# Finding stationary points on the bounded-rank variety: a geometric hurdle and a smooth workaround

The set of matrices of a certain size and rank is a smooth manifold. Unfortunately, it is not closed: this is uncomfortable for optimization. The closure of that manifold, namely, the set of matrices with bounded rank, is an algebraic variety but it is not smooth. That also is uncomfortable for optimization. Case in point, the norm of the (projected) gradient of the cost function can go to zero along a sequence even if the limit point of the sequence is not stationary. This can trick algorithms. I will characterize the geometry of this phenomenon. Then, I will discuss how lifting the problem to a smooth manifold makes it possible to converge to stationary points with certainty under mild conditions.

Joint work with Eitan Levin (CalTech) and Joe Kileel (UT Austin).