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We provide an efficient implementation of combinatorial Bernoulli factories. Such a factory turns coins of unknown biases $x$ into a random vertex of a polytope $P$ whose mean is exactly $x$. For every affine slice of the cube (the class of polytopes that admit a factory), we give a single algorithm whose expected number of coin flips and running time at every interior $x$ are polynomial in the dimension and in $1/{δ_*}$, where ${δ_*}=\min_i\{x_i,1-x_i\}$ is never supplied to the algorithm. For perfect matchings this gives $O(n^{8}{δ_*}^{-2}\log(n/{δ_*}))$ flips, while the factory of Niazadeh, Paes Leme, and Schneider [STOC 2021] spends at least $n^{2n-1}$ flips in expectation at the uniform doubly stochastic matrix. For other polytopes, our method rounds exactly given a point deep inside $P$ and the promise that $x$ is far from the boundary of $P$.

The paper presents a statistical geometric duality between the spherical squared-Hellinger distance and a hyperbolic isometric invariant of the Poincare disc under the action of the general Mobius group. Motivated by the geometric connection, we propose the usage of the L2-embedded hyperbolic isometric invariant as an alternative way to quantify statistical divergence between Gaussian measures as a contribution to information theory and machine learning.

Programming language researchers have developed a wide array of tools to reason about probabilistic programs, but it remains unclear how to apply these proof techniques to continuous-time stochastic processes. To bridge this gap, we propose a verification method for properties of continuous-time stochastic processes by reasoning about them as standard programs. To demonstrate our approach, we define a continuous semantics for a probabilistic imperative language $\mathrm{pWhile}^{\mathrm{dt}}$ by taking the Kuratowski limit of the standard semantics, which collects limit points of the standard program semantics as the time-step approaches zero. Then, we present a Hoare-style logic that uses loop invariants and decreasing functions to verify properties of continuous programs. To establish soundness, we identify a broad class of assertions that are preserved from the standard semantics to the continuous semantics. To demonstrate our approach, we verify properties of ten continuous-time stochastic processes modeled in $\mathrm{pWhile}^{\mathrm{dt}}$, including variants of Brownian motion and a stochastic hybrid system.