用线性最优传输构建概率分布的编码模型，实现高效合成与分析。
Linearized Wasserstein Barycenters: Synthesis, Analysis, Representational Capacity, and Applications
- 基于线性最优传输设计可解析求解的概率分布编码模型
- 在单位区间上能表示任意概率分布，二维情况不成立
- 适用于协方差估计和数据补全，有有限样本保证
我们提出线性巴氏编码模型（LBCM），利用线性最优传输（LOT）度量对概率测度进行分析与合成。给出了刻画LBCM中概率测度的变分问题的闭式解，并在相容测度情形下建立了LBCM与2-Wasserstein巴氏中心的等价性。开发了用于在LBCM中合成与分析测度的计算方法，并提供有限样本保证。主要理论贡献之一是识别出一种简单形式的LBCM，足以表示闭单位区间上的所有概率测度。我们证明了二维情形下的自然类比构造不成立，如何在更高维推广仍是开放问题。最后通过协方差估计和数据补全展示了LBCM的实用性。
原文摘要 · Abstract (English)
We propose the linear barycentric coding model (LBCM) which utilizes the linear optimal transport (LOT) metric for analysis and synthesis of probability measures. We provide a closed-form solution to the variational problem characterizing the probability measures in the LBCM and establish equivalence of the LBCM to the set of 2-Wasserstein barycenters in the special case of compatible measures. Computational methods for synthesizing and analyzing measures in the LBCM are developed with finite sample guarantees. One of our main theoretical contributions is to identify an LBCM, expressed in terms of a simple family, which is sufficient to express all probability measures on the closed unit interval. We show that a natural analogous construction of an LBCM in 2 dimensions fails, and we leave it as an open problem to identify the proper extension in more than 1 dimension. We conclude by demonstrating the utility of LBCM for covariance estimation and data imputation.
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