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arXiv 2003.02214cs.LGstat.ML
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面向具有指数族观测变量的潜变量模型的通用无监督优化

Generic Unsupervised Optimization for a Latent Variable Model With Exponential Family Observables

University of Oldenburg(奥尔登堡大学)

Hamid Mousavi, Jakob Drefs, Florian Hirschberger, Jörg Lücke

AI总结:

本文提出一种适用于指数族观测变量的二元潜变量通用无监督优化方法,基于EM推导通用参数更新方程,结合变分加速实现高效可扩展,可应用于多类数据及方差学习、去噪等场景。

摘要:

潜变量模型(LVMs)通过潜变量的参数化函数来表示观测变量。无监督学习中LVMs的典型例子包括概率PCA和概率SC,二者均假设通过潜变量的加权线性求和来确定观测变量高斯分布的均值。然而在许多场景下,观测变量并不服从高斯分布,因此无监督学习领域已出现针对特定非高斯观测变量的LVMs研究。即便对于特定的分布选择,参数优化也颇具挑战性,且此前仅有少数研究关注观测变量分布定义更通用的LVMs。\n本文研究适用于多种不同分布的LVMs,即观测变量可服从指数族的任意(正则)分布。提出的新型LVMs针对二元潜变量定义,采用最大化操作替代求和操作来连接潜变量与观测变量。为推导优化流程,本文采用期望最大化(EM)方法进行最大似然参数估计,推导得到一组极为简洁的参数更新方程,其对所有指数族分布均具有相同的函数形式。由此得到的通用优化方法可应用于不同类型的度量数据与离散数据,且推导得到的优化方程可与近期提出的变分加速方法结合,该加速方法同样通用适用于本文研究的LVMs。二者结合既保留了优化流程的通用性与直接适用性,更关键的是实现了高效的可扩展性。本文通过数值实验验证了解析结果,并探讨了方差结构学习、噪声类型估计与去噪等潜在应用。

英文摘要:

Latent variable models (LVMs) represent observed variables by parameterized functions of latent variables. Prominent examples of LVMs for unsupervised learning are probabilistic PCA or probabilistic SC which both assume a weighted linear summation of the latents to determine the mean of a Gaussian distribution for the observables. In many cases, however, observables do not follow a Gaussian distribution. For unsupervised learning, LVMs which assume specific non-Gaussian observables have therefore been considered. Already for specific choices of distributions, parameter optimization is challenging and only a few previous contributions considered LVMs with more generally defined observable distributions. Here, we consider LVMs that are defined for a range of different distributions, i.e., observables can follow any (regular) distribution of the exponential family. The novel class of LVMs presented is defined for binary latents, and it uses maximization in place of summation to link the latents to observables. To derive an optimization procedure, we follow an EM approach for maximum likelihood parameter estimation. We show that a set of very concise parameter update equations can be derived which feature the same functional form for all exponential family distributions. The derived generic optimization can consequently be applied to different types of metric data as well as to different types of discrete data. Also, the derived optimization equations can be combined with a recently suggested variational acceleration which is likewise generically applicable to the LVMs considered here. So, the combination maintains generic and direct applicability of the derived optimization procedure, but, crucially, enables efficient scalability. We numerically verify our analytical results and discuss some potential applications such as learning of variance structure, noise type estimation and denoising.

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