Efficient Sketching-Based Summation of Tucker Tensors

TL;DR

This paper introduces a sketching-based method for efficiently summing Tucker tensors that reduces computational cost and controls rank growth, enabling accurate low-rank approximations without forming large intermediate tensors.

Contribution

The paper presents a novel sketching framework for tensor summation in Tucker format that avoids explicit large tensor formation and leverages algebraic structures for efficiency.

Findings

Achieves substantial computational savings in tensor summation

Maintains high accuracy compared to direct summation

Effective across synthetic and real-world problems

Abstract

We present efficient, sketching-based methods for the summation of tensors in Tucker format. Leveraging the algebraic structure of Khatri-Rao and Kronecker products, our approach enables compressed arithmetic on Tucker tensors while controlling rank growth and computational cost. The proposed sketching framework avoids the explicit formation of large intermediate tensors, instead operating directly on the factor matrices and core tensors to produce accurate low-rank approximations of tensor sums. Furthermore, we analyze the computational complexity and the theoretical approximation properties of the proposed methodology. Numerical experiments demonstrate the effectiveness of our approach on four problems: two synthetic test cases, a parameter-dependent elliptic equation (commonly referred to as the cookie problem) solved via GMRES, and a one-dimensional linear transport problem discretized via high-order discontinuous Galerkin methods, where repeated tensor summation arises as a core computational bottleneck. Across these examples, the sketching-based summation achieves substantial computational savings while preserving high accuracy relative to direct summation and re-compression.