Sparse adaptive pre-conditioning for MCMC

我们将探讨如何提高马尔可夫链蒙特卡罗（MCMC）算法的效率和准确性，使用稀疏预处理和自适应MCMC方法。自适应预处理是一种非常有效的策略，可以通过学习“预处理矩阵”来加速使用马尔可夫链蒙特卡罗算法的分布探索。然而，它可能在计算上是昂贵的，因为对于d维问题需要学习O（d^2）个矩阵项，并且如果简单地完成，也可能需要O（d^3）个计算。我们将探索在预处理矩阵上施加稀疏性的策略，以便需要学习更少的矩阵条目，并且可以更有效地执行矩阵运算。我们将为MCMC开发新的理论和方法，并将新方法应用于各种应用领域。

We will explore ways to improve the efficiency and accuracy of Markov Chain Monte Carlo (MCMC) algorithms using sparse pre-conditioning and adap ve MCMC methods. Adap ve pre-conditioning is a very effective strategy to speed up the exploration of a distribution using a Markov chain Monte Carlo algorithm, by learning a 'pre-conditioning matrix'. It can, however, be computationally expensive, as O(d^2) matrix entries need to be learned for a d-dimensional problem, and O(d^3) computations may also be needed if done naively. We will explore strategies for imposing sparsity on the pre-conditioning matrix, so that fewer matrix entries need to be learned and that matrix operations can be performed more efficiently. We will develop new theory and methodology for MCMC and also apply the new methods to various application areas.