Let A be a matrix with nonnegative real entries. The PSD rank of A is the smallest integer k for which there exist k × k real PSD matrices B1, . . . , Bm, C1, . . . , Cn satisfying A(i|j) = tr(BiCj) for all i, j. This paper determines the computational complexity status of the PSD rank. Namely, we show that the problem of computing this function is polynomial-time equivalent to the existential ...

We consider the problem of recovering a symmetric, positive semidefinite (SPSD) matrix from a subset of its entries, possibly corrupted by noise. In contrast to previous matrix recovery work, we drop the assumption of a random sampling of entries in favor of a deterministic sampling of principal submatrices of the matrix. We develop a set of sufficient conditions for the recovery of a SPSD matr...

The learning of appropriate distance metrics is a critical problem in image classification and retrieval. In this work, we propose a boosting-based technique, termed BOOSTMETRIC, for learning a Mahalanobis distance metric. One of the primary difficulties in learning such a metric is to ensure that the Mahalanobis matrix remains positive semidefinite. Semidefinite programming is sometimes used t...

A real square matrix A is said to be Lyapunov diagonally semistable if there exists a positive definite diagonal matrix D, called a Lyapunov scaling factor of A, such that the matrix AD + DAT is positive semidefinite, Lyapunov diagonally semistable matrices play an important role in applications in several disciplines, and have been studied in many matrix theoretical papers, see for example [2]...

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 positive semidefinite rank, and their symmetric analogues: the completely positive rank and the completely positive semidefinite rank. We study the convergence properties of o...

Some new matrix versions of Kantorovich-Type inequalities for Hermitian matrix are proposed in this paper. We consider what happens to these inequalities when the positive definite matrix is allowed to be positive semidefinite singular or indefinite.

The paper addresses the problem of learning a regression model parameterized by a fixedrank positive semidefinite matrix. The focus is on the nonlinear nature of the search space and on scalability to high-dimensional problems. The mathematical developments rely on the theory of gradient descent algorithms adapted to the Riemannian geometry that underlies the set of fixed-rank positive semidefi...

Indefinite approximations of positive semidefinite matrices arise in many data analysis applications involving covariance matrices and correlation matrices. We propose a method for restoring positive semidefiniteness of an indefinite matrix M0 that constructs a convex linear combination S(α) = αM1 + (1− α)M0 of M0 and a positive semidefinite target matrix M1. In statistics, this construction fo...

The success of many machine learning and pattern recognition methods relies heavily upon the identification of an appropriate distance metric on the input data. It is often beneficial to learn such a metric from the input training data, instead of using a default one such as the Euclidean distance. In this work, we propose a boosting-based technique, termed BOOSTMETRIC, for learning a quadratic...

We address the rectangular matrix completion problem by lifting the unknown matrix to a positive semidefinite matrix in higher dimension, and optimizing a nonconvex objective over the semidefinite factor using a simple gradient descent scheme. WithO(μr2κ2nmax(μ, log n)) random observations of a n1×n2 μ-incoherent matrix of rank r and condition number κ, where n = max(n1, n2), the algorithm line...