DBSOP: An Efficient Heuristic for Speedy MCMC Sampling on Polytopes

Markov Chain Monte Carlo (MCMC) techniques have long been studied in computational geometry subjects whereabouts the problems to be studied are complex geometric objects which by their nature require optimized techniques to be deployed or to gain useful insights by them. MCMC approaches are directly answering to geometric problems we are attempting to answer, and how these problems could be deployed from theory to practice. Polytope which is a limited volume in n-dimensional space specified by a collection of linear inequality constraints require specific approximation. Therefore, sampling across density based polytopes can not be performed without the use of such methods in which the amount of repetition required is defined as a property of error margin. In this work we propose a simple accurate sampling approach based on the triangulation (tessellation) of a polytope. Moreover, we propose an efficient algorithm named Density Based Sampling on Polytopes (DBSOP) for speedy MCMC sampling where the time required to perform sampling is significantly lower compared to existing approaches in low dimensions with complexity $\mathcal{O}^{*}\left(n^{3}\right)$. Ultimately, we highlight possible future aspects and how the proposed scheme can be further improved with the integration of reservoir-sampling based methods resulting in more speedy and efficient solution.