Research

We present a novel technique for learning the mass matrices in samplers obtained from discretized
dynamics that preserve some energy function. Existing adaptive samplers use Riemannian preconditioning
techniques, where the mass matrices are functions of the parameters being sampled. This leads to
significant complexities in the energy reformulations and resultant dynamics, often leading to implicit
systems of equations and requiring inversion of high-dimensional matrices in the leapfrog steps. Our
approach provides a simpler alternative, by using existing dynamics in the sampling step of a
Monte Carlo EM framework, and learning the mass matrices in the M step with a novel online technique.
We also propose a way to adaptively set the number of samples gathered in the E step, using sampling
error estimates from the leapfrog dynamics. Along with a novel stochastic sampler based on
Nos'{e}-Poincar'{e} dynamics, we use this framework with standard Hamiltonian Monte Carlo (HMC) as
well as newer stochastic algorithms such as SGHMC and SGNHT, and show strong performance on synthetic
and real high-dimensional sampling scenarios; we achieve sampling accuracies comparable to Riemannian
samplers while being significantly faster.

We propose an L-BFGS optimization algorithm on Riemannian manifolds using minibatched stochastic
variance reduction techniques for fast convergence with constant step sizes, without resorting to
linesearch methods designed to satisfy Wolfe conditions. We provide a new convergence proof for
strongly convex functions without using curvature conditions on the manifold, as well as a convergence
discussion for nonconvex functions. We discuss a couple of ways to obtain the correction pairs used
to calculate the product of the gradient with the inverse Hessian, and empirically demonstrate their
use in synthetic experiments on computation of Karcher means for symmetric positive definite matrices
and leading eigenvalues of large scale data matrices. We compare our method to VR-PCA for the latter
experiment, along with Riemannian SVRG for both cases, and show strong convergence results for a range
of datasets.

We present a Monte Carlo sampler using a modified Nosé-Poincaré Hamiltonian along with Riemannian
preconditioning. Hamiltonian Monte Carlo samplers allow better exploration of the state space as
opposed to random walk-based methods, but, from a molecular dynamics perspective, may not necessarily
provide samples from the canonical ensemble. Nosé-Hoover samplers rectify that shortcoming, but the
resultant dynamics are not Hamiltonian. Furthermore, usage of these algorithms on large real-life
datasets necessitates the use of stochastic gradients, which acts as another potentially destabilizing
source of noise. In this work, we propose dynamics based on a modified Nosé-Poincaré Hamiltonian
augmented with Riemannian manifold corrections. The resultant symplectic sampling algorithm samples
from the canonical ensemble while using structural cues from the Riemannian preconditioning matrices
to efficiently traverse the parameter space. We also propose a stochastic variant using additional
terms in the Hamiltonian to correct for the noise from the stochastic gradients. We show strong
performance of our algorithms on synthetic datasets and high-dimensional Poisson factor analysis-based
topic modeling scenarios.

While most Bayesian nonparametric models in machine learning have focused on the Dirichlet process,
the beta process, or their variants, the gamma process has recently emerged as a useful nonparametric
prior in its own right. Current inference schemes for models involving the gamma process are
restricted to MCMC-based methods, which limits their scalability. In this paper, we present a
variational inference framework for models involving gamma process priors. Our approach is based on a
novel stick-breaking constructive definition of the gamma process. We prove correctness of this
stick-breaking process by using the characterization of the gamma process as a completely random
measure (CRM), and we explicitly derive the rate measure of our construction using Poisson process
machinery. We also derive error bounds on the truncation of the infinite process required for
variational inference, similar to the truncation analyses for other nonparametric models based on the
Dirichlet and beta processes. Our representation is then used to derive a variational inference
algorithm for a particular Bayesian nonparametric latent structure formulation known as the infinite
Gamma-Poisson model, where the latent variables are drawn from a gamma process prior with Poisson
likelihoods. Finally, we present results for our algorithms on nonnegative matrix factorization tasks
on document corpora, and show that we compare favorably to both sampling-based techniques and
variational approaches based on beta-Bernoulli priors.

Small-variance asymptotics provide an emerging technique for obtaining scalable combinatorial
algorithms from rich probabilistic models. We present a small-variance asymptotic analysis of the
Hidden Markov Model and its infinite-state Bayesian nonparametric extension. Starting with the
standard HMM, we first derive a “hard” inference algorithm analogous to k-means that arises when
particular variances in the model tend to zero. This analysis is then extended to the Bayesian
nonparametric case, yielding a simple, scalable, and flexible algorithm for discrete-state sequence
data with a non-fixed number of states. We also derive the corresponding combinatorial objective
functions arising from our analysis, which involve a k-means-like term along with penalties based
on state transitions and the number of states. A key property of such algorithms is that —
particularly in the nonparametric setting — standard probabilistic inference algorithms lack
scalability and are heavily dependent on good initialization. A number of results on synthetic and
real data sets demonstrate the advantages of the proposed framework.