Self-Consistent Stochastic Interpolant

• SCSI is a framework that self-consistently reconstructs underlying distributions and neural population dynamics under corrupted observations and finite-size noise.
• It employs fixed-point iterations and stochastic PDE integrations to solve inverse problems and match empirical statistical properties.
• Empirical studies show SCSI’s superior performance, stability, and efficiency compared to traditional methods in both generative modeling and neural systems.

The self-consistent stochastic interpolant (SCSI) is a methodological and theoretical framework with distinct developments in both generative modeling under black-box corruptions and in the stochastic mean-field theory of finite neuron assemblies. Across these domains, SCSI shares the core principle of self-consistently reconstructing the underlying dynamics—either of distributions or population activities—in the presence of stochasticity arising from finite sampling, measurement corruption, or network size. The following provides a technical overview of both main variants of SCSI, their operationalization, theoretical guarantees, algorithmic structure, practical implications, and empirical validation (Vinci et al., 2021, Modi et al., 11 Dec 2025).

1. Mathematical Foundations of the Self-Consistent Stochastic Interpolant

The generative modeling SCSI formulation addresses the fundamental inverse problem: given only access to corrupted samples $y \sim \mu$ and a black-box forward corruption channel $\mathcal F: \mathbb{R}^d \to \mathbb{R}^d$ with $\mu = \mathcal K \pi$ (where $\mathcal K \pi$ denotes the pushforward of the unknown clean law $\pi$ through $\mathcal F$), recover a generative model for the clean distribution.

A linear stochastic interpolant is defined via

$I_t = \alpha_t x_0 + \beta_t x_1 + \gamma_t z$

for $(x_0, x_1) \sim \nu$ (a coupling of $\pi$ and $\mu$), $z \sim \mathcal N(0, I_d)$ and schedules $\alpha_t, \beta_t, \gamma_t$ with appropriate boundary constraints. Critically, SCSI defines a transport map $\Phi$ from $\mu$ to $\pi$ such that the induced measure $\pi_\Phi = \Phi_\# \mu$ satisfies the self-consistency equation $\mathcal K \pi_\Phi = \mu$. Iteratively, the interpolant is updated using fixed-point iterations,

$$I_t^{(k+1)} = \alpha_t\,\Phi_{\Theta^{(k)}}(y) + \beta_t\,\mathcal F(\Phi_{\Theta^{(k)}}(y)) + \gamma_t z$$

where each outer update retrains the drift estimator parameters $\Theta^{(k+1)}$ on the current batch of interpolant trajectories. The process provably converges (under injectivity/stability conditions) to the fixed point where the synthetic pushforward matches the observed corrupted law (Modi et al., 11 Dec 2025).

In the stochastic population density context, SCSI extends the Fokker-Planck formalism for leaky integrate-and-fire (LIF) neuron networks by incorporating a finite-size noise source directly into the PDE for the membrane potential density $p(v, t)$. For $N$ neurons, the empirical rate

$$\nu_N(t) = \nu(t) + \eta(t), \quad \langle \eta(t) \rangle = 0, \ \langle \eta(t)\eta(t') \rangle = \frac{\nu(t)}{N}\delta(t-t') + O(N^{-2})$$

fluctuates with variance scaling as $1/N$. SCSI achieves closure by matching the power spectrum of $\nu_N$ with the renewal-theoretic result, enforcing a self-consistency relation for the coloured noise $\eta(t)$. The stochastic Fokker-Planck system is closed by a Markovian embedding of $\eta$ as a low-dimensional Ornstein–Uhlenbeck process coupled to the evolving population rate (Vinci et al., 2021).

2. Algorithmic Structure and Implementation

In the generative domain, the SCSI algorithm follows a nested optimization scheme:

• Outer loop: for each iteration $k$, fix current SI parameters $\Theta^{(k)}$.
• Inner (regression) loop: draw $y \sim \mu$, propose $x = \Phi_{\Theta^{(k)}}(y)$, simulate $y' = \mathcal F(x)$, sample $z \sim \mathcal N(0, I)$, select random $t \in [0,1]$, form the interpolant $I_t$, and update the drift and noise estimators $(b, g)$ (or unified drift $v$) via SGD to regress against analytic time derivatives of $I_t$ and sampled noise (Modi et al., 11 Dec 2025).
• After training, generative sampling is performed by running the backward SDE/ODE defined by the learned drift from $X_1 = y' \sim \mu'$ back to $X_0 \sim \pi_{\Theta^*}$.

Each SGD step involves sampling, ODE/SDE integration, and a neural network forward pass (typically a U-Net backbone with 32–70M parameters). The approach is entirely model-free regarding $\mathcal F$: it does not require differentiability or explicit likelihood evaluation and is robust to arbitrary black-box corruption channels, including non-Gaussian, nonlinear, and non-differentiable cases.

In the neural population context, the SCSI PDE system is numerically integrated:

$$\begin{aligned} \partial_t p(v, t) &= -\partial_v[(F(v) + \mu_N)p] + \frac{1}{2} \partial_v^2[\sigma_N^2 p] + \delta(v - v_\theta) \eta(t) \ d\mathbf u &= \mathbf A(\mu_N, \sigma_N) \mathbf u \,dt + \mathbf B \,dW \ \eta(t) &= \mathbf{1}^\top \mathbf u(t) + \sqrt{\frac{\nu_0}{N}} W(t) \end{aligned}$$

where $\mu_N$ and $\sigma_N$ are rates and variances driven by $\nu_N(t)$. The artificial noise $\eta(t)$ is simulated via a two-dimensional OU process, parameterized so its spectrum matches the SCSI self-consistency condition (Vinci et al., 2021).

3. Theoretical Guarantees and Analytical Properties

The SCSI scheme offers rigorous fixed-point and contraction properties under well-defined assumptions:

• Uniqueness: If $\mathcal K$ is injective and the iteration converges, then the solution matches the clean data law ($\pi_{\Theta^*} = \pi$).
• Wasserstein contraction: Provided $W_2$-Lipschitz stability, the iterates contract geometrically in $W_2$, with error bounds of $$W_2(\pi, \pi^{(k)}) \le R^{k/2} W_2(\pi, \pi^{(0)})$$ for $R < 1$.
• KL-contraction: In SDE-based SCSI, if the model class is compact, the forward drift is Lipschitz in KL, and the effective condition number $\chi$ is finite, then $KL(\pi \| \pi^{(k)})$ decays geometrically.
• Gaussian case: For additive white Gaussian noise channels, explicit iterates for the mean and covariance admit closed-form updates, with provable contraction rates outpacing classical EM in the low-SNR regime.

For finite-size spiking networks, the SCSI framework ensures:

• Correct $N^{-1}$ scaling of noise amplitude and the spectral profile of finite-size fluctuations.
• Self-consistency between renewal-theoretic spectra and the stochastic mean field, including low-frequency suppression proportional to the coefficient of variation $c_v$.
• Non-perturbative closure, retaining accuracy both in linear response and near critical dynamical transitions (Hopf bifurcation/synchronization).

4. Empirical Performance and Experimental Findings

The SCSI method has been validated on a range of synthetic and real-world inverse problems:

• Synthetic 2D data: On two-moons data with various levels of additive noise, ODE and SDE variants of SCSI maintain accurate generative modeling, with improved stability in high-noise regimes for SDE schedules.
• CIFAR-10 image reconstruction: Across random pixel masking, additive Gaussian noise, Gaussian/Motion blur, JPEG compression, and Poisson noise, SCSI achieves competitive or superior generation FID (e.g., FID ≈ 6.74 at mask $\rho=0.5$) compared to leading EM-posterior and ambient-diffusion baselines, while incurring substantially lower computational cost (86 GPU hours vs. 512) (Modi et al., 11 Dec 2025).
• Astronomical spectral data: On SDSS quasar spectra subjected to flux offsets and smoothing, SCSI attains significantly lower reconstruction MSE than the Wiener filter, accurately recovering fine spectral structure.
• Finite-size neural network validation: Across the linear, near-critical (Hopf), and post-critical (limit cycle) dynamical regimes, SCSI-reduced PDEs reproduce the rate spectra, oscillatory peaks, and finite-size broadening observed in full spiking network simulations with high fidelity (Vinci et al., 2021).

5. Comparison with Alternative Approaches

A summary of distinguishing features is provided below:

Method	Clean Data Needed	Forward-Model Grad	Generative FID (CIFAR-10, mask 50%)
Ambient Diffusion	no	yes (linear only)	18.85
EM-Posterior	no	yes (linear)	6.76
DPS	yes	yes	— (restoration LPIPS 0.0072 @$\sigma=0.1$)
SCSI	no	no	6.74

SCSI matches or exceeds the best generative FID of EM-based methods, vastly outperforms Ambient Diffusion, and maintains applicability to arbitrary, non-differentiable, or nonlinear corruption channels (e.g., JPEG, blur), whereas previous methods are limited to explicit or linear Gaussian forward models. Unlike DPS and similar score-based approaches, SCSI requires only sampling access—never gradients or likelihoods—making it more broadly applicable under constrained-computation or experimental setups (Modi et al., 11 Dec 2025).

For population density models, no alternative non-perturbative, analytically-closed frameworks have been shown to match spiking network statistics across critical regimes with finite-size scaling consistency (Vinci et al., 2021).

6. Analytical Noise Spectrum and Self-Consistency Structure

In the population density context, the SCSI noise spectrum is characterized as:

$$|\eta(\omega)|^2 = \frac{\nu_0}{N}\left[1 - \left|\frac{(i\omega + \nu_0)\rho(\omega) - \nu_0}{\nu_0 \rho(\omega) + i\omega - \nu_0}\right|^2\right],$$

with low-frequency suppression $|\eta(0)|^2 = (\nu_0/N) 4c_v^2/(1 + c_v^2)^2$ and high-frequency asymptotics $|\eta(\omega)|^2 \to \nu_0/N$. As a plausible implication, this explains the sigmoidal spectral profile and consistent suppression of low-frequency fluctuations in large but finite neural populations.

The non-perturbative SCSI closure is achieved by the coupling of the noise-generating process (e.g., $\eta(t)$ in neural models or the backward SDE in generative modeling) to the current system state, enforcing statistical match between observed empirical statistics and the model-imposed dynamics. In this sense, SCSI goes beyond linear response or perturbation, retaining consistency even far from steady-state or weak-coupling limits.

7. Practical Implementation and Scalability

For image and signal reconstruction, SCSI’s implementation utilizes a U-Net architecture (32–70M parameters), ODE integration (64 steps), and relies on minibatch SGD (64–128 samples, Adam optimizer, $5\times 10^{-4}$ learning rate). Each iteration involves only forward passing through black-box corruption channels, allowing application to domains where explicit gradients or likelihoods are unavailable (e.g., scientific imaging, heavily compressed data) (Modi et al., 11 Dec 2025).

In the neural context, one numerically integrates the closed SCSI-PDE–OU process system, with computational requirements scaling modestly with population size and enabling exploration of collective stochastic phenomena that are inaccessible to mean-field or deterministic PDE approaches.

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SCSI provides a theoretically rigorous, computationally tractable, and versatile framework for self-consistently modeling stochastic dynamics in both inverse generative problems under corrupted observations and in finite-size stochastic neuronal assemblies. Its fundamental strength lies in closure against observed statistical properties—enforced through fixed-point or spectral matching—without reliance on simplifying or perturbative assumptions, and all with high empirical fidelity to both synthetic and real-world systems (Vinci et al., 2021, Modi et al., 11 Dec 2025).