Stochastic Approximation and Memory-Limited Subspace Tracking for Poisson Streaming Data

Poisson count data is ubiquitously encountered in applications such as optical imaging, social networks, and traffic monitoring, where the data is typically modeled after a Poisson distribution and presented in a streaming fashion. Therefore, it calls for techniques to efficiently extract and track the useful information embedded therein. We consider the problem of recovering and tracking the underlying Poisson rate, where the rate vectors are assumed to lie in an unknown low-dimensional subspace, from streaming Poisson data with possibly missing entries. The recovery of the underlying subspace is posed as an expected loss minimization problem under nonnegative constraints, where the loss function is a penalized Poisson log-likelihood function. A stochastic approximation (SA) algorithm is proposed and can be implemented in an online manner. Two theoretical results are established regarding the convergence of the SA algorithm. The SA algorithm is guaranteed almost surely to converge to the same point as the original expected loss minimization problem, and the estimate converges to a local minimum. To further reduce the memory requirement and handle missing data, the SA algorithm is modified via lower bounding the log-likelihood function in a form that is decomposable and can be implemented in a memory-limited manner without storing history data. Numerical experiments are provided to demonstrate the superior performance of the proposed algorithms, compared to existing algorithms. The memory-limited SA algorithm is shown to empirically yield similar performance as the original SA algorithm at a much lower memory requirement.