Well-Behaved CPWL Interpolations

• The paper shows that enforcing at least d+1 data points per affine piece guarantees stability and rules out underdetermined interpolations.
• It presents a framework using piecewise-affine decompositions and MILP tightening strategies that improve data fitting and reduce computational time.
• The approach extends to anisotropic meshing and regular-lattice approximations, ensuring stable, artifact-free interpolations in varied applications.

Searching arXiv for relevant papers on well-behaved CPWL interpolations and related anisotropic/approximation formulations. Well-behaved CPWL interpolations are continuous piecewise-linear representations in which the affine pieces, mesh geometry, or basis structure satisfy additional regularity conditions that exclude pathological behavior. In the explicit general-dimensional formulation developed for data fitting and mixed-integer optimization, a CPWL interpolation $f:D\to\mathbb{R}$ of a data set $S=(\mathbf{x}_i,z_i)_{i=1}^N$ is well-behaved when every affine piece is exactly determined or overdetermined by the data, meaning that each piece contains at least $d+1$ data points in $\mathbb{R}^d$ under the assumption that the $\mathbf{x}_i$ are in general position (Ploussard et al., 13 Aug 2025). In adjacent CPWL literatures, analogous notions of well-behavedness arise through orphan-free anisotropic Voronoi diagrams whose duals are embedded triangulations, and through Riesz-stable box-spline spaces on nondegenerate regular lattices with optimal asymptotic error constants (Canas, 2012, Pourya et al., 5 Feb 2025).

1. Formal definition in general-dimensional CPWL interpolation

Let $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$ be a data set in $\mathbb{R}^{d+1}$, with $\mathbf{x}_i\in\mathbb{R}^d$, $z_i\in\mathbb{R}$, and with the $\mathbf{x}_i$ in general position, so that any subset of $S=(\mathbf{x}_i,z_i)_{i=1}^N$0 points is affinely independent. If $S=(\mathbf{x}_i,z_i)_{i=1}^N$1, a CPWL function on $S=(\mathbf{x}_i,z_i)_{i=1}^N$2 is a continuous function $S=(\mathbf{x}_i,z_i)_{i=1}^N$3 for which there exists a finite family of affine functions $S=(\mathbf{x}_i,z_i)_{i=1}^N$4 on compact sets $S=(\mathbf{x}_i,z_i)_{i=1}^N$5 such that $S=(\mathbf{x}_i,z_i)_{i=1}^N$6 and $S=(\mathbf{x}_i,z_i)_{i=1}^N$7 for every $S=(\mathbf{x}_i,z_i)_{i=1}^N$8 (Ploussard et al., 13 Aug 2025). The representation is intentionally non-unique: the domains need not form a strict partition, may overlap, may be empty, and may even coincide if the affine pieces agree there.

For interpolation, the condition is $S=(\mathbf{x}_i,z_i)_{i=1}^N$9 for all $d+1$0. For approximation, $d+1$1 is a CPWL $d+1$2-approximation when

$d+1$3

equivalently, when $d+1$4 interpolates the perturbed data $d+1$5 with $d+1$6. Two $d+1$7-approximations are equivalent with respect to $d+1$8 if they take the same values on all data points.

The key degree-of-freedom threshold is $d+1$9. For a given affine piece $\mathbb{R}^d$0, the paper distinguishes three cases according to the number of projected data points in $\mathbb{R}^d$1: underdetermined if fewer than $\mathbb{R}^d$2 points are interpolated, exactly determined if exactly $\mathbb{R}^d$3 points are interpolated, and overdetermined if more than $\mathbb{R}^d$4 points are interpolated. A well-behaved CPWL interpolation is then defined by the condition

$\mathbb{R}^d$5

Thus, no affine piece is allowed to interpolate fewer than $\mathbb{R}^d$6 data points. For $\mathbb{R}^d$7-approximations, the same definition is applied to the perturbed data $\mathbb{R}^d$8 (Ploussard et al., 13 Aug 2025).

This notion is structural rather than purely analytic. It does not refer to continuity, which is already built into the definition of CPWL, but to whether each affine piece is sufficiently anchored by data. The intended exclusion is a family of underdetermined pieces with arbitrary slopes or intercepts that do not affect the fitted values on $\mathbb{R}^d$9 but substantially weaken optimization formulations.

2. Existence of well-behaved versions

A central result is that well-behavedness is not a restrictive modeling assumption in the interpolation problem: every CPWL interpolation admits a well-behaved version, and the same is true for every CPWL $\mathbf{x}_i$0-approximation (Ploussard et al., 13 Aug 2025). The associated notion of a well-behaved version is precise. If $\mathbf{x}_i$1 and $\mathbf{x}_i$2 are CPWL interpolations of $\mathbf{x}_i$3, then $\mathbf{x}_i$4 is a well-behaved version of $\mathbf{x}_i$5 if $\mathbf{x}_i$6 is well-behaved and, piece by piece,

$\mathbf{x}_i$7

Each adjusted piece therefore preserves all data points originally interpolated by the corresponding piece of $\mathbf{x}_i$8, while enlarging its support on the data if necessary.

The proof strategy proceeds by eliminating underdetermined pieces one at a time. If $\mathbf{x}_i$9 already satisfies the definition, nothing must be done. Otherwise, one selects an underdetermined piece $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$0, collects its neighboring pieces $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$1 that share boundaries with $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$2, and forms the union $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$3. Using the DC structure of the global CPWL function, the neighboring configuration implies consistent inequalities of the form

$S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$4

on the neighboring domains. These constraints define a polyhedron $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$5 in the coefficient space $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$6 of the affine piece. An extremal-point argument then yields a modified affine piece $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$7 with at least $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$8 active constraints, so that $S=(\mathbf{x}_i,z_i)_{i=1,\dots,N}$9 interpolates more data points than $\mathbb{R}^{d+1}$0 while remaining compatible with its neighbors. Repeating this finite process removes all underdetermined pieces.

Two consequences are immediate. First, well-behavedness can be imposed without changing the fitted values at the sample sites: if $\mathbb{R}^{d+1}$1 is a well-behaved version of $\mathbb{R}^{d+1}$2, then $\mathbb{R}^{d+1}$3 and $\mathbb{R}^{d+1}$4 are equivalent with respect to $\mathbb{R}^{d+1}$5. Second, the restriction to well-behaved interpolations does not alter achievable interpolation or approximation quality on the data. This is the theoretical basis for safe formulation tightening in optimization models.

3. Difference-of-convex representations and MILP tightening

Every CPWL function $\mathbb{R}^{d+1}$6 admits a difference-of-convex representation

$\mathbb{R}^{d+1}$7

where $\mathbb{R}^{d+1}$8 and $\mathbb{R}^{d+1}$9 are convex CPWL functions of the form

$\mathbf{x}_i\in\mathbb{R}^d$0

with affine pieces

$\mathbf{x}_i\in\mathbb{R}^d$1

The affine domains of $\mathbf{x}_i\in\mathbb{R}^d$2 are intersections $\mathbf{x}_i\in\mathbb{R}^d$3, and on such a domain one has $\mathbf{x}_i\in\mathbb{R}^d$4 (Ploussard et al., 13 Aug 2025).

This representation underlies the baseline MILP formulation $\mathbf{x}_i\in\mathbb{R}^d$5. Its decision variables include the values $\mathbf{x}_i\in\mathbb{R}^d$6, $\mathbf{x}_i\in\mathbb{R}^d$7, $\mathbf{x}_i\in\mathbb{R}^d$8, the affine coefficients $\mathbf{x}_i\in\mathbb{R}^d$9, $z_i\in\mathbb{R}$0, absolute errors $z_i\in\mathbb{R}$1, and binary activity variables $z_i\in\mathbb{R}$2. The core constraints impose the DC identity,

$z_i\in\mathbb{R}$3

the max-of-affine inequalities,

$z_i\in\mathbb{R}$4

the big-$z_i\in\mathbb{R}$5 activation conditions,

$z_i\in\mathbb{R}$6

the coverage constraints $z_i\in\mathbb{R}$7, and the fitting bounds

$z_i\in\mathbb{R}$8

The paper considers generic linear objectives $z_i\in\mathbb{R}$9, including mean error and max error (Ploussard et al., 13 Aug 2025).

Well-behavedness is critical because it justifies six tightening strategies that remove redundant DC representations while preserving optimal solutions on the data:

• Fix one affine piece of $\mathbf{x}_i$0:

$\mathbf{x}_i$1

• Sort affine pieces:

$\mathbf{x}_i$2

• Impose at least $\mathbf{x}_i$3 points per affine domain of $\mathbf{x}_i$4 and $\mathbf{x}_i$5:

$\mathbf{x}_i$6

• Impose at least $\mathbf{x}_i$7 points per affine domain of $\mathbf{x}_i$8 by auxiliary variables $\mathbf{x}_i$9 and $S=(\mathbf{x}_i,z_i)_{i=1}^N$00 that encode the intersections $S=(\mathbf{x}_i,z_i)_{i=1}^N$01.
• Use tight big-$S=(\mathbf{x}_i,z_i)_{i=1}^N$02 values:

$S=(\mathbf{x}_i,z_i)_{i=1}^N$03

• Impose tight bounds on all continuous variables, including $S=(\mathbf{x}_i,z_i)_{i=1}^N$04, $S=(\mathbf{x}_i,z_i)_{i=1}^N$05, and all affine coefficients.

The computational study evaluates 11 combinations of these strategies on six data sets in 2D and 3D with Gurobi and the max-error objective. Tight big-$S=(\mathbf{x}_i,z_i)_{i=1}^N$06 values yield large improvements over default big-$S=(\mathbf{x}_i,z_i)_{i=1}^N$07 or indicator constraints; sorting affine pieces tends to hurt performance; and combinations that impose at least $S=(\mathbf{x}_i,z_i)_{i=1}^N$08 points per affine domain of $S=(\mathbf{x}_i,z_i)_{i=1}^N$09 and $S=(\mathbf{x}_i,z_i)_{i=1}^N$10 together with tight bounds are consistently strong. For five of six case studies, the best tightened combinations reduce solve time by factors between 3 and 23 relative to an untightened baseline, and in some cases the untightened MILP reaches the 7200s time limit whereas tightened formulations solve within minutes or less. The preprocessing required for tight big-$S=(\mathbf{x}_i,z_i)_{i=1}^N$11 values and bounds scales as $S=(\mathbf{x}_i,z_i)_{i=1}^N$12, which is significant in 3D but still reported as worthwhile overall (Ploussard et al., 13 Aug 2025).

4. Anisotropic meshes, orphan-freedom, and embedded triangulations

In anisotropic approximation, well-behaved CPWL interpolation is formulated through mesh topology rather than data anchoring. The setting is a Riemannian metric $S=(\mathbf{x}_i,z_i)_{i=1}^N$13 on $S=(\mathbf{x}_i,z_i)_{i=1}^N$14, with symmetric positive-definite square root $S=(\mathbf{x}_i,z_i)_{i=1}^N$15 defined by $S=(\mathbf{x}_i,z_i)_{i=1}^N$16. Two practical anisotropic distances are considered: $S=(\mathbf{x}_i,z_i)_{i=1}^N$17 These induce Du–Wang and Labelle–Shewchuk anisotropic Voronoi diagrams, respectively (Canas, 2012).

Here, well-behavedness centers on orphan-freedom: every Voronoi cell contains its generating site. This condition is topological but has direct numerical consequences. If the anisotropic Voronoi diagram is orphan-free, then its dual anisotropic Delaunay complex is an embedded triangulation; in 2D this means no crossings and no inverted elements. Such a triangulation is exactly the support required for standard CPWL interpolation. Once the dual mesh exists, scalar data $S=(\mathbf{x}_i,z_i)_{i=1}^N$18 at sites $S=(\mathbf{x}_i,z_i)_{i=1}^N$19 can be interpolated on each simplex $S=(\mathbf{x}_i,z_i)_{i=1}^N$20 by

$S=(\mathbf{x}_i,z_i)_{i=1}^N$21

with barycentric coordinates $S=(\mathbf{x}_i,z_i)_{i=1}^N$22 (Canas, 2012).

The paper replaces older, global orphan-freedom criteria based on an asymmetric $S=(\mathbf{x}_i,z_i)_{i=1}^N$23-net and a worst-case metric variation constant $S=(\mathbf{x}_i,z_i)_{i=1}^N$24 by more practical conditions. The site set is assumed to satisfy a $S=(\mathbf{x}_i,z_i)_{i=1}^N$25-Delone property with respect to the relevant anisotropic distance: every point is within cover radius $S=(\mathbf{x}_i,z_i)_{i=1}^N$26 of some site, and distinct sites satisfy an asymmetric packing condition at scale $S=(\mathbf{x}_i,z_i)_{i=1}^N$27. For $S=(\mathbf{x}_i,z_i)_{i=1}^N$28, the local metric variation is measured by

$S=(\mathbf{x}_i,z_i)_{i=1}^N$29

The resulting orphan-freedom conditions are explicit inequalities in $S=(\mathbf{x}_i,z_i)_{i=1}^N$30, $S=(\mathbf{x}_i,z_i)_{i=1}^N$31, and $S=(\mathbf{x}_i,z_i)_{i=1}^N$32. For Du–Wang diagrams, orphan-freedom is guaranteed if

$S=(\mathbf{x}_i,z_i)_{i=1}^N$33

For Labelle–Shewchuk diagrams, the corresponding criterion is

$S=(\mathbf{x}_i,z_i)_{i=1}^N$34

In the special case of an asymmetric $S=(\mathbf{x}_i,z_i)_{i=1}^N$35-net, the paper gives the thresholds $S=(\mathbf{x}_i,z_i)_{i=1}^N$36 for Du–Wang and $S=(\mathbf{x}_i,z_i)_{i=1}^N$37 for Labelle–Shewchuk diagrams. For piecewise-linear metrics, it also derives per-simplex formulas that allow $S=(\mathbf{x}_i,z_i)_{i=1}^N$38 to be estimated in linear time over the simplicial complex.

The significance for CPWL interpolation is specific: the paper does not derive explicit interpolation error estimates, but it guarantees the mesh validity needed for continuity, stability, and anisotropic shape control. In this literature, a well-behaved CPWL interpolation is one built on an embedded anisotropic Delaunay triangulation generated by an orphan-free anisotropic Voronoi diagram.

5. Regular lattices, box splines, and asymptotic optimality

A different notion of well-behaved CPWL interpolation arises in regular-lattice approximation. The CPWL search spaces are shift-invariant spaces generated by translates of a 2D box spline $S=(\mathbf{x}_i,z_i)_{i=1}^N$39,

$S=(\mathbf{x}_i,z_i)_{i=1}^N$40

where the grid matrix is $S=(\mathbf{x}_i,z_i)_{i=1}^N$41 and $S=(\mathbf{x}_i,z_i)_{i=1}^N$42. Every $S=(\mathbf{x}_i,z_i)_{i=1}^N$43 is CPWL on the triangulation induced by the lattice $S=(\mathbf{x}_i,z_i)_{i=1}^N$44 (Pourya et al., 5 Feb 2025).

The basis is well-behaved in two precise senses. First, the translates form a Riesz basis of $S=(\mathbf{x}_i,z_i)_{i=1}^N$45, which gives stability of the coefficient representation. Second, the space reproduces affine functions exactly, which yields second-order approximation for sufficiently smooth targets. The orthogonal projection $S=(\mathbf{x}_i,z_i)_{i=1}^N$46 defines the best $S=(\mathbf{x}_i,z_i)_{i=1}^N$47-approximation, and the error

$S=(\mathbf{x}_i,z_i)_{i=1}^N$48

admits an asymptotic expansion in the small-stepsize regime.

For lattices of fixed point density, the grid matrix is parameterized by a step size $S=(\mathbf{x}_i,z_i)_{i=1}^N$49 and angles $S=(\mathbf{x}_i,z_i)_{i=1}^N$50,

$S=(\mathbf{x}_i,z_i)_{i=1}^N$51

with $S=(\mathbf{x}_i,z_i)_{i=1}^N$52. For band-limited $S=(\mathbf{x}_i,z_i)_{i=1}^N$53 with $S=(\mathbf{x}_i,z_i)_{i=1}^N$54, the dominant approximation error on a general grid is

$S=(\mathbf{x}_i,z_i)_{i=1}^N$55

where $S=(\mathbf{x}_i,z_i)_{i=1}^N$56 depend explicitly on $S=(\mathbf{x}_i,z_i)_{i=1}^N$57. The paper further defines the asymptotic error constant

$S=(\mathbf{x}_i,z_i)_{i=1}^N$58

which isolates the dependence on lattice geometry (Pourya et al., 5 Feb 2025).

The main comparison result is that hexagonal lattices are optimal among all 2D regular lattices of equal density: $S=(\mathbf{x}_i,z_i)_{i=1}^N$59 is minimized when $S=(\mathbf{x}_i,z_i)_{i=1}^N$60 or $S=(\mathbf{x}_i,z_i)_{i=1}^N$61, with minimal value $S=(\mathbf{x}_i,z_i)_{i=1}^N$62. For the Cartesian grid $S=(\mathbf{x}_i,z_i)_{i=1}^N$63, one has $S=(\mathbf{x}_i,z_i)_{i=1}^N$64 and $S=(\mathbf{x}_i,z_i)_{i=1}^N$65. For the hexagonal grid $S=(\mathbf{x}_i,z_i)_{i=1}^N$66, one has

$S=(\mathbf{x}_i,z_i)_{i=1}^N$67

and the dominant error simplifies to

$S=(\mathbf{x}_i,z_i)_{i=1}^N$68

The disappearance of the mixed derivative term in the hexagonal case produces a rotationally isotropic leading error term. In this framework, well-behaved CPWL interpolation means stable basis structure, nondegenerate triangulation, explicit $S=(\mathbf{x}_i,z_i)_{i=1}^N$69 asymptotics, and lattice geometry that minimizes the leading constant (Pourya et al., 5 Feb 2025).

6. Geometric regularity, related notions, and terminological scope

The phrase “well-behaved” is also used in a broader geometric sense for interpolatory constructions that are not themselves finite-dimensional CPWL fitting formulations. A construction of interpolating space curves in $S=(\mathbf{x}_i,z_i)_{i=1}^N$70 uses four ingredients—local functions, blending functions, redistributing functions, and gluing functions—and proves $S=(\mathbf{x}_i,z_i)_{i=1}^N$71 geometric continuity for arbitrary prescribed order. The redistributing maps $S=(\mathbf{x}_i,z_i)_{i=1}^N$72 are piecewise linear, the local quasi-regular curves are required to be positive definite and contracted, and the final curve is described as local, free of cusps, and without self-intersection; the method is also adapted to convexity preservation, sharp corners, and sphere preservation (Hu et al., 2024). The output is not a CPWL function in the sense of affine pieces on polyhedral domains, but it supplies a geometrically controlled template for what “artifact-free” interpolation can mean.

This broader geometric viewpoint helps clarify a common misconception. In optimization-oriented CPWL fitting, well-behavedness is a combinatorial and representational restriction: every affine piece must be supported by at least $S=(\mathbf{x}_i,z_i)_{i=1}^N$73 data points (Ploussard et al., 13 Aug 2025). In anisotropic meshing, it is a topological property of Voronoi cells and their dual triangulations (Canas, 2012). In regular-lattice approximation, it is a basis-and-error property of box-spline spaces (Pourya et al., 5 Feb 2025). These are not interchangeable definitions, even though all three aim to rule out unstable or degenerate CPWL constructions.

A further terminological divergence appears in the Banach-space interpolation literature. The study of Schechter interpolators and higher-order Rochberg spaces explicitly notes that it does not use the term “CPWL”; its notion of “well-behaved” interpolation refers instead to functoriality, symmetry, hereditary $S=(\mathbf{x}_i,z_i)_{i=1}^N$74 structure, strict singularity properties, and, in the weighted Hilbertian case $S=(\mathbf{x}_i,z_i)_{i=1}^N$75, the fact that all exact sequences split and all spaces obtained are isomorphic to Hilbert spaces (Castillo et al., 2021). This is conceptually adjacent only at a very high level. The finite-dimensional CPWL topic concerns piecewise-affine approximation, triangulations, and mixed-integer formulations rather than Calderón or Kalton–Montgomery interpolation.

Taken together, these literatures suggest a stable core meaning. A CPWL interpolation is well-behaved when its local affine structure is sufficiently constrained to prevent arbitrary slopes, its supporting mesh or lattice is free of topological degeneration, and its approximation space has stability and reproducibility properties strong enough to yield reliable computation. The precise formalization depends on whether the underlying problem is data fitting, anisotropic meshing, regular-lattice approximation, or geometric curve design.