Smoothed Hierarchical Dirichlet Process: A Non-Parametric Approach to Constraint Measures

TL;DR

This paper introduces the Smoothed Hierarchical Dirichlet Process (sHDP), a non-parametric Bayesian model that incorporates temporal smoothness constraints into hierarchical mixture models, enabling better modeling of evolving distributions over time.

Contribution

The paper proposes a novel sHDP model that integrates a KL divergence-based temporal constraint into the hierarchical Dirichlet process, enhancing its ability to model time-varying mixture densities with smooth transitions.

Findings

sHDP effectively models evolving distributions in keyword data

The model maintains flexibility while enforcing smoothness constraints

Experimental results demonstrate improved performance over traditional HDP

Abstract

Time-varying mixture densities occur in many scenarios, for example, the distributions of keywords that appear in publications may evolve from year to year, video frame features associated with multiple targets may evolve in a sequence. Any models that realistically cater to this phenomenon must exhibit two important properties: the underlying mixture densities must have an unknown number of mixtures, and there must be some "smoothness" constraints in place for the adjacent mixture densities. The traditional Hierarchical Dirichlet Process (HDP) may be suited to the first property, but certainly not the second. This is due to how each random measure in the lower hierarchies is sampled independent of each other and hence does not facilitate any temporal correlations. To overcome such shortcomings, we proposed a new Smoothed Hierarchical Dirichlet Process (sHDP). The key novelty of this model is that we place a temporal constraint amongst the nearby discrete measures $\{G_j\}$ in the form of symmetric Kullback-Leibler (KL) Divergence with a fixed bound $B$. Although the constraint we place only involves a single scalar value, it nonetheless allows for flexibility in the corresponding successive measures. Remarkably, it also led us to infer the model within the stick-breaking process where the traditional Beta distribution used in stick-breaking is now replaced by a new constraint calculated from $B$. We present the inference algorithm and elaborate on its solutions. Our experiment using NIPS keywords has shown the desirable effect of the model.