68 research outputs found

    Lower bounds on matrix factorization ranks via noncommutative polynomial optimization

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    We use techniques from (tracial noncommutative) polynomial optimization to formulate hierarchies of semidefinite programming lower bounds on matrix factorization ranks. In particular, we consider the nonnegative rank, the completely positive rank, and their symmetric analogues: the positive semidefinite rank and the completely positive semidefinite rank. We study the convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples

    Exploiting ideal-sparsity in the generalized moment problem with application to matrix factorization ranks

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    We explore a new type of sparsity for the generalized moment problem (GMP) that we call ideal-sparsity. This sparsity exploits the presence of equality constraints requiring the measure to be supported on the variety of an ideal generated by bilinear monomials modeled by an associated graph. We show that this enables an equivalent sparse reformulation of the GMP, where the single (high dimensional) measure variable is replaced by several (lower-dimensional) measure variables supported on the maximal cliques of the graph. We explore the resulting hierarchies of moment-based relaxations for the original dense formulation of GMP and this new, equivalent ideal-sparse reformulation, when applied to the problem of bounding nonnegative- and completely positive matrix factorization ranks. We show that the ideal-sparse hierarchies provide bounds that are at least as good (and often tighter) as those obtained from the dense hierarchy. This is in sharp contrast to the situation when exploiting correlative sparsity, as is most common in the literature, where the resulting bounds are weaker than the dense bounds. Moreover, while correlative sparsity requires the underlying graph to be chordal, no such assumption is needed for ideal-sparsity. Numerical results show that the ideal-sparse bounds are often tighter and much faster to compute than their dense analogs.Comment: 36 pages, 3 figure

    Real Algebraic Geometry With a View Toward Moment Problems and Optimization

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    Continuing the tradition initiated in MFO workshop held in 2014, the aim of this workshop was to foster the interaction between real algebraic geometry, operator theory, optimization, and algorithms for systems control. A particular emphasis was given to moment problems through an interesting dialogue between researchers working on these problems in finite and infinite dimensional settings, from which emerged new challenges and interdisciplinary applications

    Real Algebraic Geometry with a View Toward Hyperbolic Programming and Free Probability

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    Continuing the tradition initiated in the MFO workshops held in 2014 and 2017, this workshop was dedicated to the newest developments in real algebraic geometry and polynomial optimization, with a particular emphasis on free non-commutative real algebraic geometry and hyperbolic programming. A particular effort was invested in exploring the interrelations with free probability. This established an interesting dialogue between researchers working in real algebraic geometry and those working in free probability, from which emerged new exciting and promising synergies

    Ranks of linear matrix pencils separate simultaneous similarity orbits

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    This paper solves the two-sided version and provides a counterexample to the general version of the 2003 conjecture by Hadwin and Larson. Consider evaluations of linear matrix pencils L=T0+x1T1+⋯+xmTmL=T_0+x_1T_1+\cdots+x_mT_m on matrix tuples as L(X1,
,Xm)=I⊗T0+X1⊗T1+⋯+Xm⊗TmL(X_1,\dots,X_m)=I\otimes T_0+X_1\otimes T_1+\cdots+X_m\otimes T_m. It is shown that ranks of linear matrix pencils constitute a collection of separating invariants for simultaneous similarity of matrix tuples. That is, mm-tuples AA and BB of n×nn\times n matrices are simultaneously similar if and only if the ranks of L(A)L(A) and L(B)L(B) are equal for all linear matrix pencils LL of size mnmn. Variants of this property are also established for symplectic, orthogonal, unitary similarity, and for the left-right action of general linear groups. Furthermore, a polynomial time algorithm for orbit equivalence of matrix tuples under the left-right action of special linear groups is deduced

    Real Algebraic Geometry With A View Toward Systems Control and Free Positivity

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    New interactions between real algebraic geometry, convex optimization and free non-commutative geometry have recently emerged, and have been the subject of numerous international meetings. The aim of the workshop was to bring together experts, as well as young researchers, to investigate current key questions at the interface of these fields, and to explore emerging interdisciplinary applications

    Bounding the separable rank via polynomial optimization

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    We investigate questions related to the set SEPd consisting of the linear maps ρ acting on Cd⊗Cd that can be written as a convex combination of rank one matrices of the form xx∗⊗yy∗. Such maps are known in quantum information theory as the separable bipartite states, while nonseparable states are called entangled. In particular we introduce bounds for the separable rank ranksep(ρ), defined as the smallest number of rank one states xx∗⊗yy∗ entering the decomposition of a separable state ρ. Our approach relies on the moment method and yields a hierarchy of semidefinite-based lower bounds, that converges to a parameter τsep(ρ), a natural convexification of the combinatorial parameter ranksep(ρ). A distinguishing feature is exploiting the positivity constraint ρ −xx∗⊗yy∗ 0 to impose positivity of a polynomial matrix localizing map, the dual notion of the notion of sum-of-squares polynomial matrices. Our approach extends naturally to the multipartite setting and to the real separable rank, and it permits strengthening some known bounds for the completely positive rank. In addition, we indicate how the moment approach also applies to define hierarchies of semidefinite relaxations for the set SEPd and permits to give new proofs, using only tools from moment theory, for convergence results on the DPS hierarchy from Doherty et al. (2002) [16]

    Polynomial optimization: matrix factorization ranks, portfolio selection, and queueing theory

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    Inspired by Leonhard Euler’s belief that every event in the world can be understood in terms of maximizing or minimizing a specific quantity, this thesis delves into the realm of mathematical optimization. The thesis is divided into four parts, with optimization acting as the unifying thread. Part 1 introduces a particular class of optimization problems called generalized moment problems (GMPs) and explores the moment method, a powerful tool used to solve GMPs. We introduce the new concept of ideal sparsity, a technique that aids in solving GMPs by improving the bounds of their associated hierarchy of semidefinite programs. Part 2 focuses on matrix factorization ranks, in particular, the nonnegative rank, the completely positive rank, and the separable rank. These ranks are extensively studied using the moment method, and ideal sparsity is applied (whenever possible) to enhance the bounds on these ranks and speed-up their computation. Part 3 centers around portfolio optimization and the mean-variance-skewness kurtosis (MVSK) problem. Multi-objective optimization techniques are employed to uncover Pareto optimal solutions to the MVSK problem. We show that most linear scalarizations of the MVSK problem result in specific convex polynomial optimization problems which can be solved efficiently. Part 4 explores hypergraph-based polynomials emerging from queueing theory in the setting of parallel-server systems with job redundancy policies. By exploiting the symmetry inherent in the polynomials and some classical results on matrix algebras, the convexity of these polynomials is demonstrated, thereby allowing us to prove that the polynomials attain their optima at the barycenter of the simplex.<br/

    Applied Harmonic Analysis and Data Science (hybrid meeting)

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    Data science has become a field of major importance for science and technology nowadays and poses a large variety of challenging mathematical questions. The area of applied harmonic analysis has a significant impact on such problems by providing methodologies both for theoretical questions and for a wide range of applications in signal and image processing and machine learning. Building on the success of three previous workshops on applied harmonic analysis in 2012, 2015 and 2018, this workshop focused on several exciting novel directions such as mathematical theory of deep learning, but also reported progress on long-standing open problems in the field
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