Alternating steepest descent methods for tensor completion with applications to spectromicroscopy

TL;DR

This paper introduces two novel tensor completion algorithms based on alternating steepest descent, tailored for low-rank tensor reconstruction in spectromicroscopy, demonstrating improved sampling efficiency on real data.

Contribution

The paper develops two new ASD-based tensor completion algorithms in the $igstar_M$-product framework, specifically designed for spectromicroscopy applications, with enhanced sampling efficiency.

Findings

Achieve similar reconstruction accuracy with fewer samples compared to matrix methods.

Effectively handle third-order tensors in spectromicroscopy data.

Algorithms are parallelizable and applicable to real-world data.

Abstract

In this paper we develop two new Tensor Alternating Steepest Descent algorithms for tensor completion in the low-rank $\star_{M}$-product format, whereby we aim to reconstruct an entire low-rank tensor from a small number of measurements thereof. Both algorithms are rooted in the Alternating Steepest Descent (ASD) method for matrix completion, first proposed in [J. Tanner and K. Wei, Appl. Comput. Harmon. Anal., 40 (2016), pp. 417-429]. In deriving the new methods we target the X-ray spectromicroscopy undersampling problem, whereby data are collected by scanning a specimen on a rectangular viewpoint with X-ray beams of different energies. The recorded absorptions coefficients of the mixed specimen materials are naturally stored in a third-order tensor, with spatial horizontal and vertical axes, and an energy axis. To speed the X-ray spectromicroscopy measurement process up, only a fraction of tubes from (a reshaped version of) this tensor are fully scanned, leading to a tensor completion problem. In this framework we can apply any transform (such as the Fourier transform) to the tensor tube by tube, providing a natural way to work with the $\star_{M}$-tensor algebra, and propose: (1) a tensor completion algorithm that is essentially ASD reformulated in the $\star_{M}$-induced metric space and (2) a tensor completion algorithm that solves a set of (readily parallelizable) independent matrix completion problems for the frontal slices of the transformed tensor. The two new methods are tested on real X-ray spectromicroscopy data, demonstrating that they achieve the same reconstruction error with fewer samples from the tensor compared to the matrix completion algorithms applied to a flattened tensor.