2023/05/08 by Vishwaraj Doshi, Jie Hu, Doshi, Vishwaraj +3 · 1 citation
Mathematics · Physics and Astronomy · Neuroscience · #Markov Chains and Monte Carlo Methods #Complex Network Analysis Techniques #Functional Brain Connectivity Studies
paper · pdf · doi:10.48550/arxiv.2305.05097
We consider random walks on discrete state spaces, such as general undirected graphs, where the random walkers are designed to approximate a target quantity over the network topology via sampling and neighborhood exploration in the form of Markov chain Monte Carlo (MCMC) procedures. Given any Markov chain corresponding to a target probability distribution, we design a self-repellent random walk (SRRW) which is less likely to transition to nodes that were highly visited in the past, and more likely to transition to seldom visited nodes. For a class of SRRWs parameterized by a positive real α, we prove that the empirical distribution of the process converges almost surely to the the target (stationary) distribution of the underlying Markov chain kernel. We then provide a central limit theorem and derive the exact form of the arising asymptotic co-variance matrix, which allows us to show that the SRRW with a stronger repellence (larger α) always achieves a smaller asymptotic covariance, in the sense of Loewner ordering of co-variance matrices. Especially for SRRW-driven MCMC algorithms, we show that the decrease in the asymptotic sampling variance is of the order O(1/α), eventually going down to zero. Finally, we provide numerical simulations complimentary to our theoretical results, also empirically testing a version of SRRW with α increasing in time to combine the benefits of smaller asymptotic variance due to large α, with empirically observed faster mixing properties of SRRW with smaller α.