# Nonnegative Matrix Factorization with Transform Learning

## TL;DR

This paper introduces a method to jointly learn an optimal short-time orthogonal transform along with nonnegative matrix factorization for improved audio signal decomposition, moving beyond fixed transforms.

## Contribution

It proposes a novel joint optimization framework that learns both the transform and the factorization, enhancing traditional NMF techniques.

## Key findings

- Improved audio signal decomposition results.
- Effective joint transform learning and factorization.
- Practical benefits demonstrated in experiments.

## Abstract

Traditional NMF-based signal decomposition relies on the factorization of spectral data, which is typically computed by means of short-time frequency transform. In this paper we propose to relax the choice of a pre-fixed transform and learn a short-time orthogonal transform together with the factorization. To this end, we formulate a regularized optimization problem reminiscent of conventional NMF, yet with the transform as additional unknown parameters, and design a novel block-descent algorithm enabling to find stationary points of this objective function. The proposed joint transform learning and factorization approach is tested for two audio signal processing experiments, illustrating its conceptual and practical benefits.