# Truncated Cauchy Non-negative Matrix Factorization

## TL;DR

This paper introduces Truncated CauchyNMF, a robust non-negative matrix factorization method that effectively handles outliers and noise in data, with theoretical guarantees and demonstrated success in image clustering tasks.

## Contribution

The paper proposes a novel Truncated CauchyNMF loss function for robust subspace learning, with theoretical analysis and empirical validation on corrupted datasets.

## Key findings

- Truncated CauchyNMF outperforms traditional NMF in the presence of outliers.
- The method has a proven generalization bound with convergence rate O(√(ln n)/n).
- Experimental results confirm robustness and effectiveness in image clustering.

## Abstract

Non-negative matrix factorization (NMF) minimizes the Euclidean distance between the data matrix and its low rank approximation, and it fails when applied to corrupted data because the loss function is sensitive to outliers. In this paper, we propose a Truncated CauchyNMF loss that handle outliers by truncating large errors, and develop a Truncated CauchyNMF to robustly learn the subspace on noisy datasets contaminated by outliers. We theoretically analyze the robustness of Truncated CauchyNMF comparing with the competing models and theoretically prove that Truncated CauchyNMF has a generalization bound which converges at a rate of order $O(\sqrt{{\ln n}/{n}})$, where $n$ is the sample size. We evaluate Truncated CauchyNMF by image clustering on both simulated and real datasets. The experimental results on the datasets containing gross corruptions validate the effectiveness and robustness of Truncated CauchyNMF for learning robust subspaces.