Convex and nonconvex geometries of symmetric tensor factorization

Tensors provide natural representations for massive multi-mode datasets and tensor methods also form the backbone of many machine learning, signal processing, and statistical algorithms. This work develops theories and computational methods for guaranteed overcomplete, non-orthogonal symmetric tensor factorization using convex and nonconvex optimizations. In particular, we show when the symmetric tensor factors are uniformly sampled from the unit sphere, they are provably recoverable using convex atomic norm minimization. To design scalable polynomial-time algorithms, we apply low-rank parameterization to reformulate the atomic norm regularized tensor optimization as a nonconvex program. We analyze the optimization landscape of this nonconvex program to ensure (local) convergence of gradient descent algorithms.