From latent variable models in machine learning to inverse problems in computational imaging, tensors pervade the data sciences. Often, the goal is to decompose a tensor into a particular low-rank representation, thereby recovering quantities of interest about the application at hand. In this talk, I will present a recent method for low-rank CP symmetric tensor decomposition. The key ingredients are Sylvester’s catalecticant method from classical algebraic geometry and the power method from numerical multilinear algebra. In simulations, the method is roughly one order of magnitude faster than existing CP decomposition algorithms, with similar accuracy. I will state guarantees for the relevant non-convex optimization problem, and robustness results when the tensor is only approximately low-rank (assuming an appropriate random model). Finally, if the tensor being decomposed is a higher-order moment of data points (as in multivariate statistics), our method may be performed without explicitly forming the moment tensor, opening the door to high-dimensional decompositions. This talk is based on joint works with João Pereira, Timo Klock and Tammy Kolda.
