揭示了受限玻尔兹曼机采样中混合时间的相变现象
A phase transition in sampling from Restricted Boltzmann Machines
- 通过关联吉布斯采样与动力系统，分析参数c对混合时间的影响
- 当c < c⋆ ≈ -5.87时，混合时间呈指数级增长；c = c⋆时为多项式；c > c⋆时为对数级
- 提出新等周不等式，证明临界态下分布近似对数凹，适用于概率推断研究者
受限玻尔兹曼机是一类重要的无向图模型，在深度学习和无监督学习中具有核心地位。本文证明了一参数受限玻尔兹曼机中吉布斯采样的混合时间存在相变现象：当参数c高于、等于或低于临界值c⋆≈-5.87时，混合时间分别呈对数、多项式和指数级增长。分析的关键洞察在于将吉布斯采样与一个动力系统联系起来，通过后者行为量化前者。针对临界情况c=c⋆，我们通过证明其平稳分布近乎对数凹性，建立了新的等周不等式。
原文摘要 · Abstract (English)
Restricted Boltzmann Machines are a class of undirected graphical models that play a key role in deep learning and unsupervised learning. In this study, we prove a phase transition phenomenon in the mixing time of the Gibbs sampler for a one-parameter Restricted Boltzmann Machine. Specifically, the mixing time varies logarithmically, polynomially, and exponentially with the number of vertices depending on whether the parameter $c$ is above, equal to, or below a critical value $c_\star\approx-5.87$. A key insight from our analysis is the link between the Gibbs sampler and a dynamical system, which we utilize to quantify the former based on the behavior of the latter. To study the critical case $c= c_\star$, we develop a new isoperimetric inequality for the sampler's stationary distribution by showing that the distribution is nearly log-concave.
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