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Given observations of a collection of covariates and responses $(Y, X) \in\mathbb{R}^p \times \mathbb{R}^q$, sufficient dimension reduction (SDR)techniques aim to identify a mapping $f: \mathbb{R}^q \rightarrow \mathbb{R}^k$with $k \ll q$ 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 numberof responses $p$ is also quite large, in addition to a large number $q$ ofcovariates. This leads to the challenge of fitting a succinctly parameterizedstatistical model to $Y|f(X)$, which is a problem that is usually not addressedin a traditional SDR framework. In this paper, we present a computationallytractable convex relaxation based estimator for simultaneously (a) identifyinga linear dimension reduction $f(X)$ of the covariates that is sufficient withrespect to the responses, and (b) fitting several types of structuredlow-dimensional models -- factor models, graphical models, latent-variablegraphical models -- to the conditional distribution of $Y|f(X)$. We analyze theconsistency properties of our estimator in a high-dimensional scaling regime.We also illustrate the performance of our approach on a newsgroup dataset andon a dataset consisting of financial asset prices.

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