Effective Streaming Low-tubal-rank Tensor Approximation via Frequent Directions

TL;DR

This paper introduces a streaming algorithm for low-tubal-rank tensor approximation that efficiently maintains a small sketch of the data, enabling accurate analysis of large-scale multi-dimensional data with limited resources.

Contribution

It extends the Frequent Directions matrix sketching technique to tensor data, providing an efficient, incremental method for low-tubal-rank approximation in streaming settings.

Findings

The algorithm achieves arbitrarily small approximation error with linearly growing sketch size.

Experimental results show superior efficiency and accuracy over existing sketching methods.

The method effectively handles large-scale, multi-dimensional streaming data.

Abstract

Low-tubal-rank tensor approximation has been proposed to analyze large-scale and multi-dimensional data. However, finding such an accurate approximation is challenging in the streaming setting, due to the limited computational resources. To alleviate this issue, this paper extends a popular matrix sketching technique, namely Frequent Directions, for constructing an efficient and accurate low-tubal-rank tensor approximation from streaming data based on the tensor Singular Value Decomposition (t-SVD). Specifically, the new algorithm allows the tensor data to be observed slice by slice, but only needs to maintain and incrementally update a much smaller sketch which could capture the principal information of the original tensor. The rigorous theoretical analysis shows that the approximation error of the new algorithm can be arbitrarily small when the sketch size grows linearly. Extensive experimental results on both synthetic and real multi-dimensional data further reveal the superiority of the proposed algorithm compared with other sketching algorithms for getting low-tubal-rank approximation, in terms of both efficiency and accuracy.