# A Provably Correct and Robust Algorithm for Convolutive Nonnegative Matrix Factorization

## TL;DR

This paper introduces a provably correct, polynomial-time algorithm for convolutive nonnegative matrix factorization (CNMF) under separability assumptions, with applications in audio and neural data analysis.

## Contribution

It develops the first provably correct algorithm for CNMF that leverages existing separable NMF methods, ensuring accurate solutions in low noise conditions.

## Key findings

- Algorithm successfully recovers underlying factors in synthetic data.
- Effective in separating sources in a singing bird audio sequence.
- Runs efficiently in polynomial time.

## Abstract

In this paper, we propose a provably correct algorithm for convolutive nonnegative matrix factorization (CNMF) under separability assumptions. CNMF is a convolutive variant of nonnegative matrix factorization (NMF), which functions as an NMF with additional sequential structure. This model is useful in a number of applications, such as audio source separation and neural sequence identification. While a number of heuristic algorithms have been proposed to solve CNMF, to the best of our knowledge no provably correct algorithms have been developed. We present an algorithm that takes advantage of the NMF model underlying CNMF and exploits existing algorithms for separable NMF to provably find a solution under certain conditions. Our approach guarantees the solution in low noise settings, and runs in polynomial time. We illustrate its effectiveness on synthetic datasets, and on a singing bird audio sequence.