Distributionally Robust and Multi-Objective Nonnegative Matrix Factorization
This work addresses the challenge of selecting objective functions in NMF for applications like data analysis where noise models are unknown, offering a robust solution that is incremental in nature.
The paper tackles the problem of unknown noise models in nonnegative matrix factorization (NMF) by proposing a multi-objective NMF (MO-NMF) approach that combines multiple objectives into a single weighted sum, optimized using Lagrange duality and multiplicative updates, and extends it to distributionally robust NMF (DR-NMF) to minimize the worst-case error across objectives, demonstrating effectiveness on synthetic, document, and audio datasets with robustness to noise model uncertainty.
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.