Lilja Metsälampi: Uniqueness of size-2 psd factorizations

Abstract

A positive semidefinite (PSD) factorization of size-\(k\) of a nonnegative \(p \times q\) matrix \(M\) is a collection of \(k \times k\) symmetric PSD matrices \(A_1, \dotsc, A_p, B_1, \dotsc, B_q\), such that the \((i,j)\)-th entry of \(M\) is given by the trace product of \(A_i\) and \(B_j\). The smallest \(k\) for which \(M\) admits a PSD factorization of size-\(k\) is called the PSD rank of \(M\). A matrix is of minimal PSD rank, if it is a rank- \(k(k+1)/2\) and PSD rank-\(k\) matrix. PSD factorizations originally appeared in semidefinite programming where PSD rank is related to the complexity of a semidefinite program over a convex set. Using notions from rigidity theory we study the uniqueness of PSD factorizations. We introduce \(s\)-infinitesimal motions of psd factorizations and characterize \(1\)- and \(2\)-infinitesimally rigid size-\(2\) factorizations. Lastly, we connect infinitesimal rigidity of size-\(2\) psd factorizations to uniqueness via global rigidity. This is a joint work with Kristen Dawson (SFSU), Serkan Hoşten (SFSU) and Kaie Kubjas (Aalto University).