On Convergence Properties of the EM Algorithm for Gaussian Mixtures

"Expectation-Maximization'' (EM) algorithm and gradient-based approaches for maximum likelihood learning of finite Gaussian mixtures. We show that the EM step in parameter space is obtained from the gradient via a projection matrix $P$, and we provide an explicit expression for the matrix. We then analyze the convergence of EM in terms of special properties of $P$ and provide new results analyzing the effect that $P$ has on the likelihood surface. Based on these mathematical results, we present a comparative discussion of the advantages and disadvantages of EM and other algorithms for the learning of Gaussian mixture models.

Hierarchical Mixtures of Experts and the EM Algorithm

We present a tree-structured architecture for supervised learning. The statistical model underlying the architecture is a hierarchical mixture model in which both the mixture coefficients and the mixture components are generalized linear models (GLIM's). Learning is treated as a maximum likelihood problem; in particular, we present an Expectation-Maximization (EM) algorithm for adjusting the parameters of the architecture. We also develop an on-line learning algorithm in which the parameters are updated incrementally. Comparative simulation results are presented in the robot dynamics domain.

Biologically Plausible Neural Circuits for Realization of Maximum Operations

Object recognition in the visual cortex is based on a hierarchical architecture, in which specialized brain regions along the ventral pathway extract object features of increasing levels of complexity, accompanied by greater invariance in stimulus size, position, and orientation. Recent theoretical studies postulate a non-linear pooling function, such as the maximum (MAX) operation could be fundamental in achieving such invariance. In this paper, we are concerned with neurally plausible mechanisms that may be involved in realizing the MAX operation. Four canonical circuits are proposed, each based on neural mechanisms that have been previously discussed in the context of cortical processing. Through simulations and mathematical analysis, we examine the relative performance and robustness of these mechanisms. We derive experimentally verifiable predictions for each circuit and discuss their respective physiological considerations.