DeepAI AI Chat
Log In Sign Up

Regularizers for Structured Sparsity

by   Charles A. Micchelli, et al.

We study the problem of learning a sparse linear regression vector under additional conditions on the structure of its sparsity pattern. This problem is relevant in machine learning, statistics and signal processing. It is well known that a linear regression can benefit from knowledge that the underlying regression vector is sparse. The combinatorial problem of selecting the nonzero components of this vector can be "relaxed" by regularizing the squared error with a convex penalty function like the ℓ_1 norm. However, in many applications, additional conditions on the structure of the regression vector and its sparsity pattern are available. Incorporating this information into the learning method may lead to a significant decrease of the estimation error. In this paper, we present a family of convex penalty functions, which encode prior knowledge on the structure of the vector formed by the absolute values of the regression coefficients. This family subsumes the ℓ_1 norm and is flexible enough to include different models of sparsity patterns, which are of practical and theoretical importance. We establish the basic properties of these penalty functions and discuss some examples where they can be computed explicitly. Moreover, we present a convergent optimization algorithm for solving regularized least squares with these penalty functions. Numerical simulations highlight the benefit of structured sparsity and the advantage offered by our approach over the Lasso method and other related methods.


page 9

page 13

page 28


The Trimmed Lasso: Sparsity and Robustness

Nonconvex penalty methods for sparse modeling in linear regression have ...

On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation

We present a novel algorithm that allows us to gain detailed insight int...

M-estimation with the Trimmed l1 Penalty

We study high-dimensional M-estimators with the trimmed ℓ_1 penalty. Whi...

Greedy Sparsity-Constrained Optimization

Sparsity-constrained optimization has wide applicability in machine lear...

Second order Poincaré inequalities and de-biasing arbitrary convex regularizers when p/n → γ

A new Central Limit Theorem (CLT) is developed for random variables of t...