Related papers: Positive semidefinite rank and nested spectrahedra
Padberg introduced a geometric notion of ranks for (mixed) integer rational polyhedrons and conjectured that the geometric rank of the matching polytope is one. In this work, we prove that this conjecture is true.
We study completion of partial matrices with nonnegative entries to matrices of nonnegative rank at most $r$ for some $r \in \mathbb{N}$. Most of our results are for $r \leq 3$. We show that a partial matrix with nonnegative entries has a…
We investigate border ranks of twisted powers of polynomials and smoothability of symmetric powers of algebras. We prove that the latter are smoothable. For the former, we obtain upper bounds for the border rank in general and prove that…
We investigate the properties of positive definite and positive semi-definite symmetric matrices within the framework of symmetrized tropical algebra, an extension of tropical algebra adapted to ordered valued fields. We focus on the…
In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive…
We deal with equations over free semilattice of infinite rank and prove that any infinite consistent system of equations is equivalent to its finite subsystem. Moreover, we describe irreducible algebraic sets and solve some algorithmic…
Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…
We study real ternary forms whose real rank equals the generic complex rank, and we characterize the semialgebraic set of sums of powers representations with that rank. Complete results are obtained for quadrics and cubics. For quintics we…
In this article we prove that a semialgebraic map is a branched covering if and only if its associated spectral map is a branched covering. In addition, such spectral map has a neat behavior with respect to the branching locus, the…
Polyhedra and spectrahedra over the real numbers, or more generally their images under linear maps, are respectively the feasible sets of linear and semidefinite programming, and form the family of semidefinite-representable sets. This…
We study algebraic properties of full rank 1 algebras in a general framework and derive a method to verify if one such matrix polynomial sub-algebra is bispectral. We give two examples illustrating the method. In the first one, we consider…
Positive semidefinite Hermitian matrices that are not fully specified can be completed provided their underlying graph is chordal. If the matrix is positive definite the completion can be uniquely characterized as the matrix that maximizes…
This is a foundational paper in tropical linear algebra, which is linear algebra over the min-plus semiring. We introduce and compare three natural definitions of the rank of a matrix, called the Barvinok rank, the Kapranov rank and the…
The paper addresses the problem of learning a regression model parameterized by a fixed-rank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The…
We study the computational question whether a given polytope or spectrahedron $S_A$ (as given by the positive semidefiniteness region of a linear matrix pencil $A(x)$) is contained in another one $S_B$. First we classify the computational…
In this note, simple proofs of certain well-known results involving the positive square root of positive matrices are given.
We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of $G$-invariant tensors. Here $G$ is the sub- group of the real orthogonal…
We consider partial symmetric Toeplitz matrices where a positive definite completion exists. We characterize those patterns where the maximum determinant completion is itself Toeplitz. We then extend these results with positive definite…
We study the algebraic boundary of a convex semi-algebraic set via duality in convex and algebraic geometry. We generalize the correspondence of facets of a polytope to the vertices of the dual polytope to general semi-algebraic convex…
In this paper, we study the symmetric rank of products of linear forms and an irreducible quadratic form. The main result presents a new, non-trivial lower bound for the rank, and the arguments rely on the apolarity lemma. In the special…