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Conditional Density Estimation, Latent Variable Discovery, and Optimal Transport
Communications on Pure and Applied Mathematics ( IF 3 ) Pub Date : 2020-12-28 , DOI: 10.1002/cpa.21972
Hongkang Yang 1 , Esteban G. Tabak 2
Affiliation  

A framework is proposed that addresses both conditional density estimation and latent variable discovery. The objective function maximizes explanation of variability in the data, achieved through the optimal transport barycenter generalized to a collection of conditional distributions indexed by a covariate—either given or latent—in any suitable space. Theoretical results establish the existence of barycenters, a minimax formulation of optimal transport maps, and a general characterization of variability via the optimal transport cost. This framework leads to a family of nonparametric neural network-based algorithms, the BaryNet, with a supervised version that estimates conditional distributions and an unsupervised version that assigns latent variables. The efficacy of BaryNets is demonstrated by tests on both artificial and real-world data sets. A parallel drawn between autoencoders and the barycenter framework leads to the Barycentric autoencoder algorithm (BAE). © 2020 Wiley Periodicals LLC.

中文翻译:

条件密度估计、潜在变量发现和最优传输

提出了一个解决条件密度估计和潜在变量发现的框架。目标函数最大化对数据可变性的解释,通过将最优传输重心推广到由任何合适空间中的协变量(给定或潜在)索引的条件分布集合来实现。理论结果确定了重心的存在,最优传输图的极小极大公式,以及通过最优传输成本对可变性的一般表征。这个框架导致了一系列基于非参数神经网络的算法,BaryNet,有一个估计条件分布的监督版本和一个分配潜在变量的无监督版本。BaryNets 的功效通过对人工和真实世界数据集的测试得到证明。自动编码器和重心框架之间的平行绘制导致重心自动编码器算法(BAE)。© 2020 威利期刊有限责任公司。
更新日期:2020-12-28
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