Detecting the Rank of a Symmetric Tensor

This paper deals with the problem of Canonical Polyadic (CP) decomposition of a given tensor. Standard algorithms to perform this decomposition generally require the knowledge of the rank of the sought tensor decomposition. Yet, determining the rank of a given tensor is generally hard. In this paper, we propose a method to find the rank of a symmetric tensor. We reformulate the CP decomposition problem into a truncated moment problem and we derive a sufficient condition to certify the rank of the tensor from the rank of some moment matrices associated with it. For tensors with rank not exceeding a prescribed value, this sufficient condition is also necessary. Finally, we propose to combine our rank detection procedure with existing algorithms. Experimental results show the validity of our results and provide an illustration of its practical use. Our method provides the correct rank even in the presence a moderate level of noise.