Tensor Completion with BMD Factor Nuclear Norm Minimization

TL;DR

The paper tackles third-order tensor completion under a low BM-rank assumption by proposing BM-factor tensor slicewise nuclear norm minimization (BMNN). It derives an ADMM-based solver that updates slice-wise nuclear-norm regularized factors, performs ALS-like least-squares updates, and enforces data consistency on observed entries, all while exploiting parallelizable operations on tensor slices. Empirical results on grayscale videos and hyperspectral images show BMNN achieving reconstruction comparable to HaLRTC with potential advantages in per-iteration cost and parallelizability, especially when the BM-rank is small. The work highlights the practical potential of the Bhattacharya-Mesner decomposition for scalable, structure-aware tensor completion in spatiotemporal data settings.

Abstract

This paper is concerned with the problem of recovering third-order tensor data from limited samples. A recently proposed tensor decomposition (BMD) method has been shown to efficiently compress third-order spatiotemporal data. Using the BMD, we formulate a slicewise nuclear norm penalized algorithm to recover a third-order tensor from limited observed samples. We develop an efficient alternating direction method of multipliers (ADMM) scheme to solve the resulting minimization problem. Experimental results on real data show our method to give reconstruction comparable to those of HaLRTC (Liu et al., IEEE Trans Ptrn Anal Mchn Int, 2012), a well-known tensor completion method, in about the same number of iterations. However, our method has the advantage of smaller subproblems and higher parallelizability per iteration.

Figures (5)

• Figure 1: Illustration of the BM-product of a conformable tensor triplet $\boldsymbol{\mathscr{A}},\boldsymbol{\mathscr{B}}$, and $\boldsymbol{\mathscr{C}}$ as a sum of BM-outer products of matrix slices.
• Figure 1: Comparison of the relative squared error of the basketball video: (a) Change $\lambda$ while fixing $\mu^{0} = 0.001$ and $\rho = 1.01$. (b) Change $\mu_0$ while fixing $\lambda = 0.2$ and $\rho = 1.01$. (c) Change $\rho$ while fixing $\lambda = 0.2$ and $\mu^{0} = 0.001$.
• Figure 2: (a) The $20^{\text{th}}$ frames of the original basketball video (first row) and car video (second row). (b) The corresponding frames of the 20% sampled videos. (c) Recovered results by the HaLRTC method with $\rho=10^{-6}$. (d) Recovered results by the proposed BMNN method with $\lambda=0.2$, $\mu_0 = 0.01$, and $\rho=1.05$.
• Figure 3: Comparison of the BMNN algorithm with the HaLRTC algorithm against sampling rate on two videos: (a) Basketball video. (b) Car video.
• Figure 4: Comparison of the BMNN algorithm with the HaLRTC algorithm on five hyperspectral images. Top row: original images. Second row: $30\%$ sampled images. Third row: HaLRTC recovered images. Last row: BMNN recovered images

Theorems & Definitions (4)

• Definition 1
• Definition 2
• Definition 3
• Definition 4