# Iterative Block Tensor Singular Value Thresholding for Extraction of Low Rank Component of Image Data

## TL;DR

This paper introduces a robust tensor principal component analysis method that extracts low-rank components from multi-way data by dividing tensors into blocks and applying iterative singular value thresholding, improving applications like video motion separation and face image normalization.

## Contribution

It proposes a novel block tensor singular value thresholding approach for robust TPCA, with theoretical guarantees and practical effectiveness demonstrated in real-world applications.

## Key findings

- Effective in motion separation for surveillance videos
- Improves illumination normalization for face images
- Maintains optimality similar to low-rank matrix SVD

## Abstract

Tensor principal component analysis (TPCA) is a multi-linear extension of principal component analysis which converts a set of correlated measurements into several principal components. In this paper, we propose a new robust TPCA method to extract the princi- pal components of the multi-way data based on tensor singular value decomposition. The tensor is split into a number of blocks of the same size. The low rank component of each block tensor is extracted using iterative tensor singular value thresholding method. The prin- cipal components of the multi-way data are the concatenation of all the low rank components of all the block tensors. We give the block tensor incoherence conditions to guarantee the successful decom- position. This factorization has similar optimality properties to that of low rank matrix derived from singular value decomposition. Ex- perimentally, we demonstrate its effectiveness in two applications, including motion separation for surveillance videos and illumination normalization for face images.