Variational Tensor-Product Splines

TL;DR

This work develops a variational framework for multidimensional inverse problems in which the regularization combines the M-norm of a tensor-product differential operator $L_1\otimes L_2$ with a bounded-variation component that cures the infinite-dimensional nullspace. A new 2D theory is built on a carefully constructed predual $\mathcal{C}_{L_1\otimes L_2}(\mathbb{R}^2)$ and native space $\mathcal{M}_{L_1\otimes L_2}(\mathbb{R}^2)$, enabling a sparse, separable representation of solutions as tensor products of one-dimensional splines. A Representer Theorem for general and compactly supported measurement operators shows that the solution set is weak-* compact and its extreme points are $[L_1\otimes L_2]$-splines with at most $M-N_1N_2$ knots, with localization possible when measurements are supported in a rectangle $\mathbf{K}$. The theory is illustrated with piecewise-constant and piecewise-linear spline examples, and it generalizes naturally to $D$ dimensions, establishing a principled link between spline theory and multidimensional inverse problem regularization. The framework justifies tensor-product splines as optimal representatives in this class and provides a concrete, causality-aware mechanism for null-space regularization, with practical benefits for localization and computational efficiency in inverse problems. Overall, this work offers a rigorous bridge between classical spline representations and modern multidimensional regularization strategies, with potential impact on high-dimensional imaging and related inverse problems.

Abstract

Multidimensional continuous-domain inverse problems are often solved by the minimization of a loss functional, formed as the sum of a data fidelity and a regularization. In this work, we present a new construction where the regularization is itself built as the sum of two terms: i) the M norm of the regularizing operator L1 b L2, with L1 and L2 being two one-dimensional differential operators; ii) a bounded-variation norm that regularizes on the infinite-dimensional nullspace of L1 b L2. In this construction, we show that the extreme points of the solution set are the tensor product of one-dimensional splines, with a number of atoms upper-bounded in term of the number of data points. Further, when the data of the inverse problem is localized, we reveal that the term ii) must take the form of a sum of bounded-variation norms, precomposed with partial derivative of different orders.

Figures (4)

• Figure 1: Example of a $[\mathrm{D}\otimes\mathrm{D}]$-spline of the form \ref{['eq:1.7']} with $K_1=5,K_2=5,$ and $K=10$. The dotted square represents the boundary of the unit square $[0,1]^2$. The knots and the amplitudes have been chosen randomly. The plot is made on the extended square $[-0.25,1.25]^2$ to display the canonic behaviour of the spline outside of its supporting domain $[0,1]^2.$ On the $y$ axis ($x$ axis, respectively) one can observe the localisation of the knots $z_{m'}$ (the knots $y_m$, respectively) from the colour gradient. Finally the corners inside the dotted square are the localizations of the knots $(x_{1,k},x_{2,k})$.
• Figure 2: Direct-sum decomposition of the spline in Figure \ref{['fig:1']} into its four components, according to \ref{['eq:1.7']}. The illustration is only made on the unit square $[0,1]^2$.
• Figure 3: Schematization of the spatial influence of functions in $(i)^{\star}$, $(ii)^{\star}$, and $(iii)^{\star}$.
• Figure 4: Schematic of the squares $\mathbf{K}$ and $\tilde{\mathbf{K}}$. The grey zone represents the additional area where a knot $(x_{1,k},x_{2,k})$ may be placed due to a suboptimal choice of admissible system.

Theorems & Definitions (51)

• Proposition 1
• Proposition 2
• Theorem 1
• Theorem 2
• Definition 1
• Definition 2
• Theorem 3: unser2017splines
• Theorem 4: unser2017splines
• Proposition 3
• Definition 3