Sufficient Dimension Reduction and Modeling Responses Conditioned on Covariates: An Integrated Approach via Convex Optimization

Abstract

Given observations of a collection of covariates and responses (Y,X) ∈ R^p × R^q, sufficient dimension reduction (SDR) techniques aim to identify a mapping f: R^q → R^k with k ≪ q such that Y|f(X) is independent of X. The image f(X) summarizes the relevant information in a potentially large number of covariates X that influence the responses Y. In many contemporary settings, the number of responses p is also quite large, in addition to a large number q of covariates. This leads to the challenge of fitting a succinctly parameterized statistical model to Y|f(X), which is a problem that is usually not addressed in a traditional SDR framework. In this paper, we present a computationally tractable convex relaxation based estimator for simultaneously (a) identifying a linear dimension reduction f(X) of the covariates that is sufficient with respect to the responses, and (b) fitting several types of structured low-dimensional models-factor models, graphical models, latent-variable graphical models - to the conditional distribution of Y|f(X). We analyze the consistency properties of our estimator in a high-dimensional scaling regime. We also illustrate the performance of our approach on a newsgroup dataset and on a dataset consisting of financial asset prices.

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