Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions

TL;DR

This paper compares elastic-net and $ ext{l}^1$-regularization for solving ill-posed linear inverse problems with quasi-sparse solutions, analyzing convergence rates based on solution decay and smoothness.

Contribution

It provides a theoretical analysis of elastic-net regularization, including convergence rate estimates, under assumptions on solution decay and smoothness, extending prior work on $ ext{l}^1$-regularization.

Findings

Elastic-net regularization converges under certain decay and smoothness conditions.

The convergence rate depends on the decay rate of solution coefficients and smoothness properties.

The analysis includes the influence of two parameters: the weight and regularization parameters.

Abstract

We consider the ill-posed operator equation $Ax=y$ with an injective and bounded linear operator $A$ mapping between $\ell^2$ and a Hilbert space $Y$, possessing the unique solution \linebreak $x^\dag=\{x^\dag_k\}_{k=1}^\infty$. For the cases that sparsity $x^\dag \in \ell^0$ is expected but often slightly violated in practice, we investigate in comparison with the $\ell^1$-regularization the elastic-net regularization, where the penalty is a weighted superposition of the $\ell^1$-norm and the $\ell^2$-norm square, under the assumption that $x^\dag \in \ell^1$. There occur two positive parameters in this approach, the weight parameter $\eta$ and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in $\ell^1$-regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator $A$ and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay $x^\dag_k \to 0$ for $k \to \infty$ and the classical smoothness properties of $x^\dag$ as an element in $\ell^2$.