High-Dimensional Regression with Gaussian Mixtures and Partially-Latent Response Variables

TL;DR

This paper introduces a novel probabilistic inverse regression method with mixture models and partially-latent responses, enabling effective high-dimensional data approximation and handling artifacts.

Contribution

It presents a new inverse regression approach with mixture models and partially-latent responses, along with EM algorithms for high-dimensional regression and artifact management.

Findings

Outperforms existing regression techniques on synthetic data

Effective handling of data contaminated by artifacts

Provides a tractable probabilistic framework for high-dimensional regression

Abstract

In this work we address the problem of approximating high-dimensional data with a low-dimensional representation. We make the following contributions. We propose an inverse regression method which exchanges the roles of input and response, such that the low-dimensional variable becomes the regressor, and which is tractable. We introduce a mixture of locally-linear probabilistic mapping model that starts with estimating the parameters of inverse regression, and follows with inferring closed-form solutions for the forward parameters of the high-dimensional regression problem of interest. Moreover, we introduce a partially-latent paradigm, such that the vector-valued response variable is composed of both observed and latent entries, thus being able to deal with data contaminated by experimental artifacts that cannot be explained with noise models. The proposed probabilistic formulation could be viewed as a latent-variable augmentation of regression. We devise expectation-maximization (EM) procedures based on a data augmentation strategy which facilitates the maximum-likelihood search over the model parameters. We propose two augmentation schemes and we describe in detail the associated EM inference procedures that may well be viewed as generalizations of a number of EM regression, dimension reduction, and factor analysis algorithms. The proposed framework is validated with both synthetic and real data. We provide experimental evidence that our method outperforms several existing regression techniques.