A State-Space Approach to Dynamic Nonnegative Matrix Factorization
This work addresses the challenge of improving NMF for time-series data in fields like signal processing or bioinformatics, offering a more efficient and effective method, though it appears incremental as it builds on existing NMF frameworks.
The paper tackles the problem of modeling time series with temporal dependencies in nonnegative matrix factorization (NMF) by proposing a state-space approach for dynamic NMF (D-NMF), which significantly outperforms static NMF and two state-of-the-art methods in numerical simulations and applications while requiring less memory and computational power.
Nonnegative matrix factorization (NMF) has been actively investigated and used in a wide range of problems in the past decade. A significant amount of attention has been given to develop NMF algorithms that are suitable to model time series with strong temporal dependencies. In this paper, we propose a novel state-space approach to perform dynamic NMF (D-NMF). In the proposed probabilistic framework, the NMF coefficients act as the state variables and their dynamics are modeled using a multi-lag nonnegative vector autoregressive (N-VAR) model within the process equation. We use expectation maximization and propose a maximum-likelihood estimation framework to estimate the basis matrix and the N-VAR model parameters. Interestingly, the N-VAR model parameters are obtained by simply applying NMF. Moreover, we derive a maximum a posteriori estimate of the state variables (i.e., the NMF coefficients) that is based on a prediction step and an update step, similarly to the Kalman filter. We illustrate the benefits of the proposed approach using different numerical simulations where D-NMF significantly outperforms its static counterpart. Experimental results for three different applications show that the proposed approach outperforms two state-of-the-art NMF approaches that exploit temporal dependencies, namely a nonnegative hidden Markov model and a frame stacking approach, while it requires less memory and computational power.