Effective Streaming Low-tubal-rank Tensor Approximation via Frequent Directions
This work addresses the problem of efficient tensor approximation in streaming settings for applications handling large-scale multi-dimensional data, representing an incremental advancement by adapting an existing matrix method to tensors.
The paper tackles the challenge of constructing accurate low-tubal-rank tensor approximations from streaming data with limited resources by extending the Frequent Directions matrix sketching technique to tensors using t-SVD. The result is an algorithm that maintains a small sketch capturing principal information, with theoretical analysis showing approximation error can be arbitrarily small as sketch size grows linearly, and experiments demonstrate superiority in efficiency and accuracy over other sketching methods.
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.