On the Stochastic Stability of Deep Markov Models
This work addresses stability issues in deep Markov models for researchers and practitioners in machine learning, though it is incremental as it builds on existing models.
The paper tackled the lack of stochastic stability guarantees in deep Markov models by deriving sufficient conditions for stability and proposing a stability analysis method based on neural network contraction, with empirical validation through numerical experiments.
Deep Markov models (DMM) are generative models that are scalable and expressive generalization of Markov models for representation, learning, and inference problems. However, the fundamental stochastic stability guarantees of such models have not been thoroughly investigated. In this paper, we provide sufficient conditions of DMM's stochastic stability as defined in the context of dynamical systems and propose a stability analysis method based on the contraction of probabilistic maps modeled by deep neural networks. We make connections between the spectral properties of neural network's weights and different types of used activation functions on the stability and overall dynamic behavior of DMMs with Gaussian distributions. Based on the theory, we propose a few practical methods for designing constrained DMMs with guaranteed stability. We empirically substantiate our theoretical results via intuitive numerical experiments using the proposed stability constraints.