Lower Bounds on Matrix Factorization Ranks via Noncommutative Polynomial Optimization

Abstract

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 convergence properties of our hierarchies, compare them extensively to known lower bounds, and provide some (numerical) examples.Keywords Matrix factorization ranks · Nonnegative rank · Positive semidefinite rank · Completely positive rank · Completely positive semidefinite rank · Noncommutative polynomial optimization Mathematics Subject Classification (2010) 15A48 · 15A23 · 90C22 1 Introduction Matrix factorization ranksA factorization of a matrix A ∈ R m×n over a sequence {K d } d∈N of cones that are each equipped with an inner product ·, · is a decomposition of the form A = ( X i ,Y j ) with X i ,Y j ∈ K d for all (i, j) ∈ [m] × [n], for some integer d ∈ N. Following [35], the

Bounds on entanglement dimensions and quantum graph parameters via noncommutative polynomial optimization

“…By the above result this hierarchy gives lower bounds on the smallest local dimension in which a synchronous correlation can be realized in the tensor model. However, in [15] we show that the hierarchy typically does not converge to cpsd-rank(M ) but instead (under a certain flatness condition) to a parameter ξ cpsd * (M ), which can be seen as a block-diagonal version of the completely positive semidefinite rank.…”

“…The minimal such d is its completely positive semidefinite rank, denoted cpsd-rank(M ). Completely positive semidefinite matrices are used in [25] to model quantum graph parameters and the completely positive semidefinite rank is investigated in [43,16,44,15]. By combining the proofs from [46] (see also [28]) and [41] one can show the following link between synchronous correlations and completely positive semidefinite matrices.…”

“…In [15] we use techniques from tracial polynomial optimization to define a semidefinite programming hierarchy {ξ cpsd r (M )} of lower bounds on cpsd-rank(M ). By the above result this hierarchy gives lower bounds on the smallest local dimension in which a synchronous correlation can be realized in the tensor model.…”

“…We propose an approach based on tracial optimization: starting from the observation that the trace of the d × d identity matrix gives its size d, we consider the problem of minimizing L(1) where L is a linear functional modeling the non-normalized matrix trace. This approach has been used in several recent works [51,34,15] for lower bounding factorization ranks of matrices and tensors.…”

“…We will now construct a hierarchy of lower bounds on the minimal entanglement dimension, using its formulation via A qc (P ). Our approach is based on noncommutative polynomial optimization, thus similar to the approach in [15] for bounding matrix factorization ranks. We first need some notation.…”