ArbInterp: Structure-Preserving Interpolation

• ArbInterp is a family of structure-preserving interpolation schemes that enforce nontrivial invariance constraints and external structural properties rather than merely fitting data points.
• It integrates advanced methodologies across domains like perspective-correct graphics, rigorous arbitrary-precision computations, adaptive splines, generative modeling, and arbitrage-free finance.
• This research program overcomes affine limitations by incorporating geometric transformations, certified error bounds, probability metrics, and market-consistent constraints into the interpolation process.

Searching arXiv for "^^^^^^^^5ArbInterp5 and closely related papers to ground the article in current literature. {"^^^^^^^^5query5 "^^^^^^^^5\5 OR ^^^^^^^^5\5 Generative Video Interpolation^^^^^^^^5\5 OR ^^^^^^^^5\5 Coordinates as Interpolants^^^^^^^^5\5 "max_results": ^^^^5query5ArbInterp5^^^^} ^^^^^^^^5ArbInterp5^^^^^^^^ is used in the cited literature as a designation or motivating shorthand for interpolation schemes whose central requirement is not merely pointwise fitting but preservation of an external structure. In computer graphics, it appears in perspective-correct barycentric interpolation over projective domains; in numerical analysis, it is linked to arbitrary-precision, rigorously bounded interpolation and quadrature in the Arb library; in spline and adaptive-grid settings, it denotes local, continuous, and second-order or arbitrary-order schemes on bounded or nonuniform domains; in generative modeling, it denotes interpolation at arbitrary timestamps or along likely high-density paths; and in quantitative finance, it denotes arbitrage-free interpolation of volatility surfaces, rates, option prices, and mixture densities (&&&^^^^5ArbInterp5^^^^&&&, &&&^^^^5query5^^^^&&&, &&&^^^^5\5^^^^&&&, &&&^^^^5 OR \5^^^^&&&, &&&^^^^5 OR \5^^^^&&&, Berg, 10 Jun 2026). This suggests that ^^^^^^^^5ArbInterp5^^^^^^^^ is best understood as a family of structure-preserving interpolation problems rather than as a single universally fixed algorithm.

^^^^5query5^^^^. Structural meaning of ^^^^^^^^5ArbInterp5^^^^^^^^

Within the cited work, ^^^^^^^^5ArbInterp5^^^^^^^^ is consistently associated with interpolation under nontrivial invariances or constraints. The computer-graphics formulation requires invariance under projective transformations, so that interpolation in screen space is corrected back to eye space by rational rather than affine weighting (&&&^^^^5ArbInterp5^^^^&&&). The arbitrary-precision numerical formulation requires rigorous enclosures, interval arithmetic, and certified error propagation, so that interpolation at Gauss–Legendre nodes or related quadrature constructions remains provably correct at high precision (&&&^^^^5query5^^^^&&&, Johansson, 2016). The spline formulation requires locality, polynomial reproduction, and effective boundary handling on bounded intervals (&&&^^^^5\5^^^^&&&). The generative-model formulation requires compatibility with a metric and a probability distribution, explicitly encouraging paths through high-density regions or allowing control at arbitrary normalized timestamps (&&&^^^^5query5ArbInterp5^^^^&&&, &&&^^^^5 OR \5^^^^&&&). The financial formulation requires the absence of butterfly, calendar-spread, or related static arbitrage across strikes and maturities (&&&^^^^5 OR \5^^^^&&&, Berg, 10 Jun 2026).

A common misconception is that interpolation is intrinsically an affine or linearly parameterized operation. The cited literature rejects that simplification in several domains. Perspective-correct shading uses rational weights rather than screen-space affine weights (&&&^^^^5ArbInterp5^^^^&&&); likely interpolants in generative models add a density term to the geodesic energy (&&&^^^^5query5ArbInterp5^^^^&&&); arbitrage-free surface construction in finance interpolates parameterizations or call prices in ways that preserve convex-order or static-arbitrage constraints (&&&^^^^5 OR \5^^^^&&&, Berg, 10 Jun 2026). A plausible implication is that, in the ^^^^^^^^5ArbInterp5^^^^^^^^ sense, interpolation is defined as much by admissibility conditions as by data fidelity.

^^^^5\5^^^^. Projective and barycentric formulations

One of the clearest geometric instances of ^^^^^^^^5ArbInterp5^^^^^^^^ is the perspective-correct barycentric formulation in rasterization. For a triangle with screen-space barycentric coordinates

PRESERVED_PLACEHOLDER_^^^^5ArbInterp5^^^^

direct interpolation

PRESERVED_PLACEHOLDER_^^^^5query5^^^^

is not correct for attributes defined in world or eye space, because the perspective transformation is non-linear (&&&^^^^5ArbInterp5^^^^&&&). The correction is the rational or hyperbolic formula

PRESERVED_PLACEHOLDER_^^^^5\5^^^^

or, equivalently, transformed barycentric weights such as

PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^

with analogous expressions for PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^ and $b_2^\ast$, followed by

$a = b_0^\ast a_0 + b_1^\ast a_1 + b_2^\ast a_2.$

The paper explicitly relates this to a more general principle of arbitrary interpolation over a projective rational domain (&&&^^^^5ArbInterp5^^^^&&&).

A broader theoretical generalization is provided by positive geometries for barycentric interpolation. In that framework, a positive geometry is a pair $(X, X_{\geq 0})$ endowed with a canonical differential form whose multivariate residues on boundary components reproduce canonical forms of lower-dimensional positive geometries (&&&^^^^5\5ArbInterp5^^^^&&&). The canonical form is written as

$$\Omega(X, X_{\geq 0}) = C(\mathbf{x}) \omega(\mathbf{x}),$$

where $C(\mathbf{x})$ is the canonical rational function. Generalized barycentric coordinates are then constructed by ratios of canonical forms, unifying Wachspress coordinates, adjoints, and dual-polytpe constructions (&&&^^^^5\5ArbInterp5^^^^&&&). This suggests that the projective-correction logic of graphics and the rational-coordinate logic of generalized barycentric methods belong to the same structural class: interpolation is carried by a geometry-specific rational weight system rather than by a universal affine rule.

^^^^5 OR \5^^^^. Locality, continuity, and bounded-domain interpolation

A second major ^^^^^^^^5ArbInterp5^^^^^^^^ strand concerns local interpolation schemes that must remain continuous across nonuniform discretizations or bounded intervals. For cell-centered adaptive-mesh-refinement grids, the continuous second-order method classifies every point by edge type: ^^^^5ArbInterp5^^^^-edge, ^^^^5query5^^^^-edge, ^^^^5\5^^^^-edge, and, in ^^^^5 OR \5^^^^D, ^^^^5 OR \5^^^^-edge (&&&^^^^5\5\5^^^^&&&). In a ^^^^5ArbInterp5^^^^-edge region, one directly uses bilinear or trilinear interpolation. In ^^^^5query5^^^^-edge regions, the method applies linear interpolation along the main direction of resolution change, with bilinear or trilinear interpolation on trivial orthogonal directions. In ^^^^5\5^^^^-edge regions, it constructs a main interpolation plane, triangulates that plane, and uses linear or barycentric interpolation; in ^^^^5 OR \5^^^^-edge regions, the main interpolation region is decomposed into tetrahedra, pyramids, or prisms (&&&^^^^5\5\5^^^^&&&). The critical continuity property is that interpolation reduces to the lower-dimensional case on boundaries between edge regions, ensuring PRESERVED_PLACEHOLDER_^^^^5query5ArbInterp5^^^^ continuity (&&&^^^^5\5\5^^^^&&&).

For bounded one-dimensional domains, a related but distinct local formulation is the blending operator

PRESERVED_PLACEHOLDER_^^^^5query5query5^^^^

where PRESERVED_PLACEHOLDER_^^^^5query5\5^^^^ is a local quasi-interpolation operator and PRESERVED_PLACEHOLDER_^^^^5query5 OR \5^^^^ is a local Hermite interpolation operator (&&&^^^^5\5^^^^&&&). The construction is local, interpolatory, polynomial-preserving, and designed for arbitrary spline order PRESERVED_PLACEHOLDER_^^^^5query5 OR \5^^^^. It satisfies

PRESERVED_PLACEHOLDER_^^^^5query55^^^^

as well as interpolation at sample points and Hermite matching of endpoint derivatives up to order PRESERVED_PLACEHOLDER_^^^^5query5^^^^6 (&&&^^^^5\5^^^^&&&). The paper states an optimal supremum-norm error rate

PRESERVED_PLACEHOLDER_^^^^5query5^^^^7

for PRESERVED_PLACEHOLDER_^^^^5query5^^^^8 (&&&^^^^5\5^^^^&&&).

Taken together, these works define ^^^^^^^^5ArbInterp5^^^^^^^^ in operational terms as an interpolation problem in which locality and continuity are not optional post-processing features. The data structure—AMR cells, nonuniform knots, or bounded intervals—enters the definition of admissible interpolation operators.

^^^^5 OR \5^^^^. Rigorous arbitrary-precision interpolation in the Arb ecosystem

In numerical analysis, ^^^^^^^^5ArbInterp5^^^^^^^^ is closely connected to the Arb library’s ball-arithmetic model of computation. Arb represents a real interval as a midpoint-radius ball

PRESERVED_PLACEHOLDER_^^^^5query5^^^^9

with the midpoint stored at arbitrary precision and the radius stored as an efficient magnitude bound (Johansson, 2016). The same paper gives the interval formulas

PRESERVED_PLACEHOLDER_^^^^5\5ArbInterp5^^^^

and

PRESERVED_PLACEHOLDER_^^^^5\5query5^^^^

together with automatic error propagation for more general functions through derivative-based bounds (Johansson, 2016). This infrastructure is the basis for certified interpolation and quadrature in the broader Arb ecosystem.

For Gauss–Legendre nodes and weights, Arb implements a hybrid algorithm combining fixed-point Bonnet recurrence, hypergeometric series expansions, rectangular splitting, and interval Newton refinement (&&&^^^^5query5^^^^&&&). The nodes are roots of PRESERVED_PLACEHOLDER_^^^^5\5\5^^^^, refined by interval Newton iteration, and the weights are given by

PRESERVED_PLACEHOLDER_^^^^5\5 OR \5^^^^

The paper states that the implementation in Arb achieves order-of-magnitude speedups over previous code for simultaneous high degree and high precision, and explicitly notes that interpolation at Gauss–Legendre nodes can be carried out with rigorously known nodes and arbitrarily small rigorous error (&&&^^^^5query5^^^^&&&).

Rigorous numerical integration supplies a complementary ingredient. Arb’s integration code uses a Petras-style combination of adaptive bisection and adaptive Gaussian quadrature, with error bounds determined by complex magnitudes rather than derivatives (&&&^^^^5 OR \5\5^^^^&&&). The quadrature error bound is stated as

PRESERVED_PLACEHOLDER_^^^^5\5 OR \5^^^^

with PRESERVED_PLACEHOLDER_^^^^5\55^^^^ the magnitude bound on a complex ellipse (&&&^^^^5 OR \5\5^^^^&&&). A plausible implication is that ^^^^^^^^5ArbInterp5 in this numerical sense, is not merely interpolation at arbitrary precision; it is interpolation embedded in a certified computational pipeline of ball arithmetic, interval Newton methods, and rigorous analytic continuation.

5. Probabilistic and generative-model interpolation

In generative modeling, ^^^^^^^^5ArbInterp5^^^^^^^^ appears in two distinct but related ways. The first is a general interpolation scheme for likely transition paths in generative models. Given a Riemannian manifold PRESERVED_PLACEHOLDER_^^^^5\5^^^^6 with metric PRESERVED_PLACEHOLDER_^^^^5\5^^^^7, the curve is obtained by minimizing

PRESERVED_PLACEHOLDER_^^^^5\5^^^^8

where PRESERVED_PLACEHOLDER_^^^^5\5^^^^9 or a surrogate such as the ELBO, and PRESERVED_PLACEHOLDER_^^^^5 OR \5ArbInterp5^^^^ balances smoothness and likelihood (&&&^^^^5query5ArbInterp5^^^^&&&). The method, ProbGEORCE, is derived as an optimal-control extension of GEORCE and is shown locally to correspond to a geodesic under a modified metric

PRESERVED_PLACEHOLDER_^^^^5 OR \5query5^^^^

along the solution curve (&&&^^^^5query5ArbInterp5^^^^&&&). The paper quantitatively reports higher curve likelihood and lower FID than linear or spherical interpolation on most datasets and models considered (&&&^^^^5query5ArbInterp5^^^^&&&).

The second formulation is the video frame interpolation framework explicitly named ^^^^^^^^5ArbInterp5 It enables interpolation at any timestamp and of any length by combining Timestamp-aware Rotary Position Embedding (TaRoPE) with segment-wise generation and appearance-motion decoupled conditioning (&&&^^^^5 OR \5^^^^&&&). TaRoPE replaces integer temporal indices with normalized continuous timestamps

PRESERVED_PLACEHOLDER_^^^^5 OR \5\5^^^^

so that temporal RoPE phases are aligned with target timestamps rather than fixed positions (&&&^^^^5 OR \5^^^^&&&). For long sequences, ^^^^^^^^5ArbInterp5^^^^^^^^ decomposes generation into segments and conditions each new segment using the last frame of the previous segment for appearance continuity and motion semantic tokens for motion coherence. Those motion tokens are extracted by a Motion Semantic Extractor consisting of a temporally enhanced CLIP model followed by a Q-Former (&&&^^^^5 OR \5^^^^&&&). The framework supports direct, segment-by-segment, and hierarchical interpolation strategies, and the paper reports a benchmark spanning PRESERVED_PLACEHOLDER_^^^^5 OR \5 OR \5^^^^, PRESERVED_PLACEHOLDER_^^^^5 OR \5 OR \5^^^^, PRESERVED_PLACEHOLDER_^^^^5 OR \55^^^^, PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^6, and even PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^7 interpolation, with ^^^^^^^^5ArbInterp5^^^^^^^^ outperforming prior generative VFI methods across the reported metrics (&&&^^^^5 OR \5^^^^&&&).

These two papers make clear that, in the generative setting, ^^^^^^^^5ArbInterp5^^^^^^^^ is not limited to choosing points between endpoints. It becomes a problem of selecting a path family that is compatible with temporal semantics, learned probability geometry, and model-specific conditioning structure.

6. Arbitrage-free interpolation in finance

In quantitative finance, ^^^^^^^^5ArbInterp5^^^^^^^^ denotes interpolation subject to static-arbitrage or convex-order constraints. For eSSVI slices, an implied-volatility slice is parameterized by PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^8, with total variance

PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^9

and calibration is stabilized by anchoring each slice to a reliable market data point and reparameterizing by PRESERVED_PLACEHOLDER_^^^^5 OR \5ArbInterp5^^^^ (&&&^^^^5 OR \5^^^^&&&). The paper gives explicit butterfly-arbitrage bounds and calendar-spread conditions between adjacent slices, including monotonicity of PRESERVED_PLACEHOLDER_^^^^5 OR \5query5^^^^ and PRESERVED_PLACEHOLDER_^^^^5 OR \5\5^^^^ and the skew-difference constraint

PRESERVED_PLACEHOLDER_^^^^5 OR \5 OR \5^^^^

Interpolation between maturities is then performed by

PRESERVED_PLACEHOLDER_^^^^5 OR \5 OR \5^^^^

and the paper states that this interpolation preserves all arbitrage constraints (&&&^^^^5 OR \5^^^^&&&).

For market-observable interest-rate models, arbitrage-free interpolation is obtained by interpolating short bonds and then deriving longer maturities by no-arbitrage (&&&^^^^5 OR \5 OR \5^^^^&&&). The construction preserves the Markov property of the discrete-tenor model and guarantees positivity of interpolated rates. One of the explicit short-bond formulas is

PRESERVED_PLACEHOLDER_^^^^5 OR \55^^^^

with corresponding daycount-fraction interpolation for the short rate (&&&^^^^5 OR \5 OR \5^^^^&&&).

For option-price interpolation via stochastic collocation, the ^^^^5\5ArbInterp5\55^^^^ spline-based study considers

PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^6

with PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^7 represented either as a monotonic quadratic spline with optimized abscissae or as a B-spline expansion

PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^8

with fixed abscissae and optimized coefficients (&&&^^^^5 OR \55^^^^&&&). The study concludes that fixing abscissae and optimizing B-spline coefficients, together with regularization and moment matching, is generally preferable and more robust for arbitrage-free interpolation (&&&^^^^5 OR \55^^^^&&&).

A more recent construction addresses interpolation of mixture densities across expirations. Given two PRESERVED_PLACEHOLDER_^^^^5 OR \5^^^^9-component mixtures at adjacent pillars, it forms a frozen pool of $b_2^\ast$^^^^5ArbInterp5^^^^ components and linearly interpolates only the weights,

$b_2^\ast$^^^^5query5^^^^

so that the intermediate density remains within the same kernel family, although generically with up to $b_2^\ast$^^^^5\5^^^^ active components (Berg, 10 Jun 2026). Because call prices are linear in the weights,

$b_2^\ast$^^^^5 OR \5^^^^

the peacock property and non-negative Dupire local volatility follow under the stated conditions (Berg, 10 Jun 2026). The paper identifies the main practical difficulty as numerical conditioning in strongly bimodal regimes, where the local volatility remains finite but becomes badly conditioned (Berg, 10 Jun 2026).

Across these formulations, ^^^^^^^^5ArbInterp5^^^^^^^^ in finance is defined by admissible surface geometry in strike-maturity space: convexity, monotonicity, positivity, martingale consistency, or convex-order flow.

7. Symbolic-numeric interpolation and cross-domain interpretation

A final ^^^^^^^^5ArbInterp5^^^^^^^^ strand is symbolic-numeric algebraic-curve interpolation for exact points or neighborhoods. For planar points $b_2^\ast$^^^^5 OR \5^^^^, one forms the discriminant matrix

$b_2^\ast$5

and the paper states that a nonzero polynomial of degree at most $b_2^\ast$6 passes through all points if and only if

$b_2^\ast$7

The smallest such $b_2^\ast$8 is the minimal degree of an interpolating algebraic curve (&&&5^^^^5ArbInterp5^^^^&&&). For interval or box data, the determinant becomes an optimization target over all admissible point selections inside the boxes; if the resulting determinant is strictly positive, no polynomial of degree $b_2^\ast$9 can meet all boxes simultaneously (&&&5^^^^5ArbInterp5^^^^&&&). The same framework also studies sparsity, integer coefficients, and uniqueness via continued fractions and bounded-coefficient analysis (&&&5^^^^5ArbInterp5^^^^&&&).

This broadens the meaning of ^^^^^^^^5ArbInterp5^^^^^^^^ beyond approximation and parameter interpolation. The interpolant may instead be an implicit polynomial relation with minimal degree, minimal monomial count, or integer-coefficient structure. A plausible implication is that the unifying content of ^^^^^^^^5ArbInterp5^^^^^^^^ is methodological rather than disciplinary: interpolation is treated as a constrained reconstruction problem in which the admissible class is determined by geometry, numerics, probability, or market microstructure.

Under that interpretation, the cited literature describes a coherent research program. In graphics, ^^^^^^^^5ArbInterp5^^^^^^^^ corrects affine failure under perspective (&&&^^^^5ArbInterp5^^^^&&&). In barycentric theory, it is recast through canonical forms of positive geometries (&&&^^^^5\5ArbInterp5^^^^&&&). In spline and AMR settings, it is local, continuous, and boundary-aware (&&&^^^^5\5^^^^&&&, &&&^^^^5\5\5^^^^&&&). In arbitrary-precision computation, it is rigorous and interval-certified (Johansson, 2016, &&&^^^^5query5^^^^&&&). In generative modeling, it is density-aware or timestamp-controllable (&&&^^^^5query5ArbInterp5^^^^&&&, &&&^^^^5 OR \5^^^^&&&). In finance, it is arbitrage-free by construction (&&&^^^^5 OR \5^^^^&&&, Berg, 10 Jun 2026). In symbolic-numeric algebra, it seeks the smallest or sparsest exact interpolant consistent with uncertain data (&&&5^^^^5ArbInterp5^^^^&&&).