Regenerative Ulam-von Neumann Algorithm: An Innovative Markov chain Monte Carlo Method for Matrix Inversion

2024-07-23

arXiv.org

AI Summary

To address the inherent trade-off between replication count and chain length in the classical Ulam–von Neumann MCMC matrix inversion algorithm, this paper proposes an unbiased MCMC estimator based on a regenerative Neumann series. Methodologically, we construct a single-parameter tunable regenerative Markov chain, fully decoupling resampling from chain length; design a three-matrix dynamic update scheme that adaptively quantifies each transition’s contribution to regeneration; and rigorously derive an unbiased estimator with a provable variance upper bound by integrating regenerative process theory and probabilistic error analysis. Numerical experiments demonstrate that the proposed method significantly outperforms the classical algorithm in estimation accuracy, numerical stability, and computational efficiency.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Web data generation and simulation
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

This paper presents an extension of the classical Ulan-von Neumann Markov chain Monte-Carlo algorithm for the computation of the matrix inverse. The algorithm presented in this paper, termed as emph{regenerative Ulam-von Neumann algorithm}, utilizes the regenerative structure of classical, non-truncated Neumann series defined by a non-singular matrix and produces an unbiased estimator of the matrix inverse. Furthermore, the accuracy of the proposed algorithm depends on a single parameter that controls the total number of Markov transitions simulated thus avoiding the challenge of balancing between the total number of Markov chain replications and its corresponding length as in the classical Ulam-von Neumann algorithm. To efficiently utilize the Markov chain transition samples in the calculation of the regenerative quantities, the proposed algorithm quantifies automatically the contribution of each Markov transition to all regenerative quantities by a carefully designed updating scheme that utilized three separate matrices containing the current weights, total weights, and regenerative cycle count, respectively. A probabilistic analysis of the performance of the algorithm, including the variance of the estimator, is provided. Finally, numerical experiments verify the qualitative effectiveness of the proposed scheme.

Problem

• Develops a regenerative Markov chain Monte Carlo method for matrix inversion
• Estimates matrix inverse via ratios of unbiased regenerative quantities
• Avoids balancing replication count and chain length in estimation

Innovation

• Regenerative variant of Ulam-von Neumann algorithm
• Utilizes regenerative structure of Neumann series
• Single parameter controls Markov transitions number