The Exact Solution to Rank-1 L1-norm TUCKER2 Decomposition

TL;DR

This paper introduces the first exact algorithms for rank-1 L1-norm TUCKER2 decomposition of 3-way tensors, demonstrating superior robustness to outliers compared to traditional methods through theoretical and empirical analysis.

Contribution

The paper proves the equivalence of the problem to combinatorial optimization and provides the first exact algorithms with complexity analysis for rank-1 L1-TUCKER2 decomposition.

Findings

Exact algorithms outperform standard methods on outlier-corrupted data.

Polynomial-time algorithm under mild assumptions.

Demonstrated robustness of L1-TUCKER2 in numerical studies.

Abstract

We study rank-1 {L1-norm-based TUCKER2} (L1-TUCKER2) decomposition of 3-way tensors, treated as a collection of $N$ $D \times M$ matrices that are to be jointly decomposed. Our contributions are as follows. i) We prove that the problem is equivalent to combinatorial optimization over $N$ antipodal-binary variables. ii) We derive the first two algorithms in the literature for its exact solution. The first algorithm has cost exponential in $N$; the second one has cost polynomial in $N$ (under a mild assumption). Our algorithms are accompanied by formal complexity analysis. iii) We conduct numerical studies to compare the performance of exact L1-TUCKER2 (proposed) with standard HOSVD, HOOI, GLRAM, PCA, L1-PCA, and TPCA-L1. Our studies show that L1-TUCKER2 outperforms (in tensor approximation) all the above counterparts when the processed data are outlier corrupted.