论文标题
在歧管假设下快速混合多尺度的langevin动力学
Fast Mixing of Multi-Scale Langevin Dynamics under the Manifold Hypothesis
论文作者
论文摘要
最近，图像生成的任务引起了很多关注。特别是，马尔可夫链蒙特卡洛（MCMC）兰格文动力学技术的最新经验成功促使了许多理论上的进步。尽管如此，仍然有几个出色的问题。首先，Langevin Dynamics在非凸形景观上以非常高的尺寸运行；在最坏的情况下，由于非convex优化的NP硬度，人们认为Langevin Dynamics仅在维度中的时间指数中混合。在这项工作中，我们演示了歧管假设如何允许混合时间大幅减少，从环境维度的指数到仅取决于数据的（较小）固有维度。其次，采样空间的高维度极大地损害了Langevin动力学的性能。我们利用一种多尺度的方法来帮助改善此问题，并观察到这种多分辨率算法允许在生成中进行图像质量和计算费用之间的权衡。
Recently, the task of image generation has attracted much attention. In particular, the recent empirical successes of the Markov Chain Monte Carlo (MCMC) technique of Langevin Dynamics have prompted a number of theoretical advances; despite this, several outstanding problems remain. First, the Langevin Dynamics is run in very high dimension on a nonconvex landscape; in the worst case, due to the NP-hardness of nonconvex optimization, it is thought that Langevin Dynamics mixes only in time exponential in the dimension. In this work, we demonstrate how the manifold hypothesis allows for the considerable reduction of mixing time, from exponential in the ambient dimension to depending only on the (much smaller) intrinsic dimension of the data. Second, the high dimension of the sampling space significantly hurts the performance of Langevin Dynamics; we leverage a multi-scale approach to help ameliorate this issue and observe that this multi-resolution algorithm allows for a trade-off between image quality and computational expense in generation.
