Sampling in BV-Type Spaces

TL;DR

This paper develops a rigorous framework for sampling functions of bounded variation (BV) within variational inverse problems, where pointwise BV sampling is ill-defined. It establishes that sampling functionals are continuous on the generalized BV space $\mathrm{GBV}$ but not weak$^*$-continuous, and constructs a canonical local inverse of the derivative $D$ using a unique $\,D$-admissible bi-orthogonal system, enabling a local representer theory. The authors prove existence results for regularized optimization problems with BV samples, show that extreme points of solutions are $\,D$-splines, and extend the theory to higher-order BV spaces with analogous sparsity and convergence properties. Collectively, the results provide a solid foundation for BV sampling in continuous-domain optimization, yielding sparse, structurally interpretable solution representations and practical regularization schemes for variational inverse problems and imaging applications.

Abstract

The sampling of functions of bounded variation (BV) is a long-standing problem in op- timization. The ability to sample such functions has relevance in the field of variational inverse problems, where the standard theory fails to guarantee the mere existence of solutions when the loss functional involves samples of BV functions. In this paper, we prove the continuity of sampling functionals and show that the differential operator D admits a unique local inverse. This canonical inversion enables us to formulate an existence theorem for a class of regularized optimization problems that incorporate samples of BV functions. Finally, we characterize the solution set in terms of its extreme points.

Figures (3)

• Figure 1: The kernel $g_{\phi_1}$ is constructed from the non-$[0,1]$-localized system $(1,2\mathbbm{1}_{[-0.5,0]})$. The kernel $g_{\phi_2}$ is constructed from the $[0,1]$-localized system $(1,2\mathbbm{1}_{[0,0.5]})$. The color of the graph reflects the second dimension of the kernels. The darker the graph, the smaller the value of the corresponding knot $x_k$, with the six knots $x_k$ being uniformly sampled in $[0,0.75]$.
• Figure 2: The kernel $g_{\phi}$ is constructed from the $[0,1]$-localized but non-fundamental system $(1,2\mathbbm{1}_{[0,0.5]})$. The first column presents graphs of the kernel $g_{\phi}$, while the second column presents graphs of $g_{\iota}$. In the first row, the color of the graph reflects the second dimension of the kernels, whereas in the second row it reflects the first dimension. The darker the graph, the smaller the value of the corresponding knot $x_k$, with the six knots $x_k$ being uniformly sampled in $[0,0.75]$.
• Figure 3: Schematic of the set $\overline{\mathcal{V}(\mathrm{K})}^{\mathrm{weak}^{\star}}$. The shaded area corresponds to the interior $\overset{\circ}{\mathcal{V}(\mathrm{K})}$ of the set, which is necessarily formed of minimizers. The darker (lighter, respectively) outline represents the part of the boundary that is formed by minimizers (non-minimizers, respectively).

Theorems & Definitions (34)

• Definition 1
• Theorem 1: unser2017splines
• Definition 2
• Proposition 1
• proof : Proof of Proposition \ref{['diractweak']}.
• Definition 3
• Proposition 2
• proof : Proof of Proposition \ref{['prop:localizedjustification']}
• Proposition 3: guillemet2025convergence
• Proposition 4