# Nonnegative Matrix Factorization for Music - Tuning the NMF Algorithm with Regularization Open Access

## Valyou, Jonathan (Spring 2022)

Permanent URL: https://etd.library.emory.edu/concern/etds/np193b40x?locale=en
Published

## Abstract

The mathematics behind music is a work of art in itself. Mathematicians have been utilizing mathematical tools to analyze music for decades. One such tool is Nonnegative Matrix Factorization (NMF) which has been used to decompose an audio signal into fundamental components in a source separation application. The NMF algorithm in a musical interpretation takes a spectral object known as a spectrogram represented by a matrix and separates the spectrogram into a two nonnegative sparse matrix product where one matrix takes temporal information of the sources, and the other matrix gives the frequency information of the sources. While the basic NMF algorithm excels at handling small in complexity problems with little noise, it fails to successfully separate the sources for problems with many sources or bad-quality audio data. One solution to this limitation is the implementation of regularization into the NMF algorithm. Regularization aims to induce qualities into our matrix factorization such as promoting sparsity or smoothing temporal readings that will improve the source separation accuracy. In this paper, we hope to introduce the simple NMF algorithm, display the source separation application of NMF, and demonstrate the effects of a regularized NMF algorithm.

1 Introduction........................................................................1

2 Related Works......................................................................4

3 From Recording to Spectrogram............................................7

4 Deriving the NMF Algorithm................................................11

4.1 NMF Convergence.............................................................15

5 Illustrative Examples...........................................................18

5.1 C Scale Example...............................................................18

5.2 Drum Sound Separation....................................................21

6 Regularized NMF.................................................................25

7 Experiments.......................................................................29

7.1 NMFD: Higher Complexity Audio Sample...........................29

7.2 NMFD: Symphony with Real Noise.....................................32

7.3 Regularized NMF: C Scale with Induced Noise.....................34

8 Conclusions and Future Directions.......................................38

Appendix A Example Derivation of Regularized NMF................41

Bibliography..........................................................................44