Exploring the Effectiveness of Linear Matrix Factorizations After Nonlinear Processing

Abstract

In this paper, we explore the weaknesses of using sparse coding and nonnegative matrix factorization (NMF) on data that has already been processed in a nonlinear manner. The underlying assumption of matrix factorization techniques is that an input signal can be represented as a linear combination of some set of features. This is a valid assumption in many feature extraction tasks, including several audio applications such as source separation and sound scene analysis. However, sparse coding and NMF are often used on data known to be composed of a nonlinear combination of features, such as the magnitude of an audio spectrum. This paper uses two synthetic datasets to probe the ability of linear sparse coding and NMF to discover known features that are combined nonlinearly. Even in a small dataset, common nonlinearities cause interference that prevents the algorithms from recovering the known features. However, we validate the use of NMF and sparse coding in audio applications by demonstrating that the factorization process is more effective at recovering hidden features in a dataset with harmonic structure. Finally, we show that the reconstruction error associated with modeling either dataset can be reduced by taking into account the behavior of a known nonlinearity.