Positive factor networks: A graphical framework for modeling non-negative sequential data

TL;DR

This paper introduces positive factor networks (PFNs), a new graphical framework based on coupled non-negative matrix factorization modules for modeling hierarchical, non-negative sequential data, with applications in audio analysis and language modeling.

Contribution

The paper proposes PFNs as a novel, interpretable, and scalable framework for hierarchical non-negative data modeling, leveraging existing NMF algorithms for inference and learning.

Findings

Effective target tracking with hierarchical PFNs.

Hierarchical features extracted from spectrograms.

Robustness to observation noise in state inference.

Abstract

We present a novel graphical framework for modeling non-negative sequential data with hierarchical structure. Our model corresponds to a network of coupled non-negative matrix factorization (NMF) modules, which we refer to as a positive factor network (PFN). The data model is linear, subject to non-negativity constraints, so that observation data consisting of an additive combination of individually representable observations is also representable by the network. This is a desirable property for modeling problems in computational auditory scene analysis, since distinct sound sources in the environment are often well-modeled as combining additively in the corresponding magnitude spectrogram. We propose inference and learning algorithms that leverage existing NMF algorithms and that are straightforward to implement. We present a target tracking example and provide results for synthetic observation data which serve to illustrate the interesting properties of PFNs and motivate their potential usefulness in applications such as music transcription, source separation, and speech recognition. We show how a target process characterized by a hierarchical state transition model can be represented as a PFN. Our results illustrate that a PFN which is defined in terms of a single target observation can then be used to effectively track the states of multiple simultaneous targets. Our results show that the quality of the inferred target states degrades gradually as the observation noise is increased. We also present results for an example in which meaningful hierarchical features are extracted from a spectrogram. Such a hierarchical representation could be useful for music transcription and source separation applications. We also propose a network for language modeling.