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Technical Program

Paper Detail

Paper IDD7-S2-T4.3
Paper Title An MCMC Method to Sample from Lattice Distributions
Authors Anand Jerry George, Navin Kashyap, Indian Institute of Science, India
Session D7-S2-T4: Topics in Privacy & Cryptography I
Chaired Session: Tuesday, 20 July, 22:20 - 22:40
Engagement Session: Tuesday, 20 July, 22:40 - 23:00
Abstract We introduce a Markov Chain Monte Carlo (MCMC) algorithm to generate samples from probability distributions supported on a $d$-dimensional lattice $\Lambda = \mathbf{B}\Z^d$, where $\mathbf{B}$ is a full-rank matrix. Specifically, we consider lattice distributions $P_\Lambda$ in which the probability at a lattice point is proportional to a given probability density function, $f$, evaluated at that point. To generate samples from $P_\Lambda$, it suffices to draw samples from a pull-back measure $P_{\Z^d}$ defined on the integer lattice. The probability of an integer lattice point under $P_{\Z^d}$ is proportional to the density function $\pi = |\det(\mathbf{B})|f \circ \mathbf{B}$. The algorithm we present in this paper for sampling from $P_{\Z^d}$ is based on the Metropolis-Hastings framework. In particular, we use $\pi$ as the proposal distribution and calculate the Metropolis-Hastings acceptance ratio for a well-chosen target distribution. We can use any method, denoted by \textbf{ALG}, that ideally draws samples from the probability density $\pi$, to generate a proposed state. The target distribution is chosen such that the coordinate-wise rounding of a sample drawn from the target distribution gives a sample from $P_{\Z^d}$. When \textbf{ALG} is ideal, we show that our algorithm is uniformly ergodic if $-\log(\pi)$ satisfies a gradient Lipschitz condition.