Splines are Universal Solutions of Linear Inverse Problems with Generalized-TV regularization

TL;DR

This paper proves that non-uniform splines serve as universal solutions for linear inverse problems with generalized TV regularization, regardless of measurement type, by characterizing solutions within a generalized Beppo-Levi space.

Contribution

It introduces the generalized Beppo-Levi space and demonstrates that solutions to gTV-regularized inverse problems are non-uniform splines with fewer knots than measurements, independent of measurement specifics.

Findings

Non-uniform splines are universal solutions for gTV-regularized inverse problems.

Solutions have fewer knots than the number of measurements.

Spline type depends only on the operator L, not on measurement details.

Abstract

Splines come in a variety of flavors that can be characterized in terms of some differential operator L. The simplest piecewise-constant model corresponds to the derivative operator. Likewise, one can extend the traditional notion of total variation by considering more general operators than the derivative. This leads us to the definition of the generalized Beppo-Levi space M, which is further identified as the direct sum of two Banach spaces. We then prove that the minimization of the generalized total variation (gTV) over M, subject to some arbitrary (convex) consistency constraints on the linear measurements of the signal, admits nonuniform L-spline solutions with fewer knots than the number of measurements. This shows that non-uniform splines are universal solutions of continuous-domain linear inverse problems with LASSO, L1, or TV-like regularization constraints. Remarkably, the spline-type is fully determined by the choice of L and does not depend on the actual nature of the measurements.