Conditional Interpolants

• Conditional interpolants are mathematically structured processes that interpolate between distributions while respecting observed data and latent variable constraints.
• They extend classical stochastic interpolants by using operator and data-dependent conditioning mechanisms for tasks like inpainting, superresolution, and time series prediction.
• They underpin advanced generative SDE/ODE samplers with theoretical guarantees, explicit error control, and scalable algorithmic implementations across diverse domains.

A conditional interpolant is a mathematically structured process or path that interpolates between two (or more) distributions while respecting conditioning constraints imposed by observed data, side information, or latent variables. Conditional interpolants provide the theoretical and algorithmic foundation for a wide range of modern generative modeling, bridging probability measures in continuous time or operator index and unifying flow, diffusion, and bridge/forecasting methodologies. The formalism enables exact or learned transport between conditional laws, underpins a class of generative SDE/ODE samplers, and allows the incorporation of multi-marginal constraints, local or operator-valued conditioning, function space structure, and explicit error control.

1. Mathematical Foundations of Conditional Interpolants

Conditional interpolants generalize classical stochastic interpolants—which interpolate between a base and a target distribution via couplings and mixture coefficients—by incorporating conditioning on external or latent variables and enforcing multi-marginal or pathwise constraints.

Given two random elements $(x_0, x_1)$ in a (finite- or infinite-dimensional) space, a (unconditional) stochastic interpolant is

$$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$

for scalar or operator-valued schedules $\alpha, \beta, \gamma$, and a coupling law for $(x_0,x_1)$. The conditional version introduces a side variable $z$ or external features $\xi$, leading to conditional laws $p_t(x\,|\,z)$ or $p_t(x\,|\,\xi)$. In advanced settings, the interpolation path is itself constructed to satisfy prescribed marginals at multiple timepoints, as for dynamical systems or time series with observed data at $(t_0,\ldots, t_n)$, where the interpolant $\mu(t; z)$ must satisfy $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$0 for each $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$1 (Rathod et al., 30 Jan 2026).

Operator-based generalizations replace scalar interpolation weights with linear operators or matrices $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$2 acting on the state space, defining

$$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$3

as the generalized interpolant. Conditioning is encoded via the operator path, which can absorb masking, label injection, or specification of rewards (Negrel et al., 6 Aug 2025).

Conditional stochastic interpolants also extend directly to infinite-dimensional Hilbert spaces, with laws and dynamics formulated in operator-theoretic terms and implemented via neural operators or analytically in function spaces (Yu et al., 2 Feb 2026).

2. Construction and Conditioning Mechanisms

The construction of conditional interpolants is a two-step process: (1) specifying the form of the interpolant and its associated coupling, and (2) adapting the interpolant to encode the desired conditioning.

2.1. Data-Dependent Couplings

Rather than coupling base and target independently, the conditional interpolant framework may employ data-dependent couplings. For example,

$$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$4

with $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$5 possibly Gaussian with mean a function of $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$6. This enables the interpolant to encode inpainting (mask-based conditioning), superresolution (down- and upsampling as conditioning), or any task for which the conditional dependency can be explicitly constructed (Albergo et al., 2023).

2.2. Operator and Mask Conditioning

Operator-valued interpolants admit flexible, zero-shot conditioning: in image inpainting, for example, a diagonal operator $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$7 encodes which pixels are kept as ground truth and which are drawn from noise, enabling the same architecture to handle arbitrary mask patterns without retraining (Negrel et al., 6 Aug 2025). Posterior sampling, multichannel corruption, and even path planning or reward injection can all be encoded as modifications to $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$8.

2.3. Multi-Marginal Constraints and Dynamical System Alignment

For settings with multi-point observations (time series, dynamical systems), the interpolant $$x_t = \alpha(t)\,x_0 + \beta(t)\,x_1 + \gamma(t)\,z, \qquad z \sim N(0,I),$$9 is chosen (e.g., as a B-spline of degree $\alpha, \beta, \gamma$0) to pass through all observed states at their registered times, with coefficients fit via a constrained linear system. This supports highly flexible path construction matching all observed data marginals (Rathod et al., 30 Jan 2026).

3. Associated SDE/ODE Dynamics and Training

Conditional interpolants define, or are defined by, forward and backward SDEs or ODEs whose solutions match the desired law as time or operator index progresses.

A typical forward SDE for scalar schedule interpolants is

$\alpha, \beta, \gamma$1

where $\alpha, \beta, \gamma$2 encodes side information or context (Chen et al., 2024). In general, the drift and score functions in these SDEs are given as conditional expectations, learned by minimizing square loss regression objectives of the form

$\alpha, \beta, \gamma$3

and equivalently for denoisers or other required fields (Albergo et al., 2023, Chen et al., 2024, Yu et al., 2 Feb 2026).

For operator-based interpolants, the dynamics generalize to

$\alpha, \beta, \gamma$4

with $\alpha, \beta, \gamma$5 as conditional expectations, and similar drift formulae in the SDE case (Negrel et al., 6 Aug 2025).

B-spline conditional interpolants yield closed-form analytic velocities that are exploited for flow-matching loss computation, with neural networks regressing only the easily computable B-spline time derivatives (Rathod et al., 30 Jan 2026).

4. Theoretical Guarantees and Statistical Properties

Conditional interpolants, when constructed via the quadratic regression framework or via known analytic formulae, enjoy exact law-matching when appropriately parameterized.

• Existence and uniqueness of SDE solutions is established in both finite and infinite dimensions under mild Lipschitz and integrability conditions on the interpolant and coupling law (Chen et al., 2024, Yu et al., 2 Feb 2026).
• The minimization objectives for learning drifts, denoisers, and scores are strictly convex in the function space parameterization, guaranteeing unique minimizers and statistical consistency as model capacity increases (Albergo et al., 2023, Yu et al., 2 Feb 2026).
• Exactness results: Both ODE and SDE samplers push the base law to the target or conditional law exactly, modulo approximation error in the learned functions and numerical discretization (Negrel et al., 6 Aug 2025, Chen et al., 2024).
• Explicit error bounds: Wasserstein-$\alpha, \beta, \gamma$6 error between the SDE law and the target is controlled by the integrated mean squared error of the learned drift and denoiser, amplified by the (time-local) singularity of the drift (Yu et al., 2 Feb 2026).
• B-spline conditional interpolants provide interpolation error of order $\alpha, \beta, \gamma$7 for degree-$\alpha, \beta, \gamma$8 splines over $\alpha, \beta, \gamma$9 points, outperforming linear interpolants and minimizing Runge oscillations (Rathod et al., 30 Jan 2026).

5. Algorithmic Procedures and Implementation Details

Conditional interpolants admit scalable, modular algorithmic realization, unified across model classes:

• Training: Sample mini-batches of pairs (and optionally conditioning), construct interpolated states at random times or operator indices, and regress the drift, score, and/or denoiser by square loss (Albergo et al., 2023, Chen et al., 2024, Yu et al., 2 Feb 2026).
• Sampling: Given a conditioning variable (observed past, mask, operator, etc.), initialize from the corresponding base measure, then integrate the learned SDE or ODE to produce samples from the desired conditional law.
• Operator-based and B-spline settings provide analytic expressions for velocities or interpolation paths, allowing fast evaluation and elimination of quadrature in training and sampling (Rathod et al., 30 Jan 2026, Negrel et al., 6 Aug 2025).

Conditional Diffusion Sampling (CDS) integrates conditional interpolants with tempered MCMC sampling for efficient, non-learned sampling from unnormalized targets, with a two-stage procedure: (1) parallel tempering of the initialization, and (2) analytic SDE transport (Castro-Macías et al., 5 May 2026).

6. Applications, Empirical Results, and Limitations

Conditional interpolants underpin state-of-the-art generative modeling and inference in a variety of domains:

• Time series forecasting and probabilistic prediction via recurrent interpolation and conditional diffusions, with competitive or superior CRPS and 1-Wasserstein metrics on high-dimensional benchmarks (Chen et al., 2024).
• Dynamical system trajectory modeling, with SplineFlow based on B-spline interpolants significantly reducing MSE, Wasserstein, and energy distances in deterministic/stochastic ODE and SDE settings, especially for irregular sampling or chaotic ground truth (Rathod et al., 30 Jan 2026).
• Inpainting, superresolution, and multi-task image generation via operator, mask, or data-dependent coupled conditional interpolants, with FID and PSNR/SSIM improvements on ImageNet and other datasets (Albergo et al., 2023, Negrel et al., 6 Aug 2025).
• Posterior sampling in high-dimensional Bayesian inverse problems and lattice field theory through operator-controlled conditional interpolant dynamics (Negrel et al., 6 Aug 2025, Yu et al., 2 Feb 2026).
• Scientific computing and PDE-based generative modeling in Hilbert space, with quantitative convergence and error control (Yu et al., 2 Feb 2026).

Limitations include endpoint singularities in interpolant drifts (necessitating time-changes or careful initialization), the potential for suboptimal exploration if the interpolation path crosses high-density regions, one-time overheads for fitting complex interpolant families (e.g., B-splines), and the challenge of selecting optimal interpolant parameters (knot placement, degree, time change), which often rely on cross-validation or heuristic adaptation (Rathod et al., 30 Jan 2026, Castro-Macías et al., 5 May 2026).

7. Future Directions and Open Challenges

Open challenges for conditional interpolant research include:

• Automated, data-driven learning of interpolation schedules, knot locations (for splines), and operator-valued paths to optimize over class of interpolants (Rathod et al., 30 Jan 2026, Negrel et al., 6 Aug 2025).
• State-dependent and adaptive diffusion schedules for conditional SDEs, enhancing sampler robustness and efficiency (Rathod et al., 30 Jan 2026, Chen et al., 2024).
• Geometry-aware interpolants that avoid high-energy manifold crossings in complex or singular targets, particularly relevant for scientific inference and structured multimodal distributions (Castro-Macías et al., 5 May 2026).
• End-to-end time-warp learning, adaptive interval degree selection, and joint optimization over multi-marginal constraints to scale conditional interpolant frameworks to broader classes of inverse and controlled generation problems (Rathod et al., 30 Jan 2026, Yu et al., 2 Feb 2026).
• Extension to yet broader infinite-dimensional settings and integration with emerging operator neural network architectures for PDE-constraint science applications (Yu et al., 2 Feb 2026).

Conditional interpolants thus constitute a unifying and foundational theme in modern generative modeling, enabling principled, flexible, and mathematically tractable solutions to conditional generation and inference tasks across diverse domains and scales.