On flattening of symmetric tensors and identification of latent factors

Abstract

This paper considers canonical polyadic (CP) decomposition of symmetric tensors of arbitrary even order. In earlier work [1], we showed that decomposition of such tensors is equivalent to solving a system of quadratic equations. As part of ongoing work, we further show that for almost all tensors, singular value decomposition of a certain matrix can uniquely obtain the solution to the system of quadratic equations. Our proposed algorithm is able to find the CP-decomposition, even in the regime where the CP-rank far exceeds the dimensions of the tensor (overcomplete tensors). We further show that using the symmetry of the tensor, it is possible to only use a certain type of flattening to significantly reduce the number of quadratic equations. Also, we show that the computational complexity can be reduced by a sketching technique, without any performance loss.