# L1-norm Tucker Tensor Decomposition

## TL;DR

This paper introduces L1-Tucker, a robust tensor decomposition method based on L1-norm, with algorithms that resist heavy data corruption, improving analysis of multi-way data.

## Contribution

It formulates L1-Tucker decomposition and proposes two algorithms, L1-HOSVD and L1-HOOI, with analysis of their complexity and convergence.

## Key findings

- L1-Tucker performs comparably to standard Tucker on clean data.
- L1-Tucker shows strong robustness against heavily corrupted data.
- Algorithms are effective for tensor reconstruction and classification.

## Abstract

Tucker decomposition is a common method for the analysis of multi-way/tensor data. Standard Tucker has been shown to be sensitive against heavy corruptions, due to its L2-norm-based formulation which places squared emphasis to peripheral entries. In this work, we explore L1-Tucker, an L1-norm based reformulation of standard Tucker decomposition. After formulating the problem, we present two algorithms for its solution, namely L1-norm Higher-Order Singular Value Decomposition (L1-HOSVD) and L1-norm Higher-Order Orthogonal Iterations (L1-HOOI). The presented algorithms are accompanied by complexity and convergence analysis. Our numerical studies on tensor reconstruction and classification corroborate that L1-Tucker, implemented by means of the proposed methods, attains similar performance to standard Tucker when the processed data are corruption-free, while it exhibits sturdy resistance against heavily corrupted entries.