The Best Rank-1 Approximation of Positive Tensors Based on Mean Field Approximation

Abstract

We rapidly approximate a non-negative tensor with a rank-1 tensor. Although there have been many studies on rank-1 approximation, no algorithm guarantees that the resulting rank-1 tensor is the best approximation of the input tensor in the sense of the Frobenius norm. We find that any rank-1 tensor can be represented as a product of independent distributions when the tensor is viewed as a probability distribution. This property leads to a convex optimization formulation of the rank-1 approximation of a non-negative tensor, where we minimize the KL divergence instead of the Frobenius norm from input to the output tensor by projection onto a subspace consisting of products of independent distributions. Furthermore, we obtain an analytical representation of the best rank-1 tensor in our formulation using the property that some parameters representing a tensor do not change during this projection, which makes rank-1 approximation faster. The projection onto the space of products of independent distributions is widely known as a mean-field approximation, and our approach of rank-1 tensor approximation can also be viewed as the mean-field approximation.