Slowed Interpolation Mixture

• Slowed interpolation mixtures are a class of techniques that use nonstandard, nonlinear schedules to mitigate issues like mode collapse and exposure bias in generative and diffusion models.
• They optimize performance by employing transfer formulas or mixture losses that adjust interpolation speeds, thereby preserving multi-modality and reducing numerical artifacts.
• These approaches are applied in varied contexts—from smoothed particle hydrodynamics to multifractal time series interpolation—demonstrating improved convergence and robustness.

A slowed interpolation mixture is a general term for a class of techniques that leverage nonstandard or mixture-based interpolation schedules—typically “slower” near one endpoint—to improve sampling, generative modeling, kernel estimation, or signal interpolation. It operates by constructing interpolants, model schedules, or basis functions that deviate from standard linear or uniform mixing, usually to mitigate numerical artifacts such as mode collapse, exposure bias, or instability. Across contemporary machine learning, signal processing, and computational physics, slowed interpolation mixtures appear in diverse algorithmic forms, including scalar schedule optimization in generative models, exposure-correcting mixtures in diffusion model training, basis-kernel mixtures in smoothed particle hydrodynamics, and superstatistical mixtures for multifractal time series interpolation.

1. Scalar Schedule Mixtures in Stochastic Interpolants

In generative modeling by stochastic interpolation and flow matching, the slowed interpolation mixture is instantiated as a time-indexed scalar schedule $(\alpha_t, \beta_t)$ in interpolants $I_t = \alpha_t z + \beta_t x_1$ where $x_1$ is a data sample, $z$ is a standard normal, and $t \in [0,1]$ is the schedule parameter (Chen et al., 1 Sep 2025). The key design is to choose non-linear, “slowed” $\beta_t$ such that growth is sublinear or superlinear, instead of the standard $\beta_t = t$, e.g.

$$\beta_t=\frac1M\sqrt{-\log\left(1 + (e^{-M^2}-1)t\right)}, \qquad \alpha_t = \sqrt{1-\beta_t^2}$$

for a Gaussian mixture with separation $M$. Minimizing the averaged squared Lipschitz constant of the ODE drift field,

$$A_2(\alpha,\beta) = \int_0^1 \mathbb{E}\left[\left\|\nabla b_t(I_t)\right\|_2^2 \right]\,dt$$

results in such slowed schedules, which empirically reduce mode collapse and improve few-step sampling. The principle is that, early in time, the interpolant leaves samples closer to their noise origin, slowing their movement away from initialization and hence preserving multi-modality.

A transfer formula enables the reuse of neural-net drift estimators trained under linear schedules for new, slowed schedules at inference, avoiding retraining. For high-dimensional Gaussian mixtures, such schedules exhibit sharp improvements in the ability to capture all mixture components during fast ODE/SDE integration (Chen et al., 1 Sep 2025).

2. Exposure Bias Correction in Diffusion/Likelihood Models

Slowed interpolation mixtures provide a solution to the training-sampling discrepancy in diffusion-based models, where the standard practice is to train on ground-truth interpolants at time $t$ but sample on generated states that, due to approximation error, align more closely to interpolants at a higher-noise “slowed” time $m_t < t$ (Li et al., 22 Dec 2025). The MixFlow method introduces a mixture-of-interpolants loss: for each training step $t$, sample $m_t$ from a uniform range $[(1-\gamma)t,\, t]$ (mixing range parameterized by $\gamma \sim 0.8$),

$$L_{\mathrm{MixFlow}} = \mathbb{E}_{x_0, x_1, t, m_t} \left\|u_\theta(x_{m_t}, t) - u^*(x_t, t)\right\|_2^2$$

where the input is the ground-truth interpolation at $m_t$ but the label is the target velocity at $t$. This mixture corrects exposure bias, strengthens performance especially at low sampling steps, and is easily implementable as a post-training loss without model structure changes. The empirical benefit is consistent across models and image/text-to-image tasks, as evidenced by nontrivial reductions in FID and increases in sample fidelity over ImageNet and Stable Diffusion 3.5 (Li et al., 22 Dec 2025).

3. Mixtures in Frame Interpolation and Video Generation

For video interpolation at arbitrary slow-motion rates, a related “mixture” paradigm is realized in the Mixture-of-LoRA (MoL) module of the SemFi model, which interpolates among adapters specialized for different output frame counts. Rather than a true continuous mixture, inference effectively selects or softly weights among $K$ LoRA “experts,” with the option of continuous weighting: $$\Delta W_\text{total}(N) = \Delta W_U + \sum_{k=1}^K \alpha_k(N) \cdot \Delta W_{E_k}$$ where $\alpha_k(N)$ is an indicator or softmax over the set of expert frame counts $s_k$ nearest to the requested frame rate $N$ (Hong et al., 7 Jul 2025). This enables frame interpolation at both low ("slowed") and high ("fast") speeds with high boundary fidelity and video quality, as confirmed by LPIPS/FID/PSNR metrics on the SFI-300K benchmark.

Additionally, in video frame interpolation for unknown temporal priors (Zhang et al., 2021), quadratic or curvilinear motion estimation mixtures can operate implicitly as slowed interpolation mechanisms, adapting sampling schedules to physical blur and exposure parameters. This promotes robust performance under widely variable camera and sequence statistics.

4. Kernel Mixtures in Smoothed Particle Hydrodynamics

Slowed interpolation mixtures appear as linear kernel mixes in smoothed particle hydrodynamics (SPH), used to reconcile the convergence and stability properties of low- and high-order interpolating kernels. In particular, the mixture

$$W_\mathrm{mix}(r, h) = 0.9 W_{\mathrm{Sinc},4}(r, h) + 0.1 W_{\mathrm{Sinc},9}(r, h)$$

generates a kernel with “slowed” error growth as a function of neighbor number $n_b$ while suppressing pairing instabilities that afflict pure high-order kernels (Cabezón et al., 2023). Empirical tests (e.g., Gresho–Chan vortex) demonstrate the mixture maintains high accuracy across $n_b=60$–$400$ without the need for kernel switching.

5. Gaussian Scale Mixtures for Multifractal Time Series Interpolation

In stochastic interpolation of sparsely observed time signals, a slowed interpolation mixture takes the form of a superstatistical random process generated from a Gaussian scale mixture: $$u(t) = u(t, \xi(t)),\quad\text{with}\quad u_\xi(t)\sim \mathcal{N}(0, C_\xi(t,s))$$ where the parameter process $\xi(t)$ is lognormal with slow correlation time $T_\mathrm{parameter} \gg T_\mathrm{process}$ (Lübke et al., 2022). Each point in time is assigned a local $\xi(t)$, and the path is constructed by selection from pre-simulated Gaussian processes, either via Fourier sampling or multiwavelet synthesis. The separation of time scales slows the evolution of the multifractal parameter, yielding a process that interpolates sparsely observed points while matching probabilistic small-scale regularity and multifractal scaling.

6. Practical Implementation and Effectiveness

Slowed interpolation mixtures admit algorithmic recipes that are “plug-and-play” in generation and inference. Principal guidelines include:

• Train model or estimator using standard (e.g., linear) schedules.
• At inference or post-training phase, define and use a slowed (e.g., nonlinear or mixture) interpolation schedule or kernel, typically guided by a principled numerical criterion (e.g., minimizing average squared Lipschitz drift).
• Apply a “transfer formula” or mixture weighting to avoid model retraining when changing schedules.
• For loss-based mixture approaches (e.g., MixFlow), augment training with a mixture over slowed interpolants.
• For kernel-based approaches, select kernel coefficients to optimize bias-variance and stability over target neighbor or fidelity ranges.

Empirically, slowed interpolation mixtures consistently reduce mode collapse in generative models, stabilize interpolants in physics-based simulations, and improve sample quality and adherence in high-dimensional settings (Chen et al., 1 Sep 2025, Li et al., 22 Dec 2025, Hong et al., 7 Jul 2025, Cabezón et al., 2023, Lübke et al., 2022).

7. Generalizations and Theoretical Significance

A recurring theme is the statistical equivalence of many interpolation schedules under pathwise Kullback-Leibler divergence, i.e., statistical efficiency is insensitive to scalar schedule choice if diffusion terms are tuned. Therefore, selection among mixtures or nonlinear schedules is dictated by numerical metrics—such as averaged Lipschitzness, error convergence, or empirical robustness—not by statistical theory. This insight generalizes across scalar schedule optimization, mixture-based model adaptation, and adaptive kernel techniques.

Slowed interpolation mixture designs are expected to further propagate into models for multimodal, high-dimensional, and temporally complex data where standard linear or uniform interpolation leads to instability, bias, or expressiveness loss.