InterpoLL: Interpolation Methods in Research

Updated 2 July 2026

InterpoLL is a family of advanced interpolation frameworks unifying diverse methods across deep learning, numerical verification, PDEs, and generative modeling.
It employs techniques such as representation interpolation, SOS relaxations, and Riemannian metric optimization to tackle domain-specific challenges.
Empirical studies demonstrate significant improvements in accuracy, computational efficiency, and robust handling of out-of-distribution examples.

InterpoLL refers to a set of distinct research frameworks and algorithms, each prominent in a different research community and sharing the designation "InterpoLL" or "Interpol" as a marker of interpolation-centric methodology. Across applications in machine learning, numerical analysis, PDE-constrained data assimilation, computational verification, and generative models, these methods are unified by advanced interpolation theory and efficient algorithmic realization.

1. InterpoLL for Shortcut Mitigation in Deep Learning

The "InterpoLated Learning" framework (InterpoLL) is a model-agnostic representation interpolation technique targeting the "shortcut learning" failure mode of empirical risk minimization (ERM). Shortcut learning describes the tendency of ERM-trained models to exploit spurious correlations prevalent in majority groups, thus failing on minority examples where these correlations are invalid.

Algorithm structure:

An under-parameterized auxiliary model $f_\phi$ (e.g., TinyBERT) is trained by ERM. Examples misclassified by $f_\phi$ are labeled as minority ( $g_{\min}$ ), and correctly classified examples as majority ( $g_{\max}$ ).
For each majority example, a same-label minority example is randomly selected and a new feature vector is produced by interpolation in the encoder's representation space:

$z_i = (1-\lambda)f_{\text{enc}}(x_i) + \lambda f_{\text{enc}}(x_j), \ \lambda \sim U(0, 0.5)$

Minority examples retain their original representations.

Objective: Replace the original examples in the training loss with their interpolated representations, defining a modified ERM:

$J_{\rm InterpoLL}(\theta) = \frac{1}{n}\sum_{i=1}^n \ell(f_{\rm cls}(z_i), y_i)$

Empirical findings:

On NLI and text classification tasks (e.g., MNLI→HANS, FEVER, QQP, CivilComments-WILDS), InterpoLL yields substantial OOD and minority-group accuracy gains over ERM and state-of-the-art baselines, without group labels and with negligible computational overhead.
Gains generalize across architectures (BERT, T5, GPT-2) and model scales (Korakakis et al., 7 Jul 2025).

Architecture	ERM-OOD	InterpoLL-OOD	Δ
BERT-large	74.7	80.1	+5.4
T5-3B	78.9	84.5	+5.6
GPT2-large	70.8	77.4	+6.6

Ablations demonstrate the optimality of intra-class, minority-to-majority interpolation (λ∼U(0,0.5)), robustness to label noise, and improved representational invariance to task-irrelevant shortcuts.

2. InterpoLL for Interpolant Synthesis in Numerical Verification

In formal verification, "Interpolant Synthesis for Quadratic Polynomial Inequalities" (often referenced as "INTERPOL" or interpolation schemes) introduces an SDP-based approach for generating interpolants between mutually contradictory sets of concave quadratic polynomial inequalities—fundamental for program and hybrid system analysis.

Key elements:

Extends Motzkin’s transposition theorem to concave quadratics via linearization in matrix-inner-product form and LMI constraints.
Produces an interpolant, i.e., a separating predicate implied by one conjunction but inconsistent with another, by synthesizing a sum-of-squares (SOS) witness in $O(n^3 + nm)$ time.
The algorithm is compositional: combining with equality over uninterpreted functions (EUF) via flatten–purify and congruence closure leads to mixed-theory interpolation (Gan et al., 2016).

A direct implication is that the framework unifies interpolation in classical abstract domains (octagon, polyhedral, ellipsoid) and can automate the discovery of nonlinear barrier certificates for hybrid systems.

3. InterpoLL in Adaptive Mesh Interpolation (Numerical PDEs)

In AMR-based numerical solvers, "An Efficient Second-Order Accurate and Continuous Interpolation for Block-Adaptive Grids" introduces InterpoLL as a globally continuous, second-order accurate interpolation algorithm for cell-centered AMR grids (Borovikov et al., 2014).

Algorithmic features:

Domain partitioning by n-edge: distinguishes patches by the number of coordinate directions with resolution jumps, allowing use of reduced-dimensional stencils (bilinear/trilinear for uniform regions, barycentric/1D/2D schemes at interfaces).
Guarantees $C^0$ continuity at mesh interfaces without the need for ghost-cell arrays or halo exchanges, suited to massively parallel environments.
Implementation efficiently covers applications in Lagrangian particle tracking, field-line integration for SEPs, and visualization (crack-free isosurface extraction).

Benchmarks confirm negligible computational overhead versus plain trilinear interpolation, even on highly heterogeneous mesh hierarchies.

4. InterpoLL as Linear Algebraic Framework for General Interpolation

The "Interpolations: a linear algebra approach" paper formalizes all classical interpolation (polynomial, Hermite, trigonometric, heterogeneous observations) as the solution to a dual basis inversion problem:

Let $F$ be a finite-dimensional function space; interpolation constraints (values, derivatives, etc.) are linear functionals $\ell_i \in F^*$ .
The interpolation matrix $f_\phi$ 0 is inverted to obtain the dual basis, with every interpolant expressible as $f_\phi$ 1.
Explicit formulas for dual bases in Lagrange, Hermite, and trigonometric settings are derived, covering non-uniform and mixed-observation scenarios (Arutyunov et al., 21 Jun 2026).

This decouples basis selection from constraint selection, enabling systematic, efficient algorithms for high-order interpolation, trajectory planning, and sensor fusion.

5. InterpoLL for Likelihood-Optimal Interpolation Paths in Generative Models

In the context of generative modeling, "Likely Interpolants of Generative Models" defines InterpoLL as the minimizer of a regularized geodesic energy functional:

$f_\phi$ 2

with $f_\phi$ 3, $f_\phi$ 4 and $f_\phi$ 5 a (data- or model-induced) Riemannian metric (Rygaard et al., 30 Oct 2025).

Algorithm:

Discrete relaxation via dynamic programming; updates via Pontryagin’s minimum principle and line search ensure descent.
Path remains in high-density regions of $f_\phi$ 6, avoiding the deleterious artifacts of linear or spherical interpolation in latent space.
Empirical benchmarks show higher mean log-likelihood and lower Fréchet Inception Distance (FID) across VAE, score-based, diffusion, and ControlNet models compared to classical schemes.

6. InterpoLL in Data Assimilation (PDEs)

In continuous data assimilation for dissipative PDEs (e.g., Navier-Stokes), InterpoLL denotes feedback synchronization schemes based on general interpolant observables $f_\phi$ 7 (Azouani et al., 2013):

The slave system evolves under nudging via $f_\phi$ 8, with $f_\phi$ 9 possibly composed of low Fourier modes, spatial averages, or nodal values.
Rigorous error analysis gives exponential convergence rates for solution error, contingent on spatial resolution $g_{\min}$ 0 and nudging parameter $g_{\min}$ 1 satisfying explicit inequalities.

This operationalizes synchronization in multi-scale fluid flows through theoretically quantified interpolant-driven feedback, critical for high-fidelity climate and weather assimilation.

7. InterpoLL in Statistical Data Fusion and Cloud-Scale Image Analysis

In remote sensing, optimal interpolation techniques grounded in Kalman/Bayesian updates frequently adopt the "InterpoLL" or interpolation moniker:

For Landsat time series reconstruction, per-pixel, per-band optimal interpolation fuses climatology, regression forecasts, and observed data via closed-form two-step filtering equations (Moreno-Martinez et al., 2020).
The resulting framework achieves gap-free, smooth reflectance series at massive scale, with error cross-validation demonstrating RMSE $g_{\min}$ 20.01–0.08 and close alignment with hold-out observations.

No spatial neighborhood computation is needed; the method scales efficiently in distributed computing platforms (e.g., GEE).

InterpoLL refers to a family of interpolation-driven algorithms spanning adversarial shortcut mitigation in deep models (Korakakis et al., 7 Jul 2025), global $g_{\min}$ 3 interpolation in AMR (Borovikov et al., 2014), formal verification via SOS and LMI relaxations (Gan et al., 2016), dual-function linear algebraic interpolation (Arutyunov et al., 21 Jun 2026), likelihood-optimal generative model interpolation (Rygaard et al., 30 Oct 2025), rigorous nudging in PDE assimilation (Azouani et al., 2013), and data-fusion time series smoothing (Moreno-Martinez et al., 2020). Each instantiation advances interpolation methodology to address the structural, statistical, and computational challenges specific to its domain.