Positive semidefinite rank

The positive semidefinite (psd) rank of a nonnegative real matrix M is the smallest integer k for which it is possible to find psd matrices Ai assigned to the rows of M and Bj assigned to the columns of M , of size k ˆ k, such that pi, jq-entry of M is the inner product of Ai and Bj . This is an example of a cone rank of a nonnegative matrix similar to nonnegative rank, and was introduced for studying sdp-representations of convex sets. I will present the main results and open questions we have so far on psd rank. The talk will be largely based on a recent survey written with Hamza Fawzi, Joao Gouveia, Pablo Parrilo and Richard Robinson.