Conic Sparsity: Estimation of Regression Parameters in Closed Convex Polyhedral Cones

TL;DR

This paper introduces a novel approach to estimate regression parameters within polyhedral cones by defining a new sparsity measure aligned with the cone's geometry, enabling efficient Bayesian inference under linear constraints.

Contribution

It proposes a dual form parameterization for cone-restricted parameters, generalizes spike-and-slab priors to cones, and develops an efficient posterior computation method.

Findings

Validates the sparsity measure on cone-restricted parameters.

Demonstrates the methodology on NHANES activity data.

Shows improved estimation under linear constraints.

Abstract

Statistical problems often involve linear equality and inequality constraints on model parameters. Direct estimation of parameters restricted to general polyhedral cones, particularly when one is interested in estimating low dimensional features, may be challenging. We use a dual form parameterization to characterize parameter vectors restricted to lower dimensional faces of polyhedral cones and use the characterization to define a notion of 'sparsity' on such cones. We show that the proposed notion agrees with the usual notion of sparsity in the unrestricted case and prove the validity of the proposed definition as a measure of sparsity. The identifiable parameterization of the lower dimensional faces allows a generalization of popular spike-and-slab priors to a closed convex polyhedral cone. The prior measure utilizes the geometry of the cone by defining a Markov random field over the adjacency graph of the extreme rays of the cone. We describe an efficient way of computing the posterior of the parameters in the restricted case. We illustrate the usefulness of the proposed methodology for imposing linear equality and inequality constraints by using wearables data from the National Health and Nutrition Examination Survey (NHANES) actigraph study where the daily average activity profiles of participants exhibit patterns that seem to obey such constraints.