Stochastic Interpolant Framework

• Stochastic Interpolant Framework is a measure-interpolating process that creates explicit time-continuous paths using deterministic and stochastic dynamics.
• It guarantees contractivity and regularity in high-dimensional generative models by offering sharp Lipschitz bounds and gradient field properties.
• Practical implementations leverage controlled ODE/SDE discretization techniques to ensure robust sampling and efficient model performance.

Stochastic Interpolant Framework

Stochastic interpolants are a class of measure-bridging processes that unify the dynamics of diffusion models, flow-matching, and optimal transport by defining explicit time-continuous paths between two probability measures via deterministic or stochastic dynamics. This approach facilitates the construction of generative models and sampling algorithms with provable regularity and contractivity properties, supporting both continuous- and discrete-time settings, and can be applied to high-dimensional domains, including those encountered in machine learning, physics, and Bayesian computation.

1. Mathematical Foundations of Stochastic Interpolation

Let $\mu_0, \mu_1 \in \mathcal{P}_2(\mathbb{R}^d)$ be probability measures with finite second moments. A stochastic interpolant is defined via an interpolation kernel $$I_t : \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}^d$$ satisfying

$I_0(x_0, x_1) = x_0, \quad I_1(x_0, x_1) = x_1,$

with regular dependence on $t$ and $(x_0, x_1)$. By sampling $(X_0, X_1) \sim \mu_0 \otimes \mu_1$ and setting $X_t = I_t(X_0, X_1)$, one constructs intermediate marginals $\mu_t = \text{Law}(X_t)$ forming a path from $\mu_0$ to $\mu_1$.

Practical implementations typically avoid explicit endpoint pairs. Instead, they characterize a velocity field $v_t : \mathbb{R}^d \to \mathbb{R}^d$ as

$$v_t(x) = \mathbb{E}[\partial_t I_t(X_0, X_1)\ |\ I_t(X_0, X_1) = x],$$

which fulfills the continuity equation: $\partial_t \mu_t + \nabla \cdot (\mu_t v_t) = 0.$ The ODE

$$\frac{\mathrm{d}}{\mathrm{d} t} f_t(x) = v_t(f_t(x)), \quad f_0(x) = x$$

pushes forward $\mu_0$ to $\mu_t$, with $f_1$ mapping $\mu_0$ onto $\mu_1$ (Daniels, 14 Apr 2025).

2. Contractivity and Regularity of the Flow

For $\mu_0 = \mathcal{N}(0, I_d)$ and a smooth, strongly log-concave target $\mu_1(x) = e^{-V_1(x)}$ with $\nabla^2 V_1(x) \succeq \kappa I_d$, consider the isotropic linear interpolant $I_t(x_0, x_1) = \alpha_t x_0 + \beta_t x_1$ with canonical boundary constraints.

Main Theorem (Isotropic Semi-Gaussian SI Contractivity) (Daniels, 14 Apr 2025):

1. $v_t$ is a gradient field.
2. The Jacobian $D v_t(x)$ obeys a sharp operator norm:

$$D v_t(x) \preceq \frac{\kappa\,\alpha_t \dot\alpha_t + \beta_t \dot\beta_t}{\kappa\,\alpha_t^2 + \beta_t^2}\,I_d$$

1. The flow map satisfies the global Lipschitz bound:

$$\|D f_t\|_{\mathrm{op}} \leq \sqrt{\alpha_t^2 + \frac{\beta_t^2}{\kappa}}.$$

At $t=1$, this coincides precisely with the Caffarelli constant for the optimal transport from $\mathcal{N}(0,I)$ to any $\kappa$-log-concave measure.

For more general log-concave endpoints $\mu_0 = e^{-V_0}$, $\mu_1 = e^{-V_1}$ with uniform $C^2$-bounds $\kappa_0 I \preceq \nabla^2 V_0 \preceq \eta_0 I$ and $\kappa_1 I \preceq \nabla^2 V_1 \preceq \eta_1 I$, the Lipschitz bound for $v_t$ generalizes to (Daniels, 14 Apr 2025): $$\|D v_t\|_{\mathrm{op}} \leq \frac{ \alpha_t \dot\alpha_t\,\kappa_1 + \beta_t \dot\beta_t\,\eta_0 }{ \sqrt{ \alpha_t^2 \kappa_1 + \beta_t^2 \eta_0 }\, \sqrt{\alpha_t^2\kappa_1+\beta_t^2\kappa_0 } }.$$ This yields explicit Lipschitz transport maps between arbitrary uniformly log-concave distributions, extending regularity properties beyond classical PDE transport approaches.

3. Discretization and Practical Sampling Guarantees

In practice, SI-based flows or SDEs are discretized. Discretization error is controlled directly by the global Lipschitz constant of the velocity field. For the ODE case, the explicit Euler method incurs $O(L \Delta t)$ error per step, with overall discretization error scaling as $O(h)$ (Euler) and $O(h^2)$ (Heun) with step size $h$, and $L$ the maximum Lipschitz constant (Daniels, 14 Apr 2025). In high dimensions, the error scales as $O(\sqrt{d})$, substantially reducing steps needed for fixed accuracy.

For SDE variants, error and ergodicity bounds inherit exponential dependencies on $L$, where smaller $L$ directly translates to improved mixing. Lipschitz constants from the contractivity theorem thus guide the admissible step sizes and the required complexity for stable and accurate numerical integration.

Further, the Tweedie-type representation of $v_t$ in terms of conditional expectations enables statistical estimation via kernel-density methods or modern neural parameterizations, paralleling techniques in score-based diffusion estimation.

4. Relation to Diffusion Models and Optimal Transport

Stochastic interpolation generalizes the dynamical sampling paradigm underlying diffusion models and flow-matching. Forward SIs with linear schedules coincide with reversals of Ornstein–Uhlenbeck processes, and the resulting flows can be identified with Benamou–Brenier dynamical optimal transport, except that stochastic interpolants select a specific bridging kernel rather than optimizing over all flows (Daniels, 14 Apr 2025).

In the special case where both source and target are Gaussian with commuting covariances, the SI map yields the unique optimal quadratic-cost transport. For strongly log-concave targets, the SI's contractivity matches the sharp constants of Caffarelli's optimal transport theory. Compared to classical diffusion samplers—where error control relies on uniform Lipschitz bounds for drift and score—the SI framework provides at least as strong, and in certain regimes strictly stronger, contractivity.

The SI framework thus sets a correspondence diagram:

Framework	Dynamics	Endpoint Compatibility
SI (general)	ODE/SDE, explicit kernel	General, non-Gaussian
Diffusions	SDE (OU, VP, etc.)	Mainly Gaussian base
OT (Caffarelli)	PDE gradient flow	Log-concave, Gaussian

Moreover, constructing explicit Lipschitz maps transfers Poincaré, log-Sobolev, and related functional inequalities under controlled constants.

5. Implementation and Implications for Estimation

Fitting the SI velocity reduces to a least-squares regression: $$\mathbb{E}[ \| \partial_t I_t(X_0, X_1) - \hat v_t(I_t(X_0, X_1)) \|^2 ]$$ over samples $(X_0, X_1)$. This admits both parametric (neural networks) and non-parametric estimators. For SDE-based sampling, SI's Lipschitz controls inform model capacity choice (bounding the drift norm) and time-discretization granularity. For ODE-based approaches, the contractivity guarantees enable stable integration with step-size inversely proportional to $L$, ensuring the overall TV or Wasserstein error remains tightly controlled.

For high-dimensional problems, this reduction in effective step complexity is particularly pronounced—the deterministic discretization error is $O(\sqrt{d})$, and for SDEs, mixing time bounds are exponential in $L T$—emphasizing the practical importance of contractivity in both model design and solver selection (Daniels, 14 Apr 2025).

6. Summary and Outlook

Stochastic interpolants encompass a broad range of measure-interpolating flows by coupling endpoint distributions with flexible, analytically tractable interpolations. They provide explicit regularity and contractivity via operator norm bounds that match or exceed classical optimal transport benchmarks. This framework enables provably stable and statistically efficient generative sampling in both deterministic ODE and stochastic SDE variants, leverages modern estimator design, and sits at the intersection of diffusion, flow-matching, and optimal transport. The explicit Lipschitz bounds open pathways for measure-transfer tasks, proof of functional inequalities, and robust model deployment in high dimensions (Daniels, 14 Apr 2025).

Further theoretical and computational advances are anticipated along the axes of endpoint generalization, operator interpolation, adaptive schedule selection, and application to complex scientific and machine learning systems.