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  1. 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. 
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    Free, publicly-accessible full text available June 12, 2024
  2. An assembly is a large population of neurons whose synchronous firing represents a memory, concept, word, and other cognitive category. Assemblies are believed to provide a bridge between high-level cognitive phenomena and low-level neural activity. Recently, a computational system called the \emph{Assembly Calculus} (AC), with a repertoire of biologically plausible operations on assemblies, has been shown capable of simulating arbitrary space-bounded computation, but also of simulating complex cognitive phenomena such as language, reasoning, and planning. However, the mechanism whereby assemblies can mediate {\em learning} has not been known. Here we present such a mechanism, and prove rigorously that, for simple classification problems defined on distributions of labeled assemblies, a new assembly representing each class can be reliably formed in response to a few stimuli from the class; this assembly is henceforth reliably recalled in response to new stimuli from the same class. Furthermore, such class assemblies will be distinguishable as long as the respective classes are reasonably separated — for example, when they are clusters of similar assemblies, or more generally separable with margin by a linear threshold function. To prove these results, we draw on random graph theory with dynamic edge weights to estimate sequences of activated vertices, yielding strong generalizations of previous calculations and theorems in this field over the past five years. These theorems are backed up by experiments demonstrating the successful formation of assemblies which represent concept classes on synthetic data drawn from such distributions, and also on MNIST, which lends itself to classification through one assembly per digit. Seen as a learning algorithm, this mechanism is entirely online, generalizes from very few samples, and requires only mild supervision — all key attributes of learning in a model of the brain. We argue that this learning mechanism, supported by separate sensory pre-processing mechanisms for extracting attributes, such as edges or phonemes, from real world data, can be the basis of biological learning in cortex. 
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