Sums of Squares and Sparse Semidefinite Programming

Blekherman, Grigoriy; Shu, Kevin

doi:10.1137/20m1376170

Cited by 3 publications

(14 citation statements)

References 12 publications

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“…It was shown in [3] that for a certain class of graphs, including the cycle-completable graphs defined in [1], optimizing over the G-locally PSD matrices gives a 1 + O( n g 3 ) approximation ratio, where g is the number of vertices in the smallest induced cycle with at least 4 vertices in G. This is stated precisely in Theorem 26 in [3]. We extend these results to a much wider class of graphs here.…”

Section: Introductionmentioning

confidence: 62%

“…We will denote the convex cone of PSD-completable matrices by S(G). To differ slightly from the terminology in [3], we will say that a G-partial matrix is G-locally PSD if for each clique C ⊆ V (G), the submatrix X| C is PSD. We will denote the convex cone of G-locally PSD partial matrices by P(G).…”

Section: Psd Matrix Completionmentioning

confidence: 99%

“…More recently, we have been interested in approximate PSD-completability, and whether the G-locally PSD condition is enough to guarantee that a partial matrix is approximately PSD completable. In [3], these PSD matrix completion questions were connected to the theory of sum-of-squares and nonnegative quadratic forms on certain algebraic varieties. There, quantitative results were shown on the distance between the G-locally PSD cone and the PSD-completable cone.…”

Section: Introductionmentioning

confidence: 99%

“…In [11], we strengthened the connection with the theory of nonnegative quadratic forms over algebraic varieties, which allowed us to uncover some strong structural properties of these G-locally PSD partial matrices. This paper, which combines the types of questions asked in [3] with the structural results in [11], obtains approximation guarantees for a much more general class of graphs, which we hope will prove to be useful in practical applications.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we will review the quantitative measurements of the distance between the G-locally PSD partial matrices and the PSD completable partial matrices in [3]. We will define our class of thickened graphs in 2.4 and we will give our quantitative results in Theorem 3.1.…”

Section: Introductionmentioning

confidence: 99%

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Approximate PSD-Completion for Generalized Chordal Graphs

Shu

2021

Preprint

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Recently, there has been interest in the question of whether a partial matrix in which many of the fully defined principal submatrices are PSD is approximately PSD completable. These questions are related to graph theory because we can think of the entries of a symmetric matrix as corresponding to the edges of a graph.We first introduce a family of graphs, which we call thickened graphs; these contain both triangle-free and chordal graphs, and can be viewed as the result of replacing the edges of a graph by an arbitrary chordal graph. We believe these graphs might be of independent interest.We then show that for a class of graphs including thickened graphs, it is possible to get quantitative bounds on how well the property of having these principal submatrices being PSD approximates the PSD-completability property. These bounds frequently only depend on the size of the smallest cycle of size at least 4 in the graph. We introduce some tools that allow us to better control the quality of these approximations and indicate how these approximations can be used to improve the performance of semidefinite programs. The tools we use in this paper are an interesting mix of algebraic topology, structural graph theory, and spectral analysis.

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Section: Introductionmentioning

confidence: 62%

Section: Psd Matrix Completionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Approximate PSD-Completion for Generalized Chordal Graphs

Shu

2021

Preprint

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Bilinear matrix inequalities and polynomials in several freely noncommuting variables

Balasubramanian

Hotwani

McCullough

2023

Journal of Mathematical Analysis and Applications

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Extreme Nonnegative Quadratics over Stanley Reisner Varieties

Shu

2021

Preprint

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We consider the convex geometry of the cone of nonnegative quadratics over Stanley-Reisner varieties. Stanley-Reisner varieties (which are unions of coordinate planes) are amongst the simplest real projective varieties, so this is potentially a starting point that can generalize to more complicated real projective varieties. This subject has some suprising connections to algebraic topology and category theory, which we exploit heavily in our work.These questions are also valuable in applied math, because they directly translate to questions about positive semidefinite (PSD) matrices. In particular, this relates to a long line of work concerning the extent to which it is possible to approximately check that a matrix is PSD by checking that some principle submatrices are PSD, or to check if a partial matrix can be approximately completed to full PSD matrix.We systematize both these practical and theoretical questions using a framework based on algebraic topology, category theory, and convex geometry. As applications of this framework we are able to classify the extreme nonnegative quadratics over many Stanley-Reisner varieties. We plan to follow these structural results with a paper that is more focused on quantitative questions about PSD matrix completion, which have applications in sparse semidefinite programming.

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Sums of Squares and Sparse Semidefinite Programming

Cited by 3 publications

References 12 publications

Approximate PSD-Completion for Generalized Chordal Graphs

Approximate PSD-Completion for Generalized Chordal Graphs

Bilinear matrix inequalities and polynomials in several freely noncommuting variables

Extreme Nonnegative Quadratics over Stanley Reisner Varieties

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