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 A_i assigned to the rows of M and B_j assigned to the columns of M, of size k x k, such that (i,j)-entry of M is the inner product of A_i and B_j. 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.

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