Randomized algorithms for streaming low-rank approximation in tree tensor network format

TL;DR

This paper introduces the TTNN algorithm for streaming low-rank tensor approximation in tree tensor network format, combining randomized, single-pass, and parallel features with theoretical error bounds and practical efficiency.

Contribution

It extends streamable tensor approximation methods to tree tensor networks, providing new algorithms with error guarantees and parallelizable structure.

Findings

Efficient low-rank tensor approximation in tree format

Deterministic and probabilistic error bounds established

Algorithms outperform existing methods in numerical experiments

Abstract

In this work, we present the tree tensor network Nystr\"om (TTNN), an algorithm that extends recent research on streamable tensor approximation, such as for Tucker and tensor-train formats, to the more general tree tensor network format, enabling a unified treatment of various existing methods. Our method retains the key features of the generalized Nystr\"om approximation for matrices, that is randomized, single-pass, streamable, and cost-effective. Additionally, the structure of the sketching allows for parallel implementation. We provide a deterministic error bound for the algorithm and, in the specific case of Gaussian dimension reduction maps, also a probabilistic one. We also introduce a sequential variant of the algorithm, referred to as sequential tree tensor network Nystr\"om (STTNN), which offers better performance for dense tensors. Furthermore, both algorithms are well-suited for the recompression or rounding of tensors in the tree tensor network format. Numerical experiments highlight the efficiency and effectiveness of the proposed methods.