Lipschitz Parametrization of Probabilistic Graphical Models
This work provides theoretical insights for improving probabilistic models in machine learning, though it appears incremental as it builds on existing Lipschitz continuity concepts.
The paper demonstrates that the log-likelihood of probabilistic graphical models is Lipschitz continuous with respect to lp-norm parameters, leading to bounds on Kullback-Leibler divergence and generalization ability, with preliminary applications in activity recognition and temporal segmentation.
We show that the log-likelihood of several probabilistic graphical models is Lipschitz continuous with respect to the lp-norm of the parameters. We discuss several implications of Lipschitz parametrization. We present an upper bound of the Kullback-Leibler divergence that allows understanding methods that penalize the lp-norm of differences of parameters as the minimization of that upper bound. The expected log-likelihood is lower bounded by the negative lp-norm, which allows understanding the generalization ability of probabilistic models. The exponential of the negative lp-norm is involved in the lower bound of the Bayes error rate, which shows that it is reasonable to use parameters as features in algorithms that rely on metric spaces (e.g. classification, dimensionality reduction, clustering). Our results do not rely on specific algorithms for learning the structure or parameters. We show preliminary results for activity recognition and temporal segmentation.