3,584 research outputs found
Convergence of Langevin MCMC in KL-divergence
Langevin diffusion is a commonly used tool for sampling from a given
distribution. In this work, we establish that when the target density is
such that is smooth and strongly convex, discrete Langevin
diffusion produces a distribution with in
steps, where is the dimension of the sample
space. We also study the convergence rate when the strong-convexity assumption
is absent. By considering the Langevin diffusion as a gradient flow in the
space of probability distributions, we obtain an elegant analysis that applies
to the stronger property of convergence in KL-divergence and gives a
conceptually simpler proof of the best-known convergence results in weaker
metrics
The permutation action of finite symplectic groups of odd characteristic on their standard modules
Motivated by the incidence problems between points and flats of a symplectic
polar space, we study a large class of submodules of the space of functions on
the standard module of a finite symplectic group of odd characteristic. Our
structure results on this class of submodules allow us to determine the
-ranks of the incidence matrices between points and flats of the symplectic
polar space. In particular, we give an explicit formula for the -rank of the
generalized quadrangle , where is an odd prime power.
Combined with the earlier results of Sastry and Sin on the 2-rank of , it completes the determination of the -ranks of .Comment: 22 page
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