Efficient Construction of Tensor Ring Representations from Sampling

Abstract

In this paper we propose an efficient method to compress a high dimensional function into a tensor ring format, based on alternating least squares (ALS). Since the function has size exponential in $d$, where $d$ is the number of dimensions, we propose an efficient sampling scheme to obtain $O(d)$ important samples in order to learn the tensor ring. Furthermore, we devise an initialization method for ALS that allows fast convergence in practice. Numerical examples show that to approximate a function with similar accuracy, the tensor ring format provided by the proposed method has fewer parameters than the tensor-train format and also better respects the structure of the original function.