Phase Transitions and Sample Complexity in Bayes-Optimal Matrix Factorization

Computational Complexity and Phase Transitions

Phase transitions in combinatorial problems have recently been shown [2] to be useful in locating “hard” instances of combinatorial problems. The connection between computational complexity and the existence of phase transitions has been addressed in Statistical Mechanics [2] and Artificial Intelligence [3], but not studied rigorously. We take a first step in this direction by investigating the...

The complexity of positive semidefinite matrix factorization

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 ...

On the complexity of nonnegative matrix factorization

Nonnegative matrix factorization (NMF) has become a prominent technique for the analysis of image databases, text databases and other information retrieval and clustering applications. In this report, we define an exact version of NMF. Then we establish several results about exact NMF: (1) that it is equivalent to a problem in polyhedral combinatorics; (2) that it is NP-hard; and (3) that a pol...

Phase Transitions in the Complexity of Counting

Quantum phase transitions in matrix product systems.

We investigate quantum phase transitions (QPTs) in spin chain systems characterized by local Hamiltonians with matrix product ground states. We show how to theoretically engineer such QPT points between states with predetermined properties. While some of the characteristics of these transitions are familiar, like the appearance of singularities in the thermodynamic limit, diverging correlation ...