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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 commonly-used 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 ill-conditioned, non-smooth and constrained distributions in very high dimension efficiently in practice. 

Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+\epsilon)$-approximate cut-preserving 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 $c$-edge-connectivity queries for $c \ge 4$ in polylogarithmic time per query as well as more efficient algorithms for survivable network design on bounded treewidth graphs.