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We study the convergence rate of discretized Riemannian Hamiltonian Monte Carlo on sampling from distributions in the form of e^{−f(x)} on a convex body M ⊂ R^n. We show that for distributions in the form of e−^{a x} on a polytope with m constraints, the convergence rate of a family of commonlyused integrators is independent of ∥a∥_2 and the geometry of the polytope. In particular, the implicit midpoint method (IMM) and the generalized Leapfrog method (LM) have a mixing time of mn^3 to achieve ϵ total variation distance to the target distribution. These guarantees are based on a general bound on the convergence rate for densities of the form e^{−f(x)} in terms of parameters of the manifold and the integrator. Our theoretical guarantee complements the empirical results of our old result, which shows that RHMC with IMM can sample illconditioned, nonsmooth and constrained distributions in very high dimension efficiently in practice.more » « lessFree, publiclyaccessible full text available June 12, 2024

null (Ed.)Graph compression or sparsification is a basic informationtheoretic and computational question. A major open problem in this research area is whether $(1+\epsilon)$approximate cutpreserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we initiate the study of a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call \emph{connectivity$c$ mimicking network}, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity$c$ mimicking networks of size $O(kc^4)$ exist and can be found in time $m(c\log n)^{O(c)}$. We also give a separate algorithm that constructs such graphs of size $k \cdot O(c)^{2c}$ in time $mc^{O(c)}\log^{O(1)}n$. These results lead to the first offline data structures for answering fully dynamic $c$edgeconnectivity queries for $c \ge 4$ in polylogarithmic time per query as well as more efficient algorithms for survivable network design on bounded treewidth graphs.more » « less