Stochastic Collapsed Variational Inference for Sequential Data

Stochastic variational inference for collapsed models has recently been successfully applied to large scale topic modelling. In this paper, we propose a stochastic collapsed variational inference algorithm in the sequential data setting. Our algorithm is applicable to both finite hidden Markov models and hierarchical Dirichlet process hidden Markov models, and to any datasets generated by emission distributions in the exponential family. Our experiment results on two discrete datasets show that our inference is both more efficient and more accurate than its uncollapsed version, stochastic variational inference.Comment: NIPS Workshop on Advances in Approximate Bayesian Inference, 201

Expanding translates of shrinking submanifolds in homogeneous spaces and Diophantine approximation

On the space $\mathcal{L}_{n+1}$ of unimodular lattices in $\mathbb{R}^{n+1}$, we consider the action of $a(t)={\rm diag}(t^n,t^{-1},\ldots,t^{-1})\in {\rm SL}(n+1,\mathbb{R})$ for $t>1$. Let $M$ be a nondegenerate $C^{n+1}$-submanifold of an expanding horospherical leaf in $\mathcal{L}_{n+1}$. We prove that for almost every $x\in M$, the shrinking balls in $M$ of radii $t^{-1}$ around $x$ get asymptotically equidistributed in $\mathcal{L}_{n+1}$ under the action of $a(t)$ as $t\to\infty$. This result implies non-improvability of Dirichlet's Diophantine approximation theorem for almost every point on a nondegenerate $C^{n+1}$-submanifold of $\mathbb{R}^n$, answering a question of Davenport and Schmidt (1969).Comment: 16 page