Representation Interpolation

• Representation interpolation is the systematic construction of new representations from existing data, ensuring specific values, derivatives, and symmetries.
• It integrates diverse methods—from classical dual-space and spline techniques to modern neural diffusion and latent interpolation—for robust data synthesis.
• Its practical applications span numerical analysis, signal processing, machine learning, quantum integrable systems, and logical reasoning in AI.

Representation interpolation refers to the systematic construction of new data, functions, or algebraic objects from known samples, observations, or representations, ensuring that the resulting structure matches specified values, properties, or symmetries at prespecified sites. This concept unifies a spectrum of methods across algebraic, analytical, probabilistic, and geometric settings, from classical polynomial and spline interpolation to modern implicit neural representations and categorical interpolation in representation theory. The methodologies range from explicit construction using dual bases in function spaces, through probabilistic generative models, to category-theoretic universal properties, and find applications in numerical analysis, signal processing, machine learning, mathematical physics, and knowledge representation.

1. Linear Algebraic and Dual-Space Foundations

Finite-dimensional function spaces provide a canonical setting for interpolation. Let $F$ be a vector space of candidate representations (e.g., polynomials of degree at most $n$, splines, or trigonometric polynomials). Interpolation is formalized via a set of $N$ linear functionals $\{\delta_i\}_{i=1}^N \subset F^*$ representing measurement or sampling conditions. For any $\alpha = (\alpha_1, \dots, \alpha_N)$, the problem is to construct $p \in F$ such that $\delta_i(p) = \alpha_i$.

The fundamental result is that if $\{\delta_i\}$ forms a basis of $F^*$, there exists a unique $p$ satisfying the constraint, which can be written as a linear combination $n$0, where each $n$1 is the dual basis element defined by $n$2. Classic instances include:

• Lagrange interpolation: $n$3 yields Lagrange polynomials as dual basis functions.
• Hermite interpolation: $n$4 for values and derivatives, producing Hermite polynomials.
• Trigonometric interpolation: $n$5 with $n$6 a space of trigonometric polynomials, using discrete orthogonality and cardinal functions.

This dual-space approach extends to mixed data types (function and derivative values), degeneracy analysis (ill-conditioning via the vanishing of Vandermonde-type determinants), and generalized bases, such as B-splines and radial basis functions. It decouples interpolation constraints from the choice of function basis, enabling efficient computation and modularity in real-time or heterogeneous-sensor environments (Arutyunov et al., 21 Jun 2026).

2. Spline Interpolation and Osculatory Representations

Hermite osculatory interpolation demands higher smoothness and the matching of multiple derivatives at prescribed nodes. Representation interpolation in this context is performed in structured spline spaces constructed over refined partitions of an interval. For example, refined knot sets and mixed smoothness conditions allow the construction of spaces of splines $n$7 whose dimension matches the total number of value and derivative conditions.

The key development is the explicit construction of partition-of-unity, nonnegative, locally supported B-spline-like basis functions $n$8. Each $n$9 is dual to the Hermite derivative conditions at node $N$0 and is constructed via localized Bernstein polynomials with careful support to guarantee the satisfaction of all interpolation and smoothness constraints. The basis satisfies:

• Locality and nonnegativity,
• Partition of unity,
• Hermite-exactness in matching derivatives up to order $N$1 at each $N$2.

These bases enable both exact Hermite interpolation (via direct solution of a local nondegenerate linear system involving the derivative moments) and quasi-interpolation (approximate representations via blossoming or polarization, reproducing all low-degree polynomials). The approach yields high-order convergence and improved stability relative to classical methods (Boushabi et al., 2024).

3. Neural, Probabilistic, and Latent Representation Interpolation

Recent developments leverage implicit neural representations, probabilistic generative models, and decomposition-in-latent-space as powerful tools for interpolating high-dimensional, structured data.

(a) Implicit Neural Representations and Diffusion Guidance

DIER exemplifies the synthesis of coordinate-based neural architectures and denoising diffusion probabilistic models for high-fidelity interpolation of empirical Green's function (EGF) fields in seismology. The network $N$3, conditioned on spatial coordinates $N$4, produces the full waveform $N$5; its parameters are trained as the reverse kernel of a coordinate-conditional diffusion process $N$6. The diffusion prior ensures phase-coherent, dispersion-preserving interpolation unattainable with pointwise methods like RBFs. At deployment, unsampled wavefields at $N$7 are synthesized deterministically by running the reverse diffusion with $N$8 as condition (Chen et al., 9 Jan 2026).

(b) Latent Space Interpolation for Controllable and Disentangled Representations

Controllable Interpolation Regularization (CIR) systematically enforces global convexity and attribute disentanglement within the latent space of variational or invertible encoders by explicit interpolation between codes. By locally mixing only specific semantic subspaces and enforcing stability under re-encoding, the model regularizes both attribute independence and on-manifold quality, as measured by mutual information, attribute co-prediction, block-wise latent correlations, and adversarial image quality assessments. CIR improves manipulation controllability and enhances downstream synthesis capabilities (Ge et al., 2021).

(c) Latent Interpolation in Counterfactual and Causal Machine Learning

The CAT framework introduces representation interpolation within deep transformer models for counterfactual adversarial training. For each sample, layer-wise hidden representations are interpolated (with trainable mixing weights) with those from other samples to construct counterfactuals. Adversarial objectives force these interpolants to cross the classifier's decision boundary with minimal displacement. Subsequent counterfactual risk minimization uses importance-weighted sampling to emphasize causal, rather than merely correlational, features in the representation (Wang et al., 2021).

(d) Learned Latent Code Interpolation for View Synthesis

In VIINTER, a family of neural implicit functions parametrized by learned code vectors $N$9 is trained for view-dependent image synthesis. Interpolation between latent codes, enforced to lie on a convex, normalized shell (e.g., $\{\delta_i\}_{i=1}^N \subset F^*$0 unit norm), allows high-quality view morphing without explicit 3D modeling or correspondences. Training incorporates reconstruction and perceptual interpolation losses to align feature-space interpolates with the desired convex combination of view features (Feng et al., 2022).

4. Trigonometric, Spline, and Spectral Representations

Representation interpolation in periodic or oscillatory function spaces is realized through several distinct forms:

• Trigonometric polynomial interpolation: Coefficient-based representations, often computed via discrete Fourier transforms, exploit global spectral accuracy and symmetry.
• Trigonometric B-splines: Localized periodic basis functions, parameterized by smoothness, support adaptive, band-limited, and boundary-flexible interpolation.
• Fundamental (cardinal) splines: Shifted, interpolation kernels built from Fourier series or their even/odd restrictions enable direct sampling implementations.

Each form has computational and structural implications—spectral forms maximize global accuracy; B-splines support locality and adaptivity; fundamental splines enable shift-invariance and sampling-based constructions (Denysiuk et al., 2024).

5. Interpolation in Knowledge Representation and Logic

Semantic and syntactic interpolation in logic, notably Craig interpolation (CI) and uniform interpolation (UI), is a foundational tool in description logics, automated reasoning, and knowledge representation. Given formulas or ontologies $\{\delta_i\}_{i=1}^N \subset F^*$1 and $\{\delta_i\}_{i=1}^N \subset F^*$2 with $\{\delta_i\}_{i=1}^N \subset F^*$3, a CI is a formula entailed by $\{\delta_i\}_{i=1}^N \subset F^*$4, entailing $\{\delta_i\}_{i=1}^N \subset F^*$5, and containing only shared signature symbols. UI (or "forgetting") seeks the strongest consequence of $\{\delta_i\}_{i=1}^N \subset F^*$6 expressible in a restricted vocabulary.

Key theoretical facts:

• In ALC, CI always exists (Craig Interpolation Property, CIP), but exact UI may fail.
• Complexity of checking existence and constructing interpolants is high: up to 2ExpTime-complete in ALC.
• Practical methods include resolution with role propagation and Ackermann elimination (Lethe), second-order quantifier elimination, and tableau-based procedures.
• Applications span explainable AI, privacy-preserving ontology design, modularization, concept learning, and abduction.

These interpolative methods function as representation restrictors or summarizers, maintaining logical entailments across symbol sets, and are vital in scalable reasoning and modular knowledge systems (Jung et al., 9 Dec 2025).

6. Categorical and Representation-Theoretic Interpolation

Representation interpolation in the categorical and algebraic sense involves functorial constructions that "interpolate" representation categories across parameters, often using universal properties. For example, Kriz's categorical interpolation of oscillator representations connects the symmetric tensor category $\{\delta_i\}_{i=1}^N \subset F^*$7 with a universal splitting category in which a twisted group Azumaya algebra becomes split. The existence of the interpolation category is governed by the triviality of a specific Picard group invariant $\{\delta_i\}_{i=1}^N \subset F^*$8; its construction involves monoidal enlargements and diagrammatic (rather than spectral-theoretic) arguments. Such categorical interpolation underlies modern advances in the theory of tensor categories, quantum groups, and the algebraic structure of integrable systems (Snowden, 2024).

In quantum integrable systems, representation-theoretic interpolation emerges via the classification of symmetric polynomials (e.g., $\{\delta_i\}_{i=1}^N \subset F^*$9-Whittaker polynomials) determined uniquely by representation-theoretic vanishing conditions (interpolation grids) and degree constraints. Intertwining operators, rather than $\alpha = (\alpha_1, \dots, \alpha_N)$0-matrices, provide the structural underpinning for the construction of novel interpolation families within the space of symmetric functions, with applications to stochastic models and quantum Hamiltonians (Korotkikh, 2022).

7. Outlook and Contextual Significance

Representation interpolation synthesizes structural properties, functional data, or algebraic constraints within a unifying paradigm of matching prescribed information—be it values, derivatives, spectra, logical entailments, or categorical morphisms—across domains. Its span includes

• High-dimensional generative modeling (implicit neural representation, diffusion models),
• Modular and explainable reasoning (logic, description logics, forgetting),
• Stable and adaptive numerical schemes (B-splines, trigonometric splines),
• Category-theoretic advances (universal splitting, Picard group obstructions),
• Symmetric function theory and quantum integrable systems.

The trajectory of research is toward increasingly expressive and structurally-aware interpolation, leveraging geometric, algebraic, and probabilistic priors, with implications for scalable learning, automated reasoning, and the synthesis of generalized function and representation spaces. Open directions involve improved scalability, universality, and robustness of interpolants in complex domains, as well as bridges between classical and machine-learning-based approaches to high-dimensional representation interpolation (Chen et al., 9 Jan 2026, Arutyunov et al., 21 Jun 2026, Jung et al., 9 Dec 2025, Boushabi et al., 2024, Denysiuk et al., 2024, Snowden, 2024, Feng et al., 2022, Korotkikh, 2022, Ge et al., 2021, Wang et al., 2021, Alpay et al., 2011).