# Distributionally Robust and Multi-Objective Nonnegative Matrix Factorization

## TL;DR

This paper introduces a distributionally robust multi-objective NMF framework that optimizes for the worst-case error across multiple objectives, enhancing robustness when the noise model is unknown.

## Contribution

The paper proposes a novel multi-objective NMF formulation with a dual optimization approach for distributional robustness, using a simple multiplicative update algorithm.

## Key findings

- DR-NMF is robust to unknown noise models.
- The approach effectively minimizes the maximum error across objectives.
- Results on synthetic, document, and audio data demonstrate its effectiveness.

## Abstract

Nonnegative matrix factorization (NMF) is a linear dimensionality reduction technique for analyzing nonnegative data. A key aspect of NMF is the choice of the objective function that depends on the noise model (or statistics of the noise) assumed on the data. In many applications, the noise model is unknown and difficult to estimate. In this paper, we define a multi-objective NMF (MO-NMF) problem, where several objectives are combined within the same NMF model. We propose to use Lagrange duality to judiciously optimize for a set of weights to be used within the framework of the weighted-sum approach, that is, we minimize a single objective function which is a weighted sum of the all objective functions. We design a simple algorithm based on multiplicative updates to minimize this weighted sum. We show how this can be used to find distributionally robust NMF (DR-NMF) solutions, that is, solutions that minimize the largest error among all objectives, using a dual approach solved via a heuristic inspired from the Frank-Wolfe algorithm. We illustrate the effectiveness of this approach on synthetic, document and audio data sets. The results show that DR-NMF is robust to our incognizance of the noise model of the NMF problem.