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A Distributed Framework for the Construction of Transport Maps

机译：运输地图构建的分布式框架

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摘要

The need to reason about uncertainty in large, complex, and multimodal data sets has become increasingly common across modern scientific environments. The ability to transform samples from one distribution P to another distribution Q enables the solution to many problems in machine learning (e.g., Bayesian inference, generative modeling) and has been actively pursued from theoretical, computational, and application perspectives across the fields of information theory, computer science, and biology. Performing such transformations in general still leads to computational difficulties, especially in high dimensions. Here, we consider the problem of computing such "measure transport maps" with efficient and parallelizable methods. Under the mild assumptions that P need not be known but can be sampled from and that the density of Q is known up to a proportionality constant, and that Q is log-concave, we provide in this work a convex optimization problem pertaining to relative entropy minimization. We show how an empirical minimization formulation and polynomial chaos map parameterization can allow for learning a transport map between P and Q with distributed and scalable methods. We also leverage findings from nonequilibrium thermodynamics to represent the transport map as a composition of simpler maps, each of which is learned sequentially with a transport cost regularized version of the aforementioned problem formulation. We provide examples of our framework within the context of Bayesian inference for the Boston housing data set and generative modeling for handwritten digit images from the MNIST data set.
机译：在现代科学环境中，越来越需要对大型，复杂和多模式数据集中的不确定性进行推理。将样本从一种分布P转换为另一种分布Q的能力可以解决机器学习中的许多问题（例如，贝叶斯推理，生成建模），并且已经在信息理论领域中从理论，计算和应用角度进行了积极探索。 ，计算机科学和生物学。通常，执行这样的转换仍然会导致计算困难，尤其是在高维度上。在这里，我们考虑使用有效且可并行化的方法来计算此类“度量传输图”的问题。在一个温和的假设下，P不需要知道，但可以从中采样，并且Q的密度在比例常数之前都是已知的，并且Q是对数凹的，我们在这项工作中提供了一个与相对熵有关的凸优化问题。最小化。我们展示了经验最小化公式和多项式混沌图参数化如何允许使用分布式和可扩展方法学习P和Q之间的传输图。我们还利用非平衡热力学的发现将运输图表示为更简单的图的组成，每个图都使用上述问题公式的运输成本正则化形式顺序学习。我们在波士顿住房数据集的贝叶斯推断和MNIST数据集的手写数字图像生成模型的上下文中，提供了我们框架的示例。

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  • 来源
    《Neural computation》 |2019年第4期|613-652|共40页
  • 作者

    Mesa Diego A.; Tantiongloc Justin; Mendoza Marcela; Kim Sanggyun; Coleman Todd P.;

  • 作者单位

    Vanderbilt Univ, Dept Elect Engn & Comp Sci, Nashville, TN 37205 USA|Vanderbilt Univ, Dept Biomed Informat, Nashville, TN 37205 USA;

    Univ Calif San Diego, Dept Comp Sci & Engn, La Jolla, CA 92093 USA;

    Univ Calif San Diego, Dept Bioengn, La Jolla, CA 92093 USA;

    Stanford Univ, Dept Psychiat & Behav Sci, Palo Alto, CA 94304 USA;

    Univ Calif San Diego, Dept Bioengn, La Jolla, CA 92093 USA;

  • 收录信息 美国《科学引文索引》(SCI);美国《化学文摘》(CA);
  • 原文格式 PDF
  • 正文语种 eng
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