Michelle Baker

Michelle Baker, Sara Pollock

In this paper, we apply acceleration to the inverse-free preconditioned Krylov subspace method introduced by Golub and Ye, which solves the symmetric generalized eigenvalue problem for the algebraically smallest eigenvalue. As the method is an improvement on steepest descent, we consider acceleration based on Nesterov accelerated steepest descent and Polyak's heavy-ball method. We extend acceleration to the block version of the Krylov subspace method and prove convergence for a more generalized choice of subspace. We present numerical results demonstrating the effect of fixed and safeguarded-adaptive choice of the momentum parameter, which show convergence in fewer outer iterations compared with LOBPCG with the same subspace size and generally fewer iterations than the base method when solving for multiple clustered eigenvalues with small dimension size. We also provide an explanation for the acceleration seen from implementing Polyak's heavy-ball method, including justifying the given parameter range.

Michelle Baker, Terrance E. Boult

This paper analyzes the circumstances under which Bayesian networks can be pruned in order to reduce computational complexity without altering the computation for variables of interest. Given a problem instance which consists of a query and evidence for a set of nodes in the network, it is possible to delete portions of the network which do not participate in the computation for the query. Savings in computational complexity can be large when the original network is not singly connected. Results analogous to those described in this paper have been derived before [Geiger, Verma, and Pearl 89, Shachter 88] but the implications for reducing complexity of the computations in Bayesian networks have not been stated explicitly. We show how a preprocessing step can be used to prune a Bayesian network prior to using standard algorithms to solve a given problem instance. We also show how our results can be used in a parallel distributed implementation in order to achieve greater savings. We define a computationally equivalent subgraph of a Bayesian network. The algorithm developed in [Geiger, Verma, and Pearl 89] is modified to construct the subgraphs described in this paper with O(e) complexity, where e is the number of edges in the Bayesian network. Finally, we define a minimal computationally equivalent subgraph and prove that the subgraphs described are minimal.