applies in particular to the Hadamard or the discrete Fourier transform matrix of size 2 L .

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Section: Discussionmentioning

confidence: 99%

“…where P N ∈ B N ×N is the permutation matrix which sorts the odd indices, then the even indices (e.g., for N = 4, it permutes [1,2,3,4] to [1,3,2,4]), and…”

mentioning

confidence: 99%

Efficient Identification of Butterfly Sparse Matrix Factorizations

Zheng,

Riccietti,

Gribonval

2021

Preprint

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“…More related to the sparse matrix factorization problem, stability in deep structured linear networks under sparsity constraints has been studied with the tensorial lifting approach [21], but in contrary to our framework, permutation ambiguities were not taken into account in that work. As our approach relies on matrix decomposition into rank-one matrices, one perspective in continuation to our work is to exploit existing stability results on rank-one matrix completability [3,6] to study stability in sparse matrix factorization.…”

Section: Discussionmentioning

confidence: 99%

Identifiability in Two-Layer Sparse Matrix Factorization

Zheng,

Riccietti,

Gribonval

2021

Preprint

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Sparse matrix factorization is the problem of approximating a matrix Z by a product of L sparse factors X (L) X (L−1) . . . X (1) . This paper focuses on identifiability issues that appear in this problem, in view of better understanding under which sparsity constraints the problem is well-posed. We give conditions under which the problem of factorizing a matrix into two sparse factors admits a unique solution, up to unavoidable permutation and scaling equivalences. Our general framework considers an arbitrary family of prescribed sparsity patterns, allowing us to capture more structured notions of sparsity than simply the count of nonzero entries. These conditions are shown to be related to essential uniqueness of exact matrix decomposition into a sum of rank-one matrices, with structured sparsity constraints. A companion paper further exploits these conditions to derive identifiability properties in multilayer sparse matrix factorization of some well-known matrices like the Hadamard or the discrete Fourier transform matrices.

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“…We also implemented a two levels sum-of-squares (SoS) hierarchy [31,32]. SoS-type algorithms reduce polynomial optimization problems to SDPs by introducing new variables for each monomial appearing in the cost function.…”

Section: Convex Relaxations For Binary Phase Retrievalmentioning

confidence: 99%

Sparse multi-reference alignment: sample complexity and computational hardness

Bendory¹,

Mickelin²,

Singer³

2021

Preprint

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Motivated by the problem of determining the atomic structure of macromolecules using single-particle cryo-electron microscopy (cryo-EM), we study the sample and computational complexities of the sparse multi-reference alignment (MRA) model: the problem of estimating a sparse signal from its noisy, circularly shifted copies. Based on its tight connection to the crystallographic phase retrieval problem, we establish that if the number of observations is proportional to the square of the variance of the noise, then the sparse MRA problem is statistically feasible for sufficiently sparse signals. To investigate its computational hardness, we consider three types of computational frameworks: projection-based algorithms, bispectrum inversion, and convex relaxations. We show that a state-of-the-art projection-based algorithm achieves the optimal estimation rate, but its computational complexity is exponential in the sparsity level. The bispectrum framework provides a statisticalcomputational trade-off: it requires more observations (so its estimation rate is suboptimal), but its computational load is provably polynomial in the signal's length. The convex relaxation approach provides polynomial-time algorithms (with a large exponent) that recover sufficiently sparse signals at the optimal estimation rate. We conclude the paper by discussing potential statistical and algorithmic implications for cryo-EM.

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Stable Rank-One Matrix Completion is Solved by the Level 2 Lasserre Relaxation

Cited by 4 publications

References 34 publications

Efficient Identification of Butterfly Sparse Matrix Factorizations

Efficient Identification of Butterfly Sparse Matrix Factorizations

Identifiability in Two-Layer Sparse Matrix Factorization

Sparse multi-reference alignment: sample complexity and computational hardness

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