A lava attack on the recovery of sums of dense and sparse signals

TL;DR

This paper introduces lava, a new penalization method for high-dimensional signals that are a combination of sparse and dense components, outperforming traditional methods like lasso and ridge in various models.

Contribution

The paper proposes lava, a novel computationally efficient penalization method tailored for the combined sparse+dense signal model, with theoretical risk analysis and empirical validation.

Findings

Lava dominates lasso and ridge in risk performance.

Analytic risk expressions derived for Gaussian models.

Simulation shows lava's superior performance with data-dependent penalties.

Abstract

Common high-dimensional methods for prediction rely on having either a sparse signal model, a model in which most parameters are zero and there are a small number of non-zero parameters that are large in magnitude, or a dense signal model, a model with no large parameters and very many small non-zero parameters. We consider a generalization of these two basic models, termed here a "sparse+dense" model, in which the signal is given by the sum of a sparse signal and a dense signal. Such a structure poses problems for traditional sparse estimators, such as the lasso, and for traditional dense estimation methods, such as ridge estimation. We propose a new penalization-based method, called lava, which is computationally efficient. With suitable choices of penalty parameters, the proposed method strictly dominates both lasso and ridge. We derive analytic expressions for the finite-sample risk function of the lava estimator in the Gaussian sequence model. We also provide an deviation bound for the prediction risk in the Gaussian regression model with fixed design. In both cases, we provide Stein's unbiased estimator for lava's prediction risk. A simulation example compares the performance of lava to lasso, ridge, and elastic net in a regression example using feasible, data-dependent penalty parameters and illustrates lava's improved performance relative to these benchmarks.