Rank Minimization Using Sums of Squares of Nonnegative Matrices

AbstractWe analyze the global convergence of the power iterates for the computation of a general mixed-subordinate matrix norm. We prove a new global convergence theorem for a class of entrywise nonnegative matrices that generalizes and improves a well-known results for mixed-subordinate $$\ell ^p$$ ℓ p matrix norms. In particular, exploiting the Birkoff–Hopf contraction ratio of nonnegative matrices, we obtain novel and explicit global convergence guarantees for a range of matrix norms whose computation has been recently proven to be NP-hard in the general case, including the case of mixed-subordinate norms induced by the vector norms made by the sum of different $$\ell ^p$$ ℓ p -norms of subsets of entries.

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Positive Semidefinite Matrix Factorization: A Connection with Phase Retrieval and Affine Rank Minimization

IEEE Transactions on Signal Processing ◽

10.1109/tsp.2021.3071293 ◽

2021 ◽

pp. 1-1

Author(s):

Dana Lahat ◽

Yanbin Lang ◽

Vincent Y. F. Tan ◽

Cedric Fevotte

Keyword(s):

Matrix Factorization ◽

Phase Retrieval ◽

Positive Semidefinite ◽

Positive Semidefinite Matrix ◽

Rank Minimization ◽

Semidefinite Matrix

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Low-rank factorization for rank minimization with nonconvex regularizers

Computational Optimization and Applications ◽

10.1007/s10589-021-00276-5 ◽

2021 ◽

Author(s):

April Sagan ◽

John E. Mitchell

Keyword(s):

Low Rank ◽

Rank Minimization ◽

Rank Factorization

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On the similarity to nonnegative and Metzler Hessenberg forms

Special Matrices ◽

10.1515/spma-2020-0140 ◽

2021 ◽

Vol 10 (1) ◽

pp. 1-8

Author(s):

Christian Grussler ◽

Anders Rantzer

Keyword(s):

Standard Form ◽

Complete Characterization ◽

Nonnegative Matrices ◽

Positive Systems ◽

Third Order ◽

Hessenberg Form ◽

Hessenberg Matrices ◽

Open Question ◽

Order Continuous

Abstract We address the issue of establishing standard forms for nonnegative and Metzler matrices by considering their similarity to nonnegative and Metzler Hessenberg matrices. It is shown that for dimensions n 3, there always exists a subset of nonnegative matrices that are not similar to a nonnegative Hessenberg form, which in case of n = 3 also provides a complete characterization of all such matrices. For Metzler matrices, we further establish that they are similar to Metzler Hessenberg matrices if n 4. In particular, this provides the first standard form for controllable third order continuous-time positive systems via a positive controller-Hessenberg form. Finally, we present an example which illustrates why this result is not easily transferred to discrete-time positive systems. While many of our supplementary results are proven in general, it remains an open question if Metzler matrices of dimensions n 5 remain similar to Metzler Hessenberg matrices.

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Optimization Over the Boolean Hypercube Via Sums of Nonnegative Circuit Polynomials

Foundations of Computational Mathematics ◽

10.1007/s10208-021-09496-x ◽

2021 ◽

Author(s):

Mareike Dressler ◽

Adam Kurpisz ◽

Timo de Wolff

Keyword(s):

Computer Science ◽

Affine Transformation ◽

Optimization Problems ◽

Polynomial Optimization ◽

Theoretical Computer Science ◽

Sums Of Squares ◽

Theoretical Computer ◽

Complexity Bounds ◽

Polynomial Optimization Problems ◽

Transformation Of Variables

AbstractVarious key problems from theoretical computer science can be expressed as polynomial optimization problems over the boolean hypercube. One particularly successful way to prove complexity bounds for these types of problems is based on sums of squares (SOS) as nonnegativity certificates. In this article, we initiate optimization problems over the boolean hypercube via a recent, alternative certificate called sums of nonnegative circuit polynomials (SONC). We show that key results for SOS-based certificates remain valid: First, for polynomials, which are nonnegative over the n-variate boolean hypercube with constraints of degree d there exists a SONC certificate of degree at most $$n+d$$ n + d . Second, if there exists a degree d SONC certificate for nonnegativity of a polynomial over the boolean hypercube, then there also exists a short degree d SONC certificate that includes at most $$n^{O(d)}$$ n O ( d ) nonnegative circuit polynomials. Moreover, we prove that, in opposite to SOS, the SONC cone is not closed under taking affine transformation of variables and that for SONC there does not exist an equivalent to Putinar’s Positivstellensatz for SOS. We discuss these results from both the algebraic and the optimization perspective.

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