# Fast Randomized Algorithms for t-Product Based Tensor Operations and Decompositions with Applications to Imaging Data

## TL;DR

This paper introduces fast randomized algorithms for tensor operations based on the t-product, providing efficient low-rank approximations and decompositions with error guarantees, demonstrated on imaging and signal processing applications.

## Contribution

It develops novel randomized algorithms for tensor low-rank approximation and decomposition using the t-product, with theoretical error guarantees and practical efficiency.

## Key findings

- Algorithms achieve significant speed-up over traditional methods.
- High accuracy in tensor completion tasks.
- Effective in imaging applications like mass spectrometry and video recovery.

## Abstract

Tensors of order three or higher have found applications in diverse fields, including image and signal processing, data mining, biomedical engineering and link analysis, to name a few. In many applications that involve for example time series or other ordered data, the corresponding tensor has a distinguishing orientation that exhibits a low tubal structure. This has motivated the introduction of the tubal rank and the corresponding tubal singular value decomposition in the literature. In this work, we develop randomized algorithms for many common tensor operations, including tensor low-rank approximation and decomposition, together with tensor multiplication. The proposed tubal focused algorithms employ a small number of lateral and/or horizontal slices of the underlying 3-rd order tensor, that come with {\em relative error guarantees} for the quality of the obtained solutions. The performance of the proposed algorithms is illustrated on diverse imaging applications, including mass spectrometry data and image and video recovery from incomplete and noisy data. The results show both good computational speed-up vis-a-vis conventional completion algorithms and good accuracy.