Performance of the low-rank tensor-train SVD (TT-SVD) for large dense tensors on modern multi-core CPUs

TL;DR

This paper introduces an efficient TT-SVD algorithm optimized for modern multi-core CPUs, significantly improving the performance of low-rank tensor approximations compared to existing implementations.

Contribution

The authors develop a TT-SVD algorithm based on QR and matrix multiplication techniques, analyze its performance with the Roofline model, and demonstrate its efficiency on shared and distributed-memory systems.

Findings

Common TT-SVD implementations have severe performance issues.

The proposed algorithm achieves near-memory bandwidth limits.

A dedicated tensor factorization library could enable faster low-rank approximations.

Abstract

There are several factorizations of multi-dimensional tensors into lower-dimensional components, known as `tensor networks'. We consider the popular `tensor-train' (TT) format and ask: How efficiently can we compute a low-rank approximation from a full tensor on current multi-core CPUs? Compared to sparse and dense linear algebra, kernel libraries for multi-linear algebra are rare and typically not as well optimized. Linear algebra libraries like BLAS and LAPACK may provide the required operations in principle, but often at the cost of additional data movements for rearranging memory layouts. Furthermore, these libraries are typically optimized for the compute-bound case (e.g.\ square matrix operations) whereas low-rank tensor decompositions lead to memory bandwidth limited operations. We propose a `tensor-train singular value decomposition' (TT-SVD) algorithm based on two building blocks: a `Q-less tall-skinny QR' factorization, and a fused tall-skinny matrix-matrix multiplication and reshape operation. We analyze the performance of the resulting TT-SVD algorithm using the Roofline performance model. In addition, we present performance results for different algorithmic variants for shared-memory as well as distributed-memory architectures. Our experiments show that commonly used TT-SVD implementations suffer severe performance penalties. We conclude that a dedicated library for tensor factorization kernels would benefit the community: Computing a low-rank approximation can be as cheap as reading the data twice from main memory. As a consequence, an implementation that achieves realistic performance will move the limit at which one has to resort to randomized methods that only process part of the data.