Representer Theorems for Sparsity-Promoting <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math> </inline-formula> Regularization

Abstract

We present a theoretical analysis and comparison of the effect of l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> versus l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> regularization for the resolution of ill-posed linear inverse and/or compressed sensing problems. Our formulation covers the most general setting where the solution is specified as the minimizer of a convex cost functional. We derive a series of representer theorems that give the generic form of the solution depending on the type of regularization. We start with the analysis of the problem in finite dimensions and then extend our results to the infinite-dimensional spaces l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (Z) and l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> (Z). We also consider the use of linear transformations in the form of dictionaries or regularization operators. In particular, we show that the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> solution is forced to live in a predefined subspace that is intrinsically smooth and tied to the measurement operator. The l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> solution, on the other hand, is formed by adaptively selecting a subset of atoms in a dictionary that is specified by the regularization operator. Beside the proof that l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> solutions are intrinsically sparse, the main outcome of our investigation is that the use of l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> regularization is much more favorable for injecting prior knowledge: it results in a functional form that is independent of the system matrix, while this is not so in the l <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> scenario.