Hierarchical Optimal Transport for Multimodal Distribution Alignment
Hierarchical Optimal Transport for Multimodal Distribution Alignment
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发表时间：
2019-06
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通讯作者：
John Lee;M. Dabagia;Eva L. Dyer;C. Rozell
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作者：
John Lee;M. Dabagia;Eva L. Dyer;C. Rozell
在许多机器学习应用中，有必要通过对齐来有意义地聚合不同但相关的数据集。基于最优传输（OT）的方法将对齐作为发散最小化问题：目的是使用Wasserstein距离作为发散度量来转换源数据集以匹配目标数据集。我们引入了一个层次化的配方OT利用数据中的聚类结构，以提高在嘈杂，模糊，或多模态设置对齐。为了解决这个问题的数值，我们提出了一个分布式ADMM算法，也利用了Sinkhorn距离，因此它有一个有效的计算复杂度，规模与最大集群的大小平方。当两个数据集之间的转换是单一的，我们提供的性能保证，描述何时以及如何对齐的集群对应关系可以恢复与我们的配方，以及提供最坏情况下的数据集几何这样的策略。我们将这种方法应用于将数据建模为低秩高斯混合物的合成数据集，并研究数据的不同几何属性对对齐的影响。接下来，我们将我们的方法应用于神经解码应用程序，其目标是预测猕猴初级运动皮层神经元群体的运动方向和瞬时速度。我们的研究结果表明，当聚类结构存在于数据集中，并且在试验或时间点上是一致的，利用这种结构的分层对齐策略可以显着改善跨域对齐。
In many machine learning applications, it is necessary to meaningfully aggregate, through alignment, different but related datasets. Optimal transport (OT)-based approaches pose alignment as a divergence minimization problem: the aim is to transform a source dataset to match a target dataset using the Wasserstein distance as a divergence measure. We introduce a hierarchical formulation of OT which leverages clustered structure in data to improve alignment in noisy, ambiguous, or multimodal settings. To solve this numerically, we propose a distributed ADMM algorithm that also exploits the Sinkhorn distance, thus it has an efficient computational complexity that scales quadratically with the size of the largest cluster. When the transformation between two datasets is unitary, we provide performance guarantees that describe when and how well aligned cluster correspondences can be recovered with our formulation, as well as provide worst-case dataset geometry for such a strategy. We apply this method to synthetic datasets that model data as mixtures of low-rank Gaussians and study the impact that different geometric properties of the data have on alignment. Next, we applied our approach to a neural decoding application where the goal is to predict movement directions and instantaneous velocities from populations of neurons in the macaque primary motor cortex. Our results demonstrate that when clustered structure exists in datasets, and is consistent across trials or time points, a hierarchical alignment strategy that leverages such structure can provide significant improvements in cross-domain alignment.