Sampling from convex sets with a cold start using multiscale decompositions
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Running a random walk in a convex body K⊆ℝ^n is a standard approach to sample approximately uniformly from the body. The requirement is that from a suitable initial distribution, the distribution of the walk comes close to the uniform distribution π_K on K after a number of steps polynomial in n and the aspect ratio R/r (i.e., when rB_2 ⊆ K ⊆ RB_2). Proofs of rapid mixing of such walks often require the probability density η_0 of the initial distribution with respect to π_K to be at most poly(n): this is called a "warm start". Achieving a warm start often requires non-trivial pre-processing before starting the random walk. This motivates proving rapid mixing from a "cold start", wherein η_0 can be as high as exp(poly(n)). Unlike warm starts, a cold start is usually trivial to achieve. However, a random walk need not mix rapidly from a cold start: an example being the well-known "ball walk". On the other hand, Lovász and Vempala proved that the "hit-and-run" random walk mixes rapidly from a cold start. For the related coordinate hit-and-run (CHR) walk, which has been found to be promising in computational experiments, rapid mixing from a warm start was proved only recently but the question of rapid mixing from a cold start remained open. We construct a family of random walks inspired by classical decompositions of subsets of ℝ^n into countably many axis-aligned dyadic cubes. We show that even with a cold start, the mixing times of these walks are bounded by a polynomial in n and the aspect ratio. Our main technical ingredient is an isoperimetric inequality for K for a metric that magnifies distances between points close to the boundary of K. As a corollary, we show that the CHR walk also mixes rapidly both from a cold start and from a point not too close to the boundary of K.
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