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Recently, the task of image generation has attracted much attention. Inparticular, the recent empirical successes of the Markov Chain Monte Carlo(MCMC) technique of Langevin Dynamics have prompted a number of theoreticaladvances; despite this, several outstanding problems remain. First, theLangevin Dynamics is run in very high dimension on a nonconvex landscape; inthe worst case, due to the NP-hardness of nonconvex optimization, it is thoughtthat Langevin Dynamics mixes only in time exponential in the dimension. In thiswork, we demonstrate how the manifold hypothesis allows for the considerablereduction of mixing time, from exponential in the ambient dimension todepending only on the (much smaller) intrinsic dimension of the data. Second,the high dimension of the sampling space significantly hurts the performance ofLangevin Dynamics; we leverage a multi-scale approach to help ameliorate thisissue and observe that this multi-resolution algorithm allows for a trade-offbetween image quality and computational expense in generation.

Fast Mixing of Multi-Scale Langevin Dynamics under the Manifold Hypothesis