Necessary and sufficient conditions of solution uniqueness in $\ell_1$ minimization

TL;DR

This paper establishes necessary and sufficient conditions for the uniqueness of solutions in various $oldsymbol{ extit{ extsc{l}}_1}$ minimization models, unifying and extending previous results across multiple convex optimization frameworks.

Contribution

It provides a unified set of necessary and sufficient conditions for solution uniqueness in a broad class of $oldsymbol{ extit{ extsc{l}}_1}$ models, including basis pursuit and Lasso.

Findings

Conditions are necessary and sufficient for solution uniqueness.

Full column rank of $A_I$ is essential for uniqueness.

Numerical methods to verify uniqueness are discussed.

Abstract

This paper shows that the solutions to various convex $\ell_1$ minimization problems are \emph{unique} if and only if a common set of conditions are satisfied. This result applies broadly to the basis pursuit model, basis pursuit denoising model, Lasso model, as well as other $\ell_1$ models that either minimize $f(Ax-b)$ or impose the constraint $f(Ax-b)\leq\sigma$, where $f$ is a strictly convex function. For these models, this paper proves that, given a solution $x^*$ and defining $I=\supp(x^*)$ and $s=\sign(x^*_I)$, $x^*$ is the unique solution if and only if $A_I$ has full column rank and there exists $y$ such that $A_I^Ty=s$ and $|a_i^Ty|_\infty<1$ for $i\not\in I$. This condition is previously known to be sufficient for the basis pursuit model to have a unique solution supported on $I$. Indeed, it is also necessary, and applies to a variety of other $\ell_1$ models. The paper also discusses ways to recognize unique solutions and verify the uniqueness conditions numerically.