MM-SOLD: Moment-Matched Score-Smoothed Langevin Dynamics

• The paper introduces a training-free generative sampler that replaces learned neural scores with closed-form scores from a diffusion-induced Gaussian mixture model.
• It employs overdamped Langevin dynamics with a constrained interacting particle system to strictly match empirical mean and covariance, preserving data geometry.
• Experiments demonstrate MM-SOLD’s robust performance on low-dimensional and latent-space image tasks, outperforming baseline diffusion and approximate-score methods.

Searching arXiv for the cited MM-SOLD and related Langevin robustness papers. Moment-Matched Score-Smoothed Overdamped Langevin Dynamics (MM-SOLD) is a training-free generative sampler built on an interacting particle system that replaces a learned neural score with a closed-form score derived from a diffusion-induced Gaussian mixture model, then modifies overdamped Langevin dynamics by enforcing empirical mean and covariance constraints throughout the sampling trajectory. In the formulation introduced in “Training-Free Generative Sampling via Moment-Matched Score Smoothing,” MM-SOLD combines score smoothing, overdamped Langevin dynamics, and hard moment constraints so that, in the large-particle limit, the empirical particle density converges to a deterministic limit whose one-particle stationary marginal is a Gibbs–Boltzmann density obtained by exponentially tilting a naive score-smoothed diffusion target; the mean and covariance of this limiting law agree with the empirical moments of the training data (Yao et al., 14 May 2026). The method is positioned in relation to both diffusion models, which learn annealed scores, and to approximate-score Langevin methods, whose robustness properties can be strongly negative when only $L^2$ or $L^p$ score accuracy is known (Cao et al., 11 Mar 2026).

1. Definition and conceptual setting

MM-SOLD targets the same kind of distributions as diffusion / score-based generative models, but does not train a neural network score. Instead, it uses the closed-form empirical score of the Gaussian mixture model induced by a forward diffusion initialized at the training data; smooths that empirical score, or equivalently smooths the log-density, to obtain a “naive score-smoothed diffusion target”; runs overdamped Langevin dynamics on this smoothed potential for a collection of particles; and constrains the particle system so that its empirical mean and covariance match those of the training data at every step (Yao et al., 14 May 2026).

The construction begins from the empirical distribution

$$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$

with forward Ornstein–Uhlenbeck-type diffusion

$$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$

For a finite dataset, this yields an empirical forward density

$$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$

whose score admits the closed-form expression

$$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$

where

$$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$

For small $t$, $\beta_t$ is small, so $w_i^t(z)$ is concentrated on the nearest neighbor; this reproduces training samples and does not generate novel ones (Yao et al., 14 May 2026).

The central motivation is that the exact empirical score for a finite dataset makes the reverse diffusion memorize, whereas neural diffusion models generalize because training implicitly smooths this empirical score. However, naive score smoothing is scale-sensitive: too little smoothing preserves memorization, while too much smoothing collapses the distribution toward barycenters of training samples. MM-SOLD addresses this by enforcing global empirical moments while smoothing: smoothing interpolates locally, but moment constraints prevent global barycentric collapse and preserve the overall geometry of the data (Yao et al., 14 May 2026).

2. Score smoothing and the naive target

The score smoothing step averages the empirical score against a kernel $L^p$0. The smoothed score is defined by

$L^p$1

For an additive kernel with bandwidth $L^p$2, $L^p$3 with $L^p$4 and zero mean, this becomes

$L^p$5

A Monte Carlo approximation is

$L^p$6

Since $L^p$7 is independent of $L^p$8,

$L^p$9

If one runs the reverse SDE with this smoothed score and enough corrector steps, and stops at time $$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$0, the sampling target becomes, up to normalization,

$$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$1

For small $$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$2, $$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$3 is well approximated by an isotropic GMM $$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$4 with small component standard deviation $$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$5, so the naive score-smoothed target is

$$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$6

Writing

$$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$7

one has

$$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$8

which is the invariant law of unconstrained overdamped Langevin dynamics

$$\hat\pi_{\mathrm{data}} = \frac{1}{N}\sum_{i=1}^N \delta_{x_i},$$9

This is the “naive score-smoothed diffusion target” whose distortion MM-SOLD wants to correct (Yao et al., 14 May 2026).

The paper describes this log-domain smoothing as known to adapt to data manifolds, but also as very sensitive to the smoothing bandwidth $$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$0. It further states that isotropic kernels without manifold information cause barycentric collapse when $$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$1 is too large. A plausible implication is that MM-SOLD should be understood not as a rejection of score smoothing, but as a constrained correction of its global geometric bias (Yao et al., 14 May 2026).

3. Constrained interacting-particle formulation

MM-SOLD modifies unconstrained overdamped Langevin dynamics by running many particles $$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$2 in parallel and imposing constraints that the empirical mean and covariance of all particles equal those of the data at all times. From the training data, the empirical moments are

$$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$3

with Cholesky factor

$$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$4

For $$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$5 particles stacked in $$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$6, the empirical particle moments are

$$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$7

and MM-SOLD enforces

$$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$8

Introducing whitened coordinates

$$\mathrm{d} X_t = -\alpha X_t \,\mathrm{d}t + \sqrt{2}\,\mathrm{d}B_t,\quad X_0\sim\pi_{\mathrm{data}},\ \alpha\ge 0.$$9

these constraints become

$$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$0

Thus $$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$1 lies on the centered and scaled Stiefel-type manifold

$$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$2

Necessarily $$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$3 (Yao et al., 14 May 2026).

In $$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$4-space, the one-particle potential becomes

$$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$5

For $$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$6 particles $$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$7, define

$$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$8

The corresponding constrained Gibbs equilibrium on $$\hat p_t(z) = \frac{1}{N}\sum_{i=1}^N \mathcal N\!\left(z\,\middle|\,e^{-\alpha t}x_i, \beta_t I_d\right),\quad \beta_t = \frac{1-e^{-2\alpha t}}{\alpha},$$9 is

$$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$0

where $$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$1 is the uniform measure on $$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$2 (Yao et al., 14 May 2026).

The tangent projection machinery is explicit. The centering projection is

$$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$3

the Stiefel tangent projection is

$$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$4

and the full tangent projection is

$$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$5

After each Langevin step, retraction back to $$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$6 is performed by centering $$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$7, taking reduced QR $$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$8, and setting

$$\nabla\log \hat p_t(z) = \frac{1}{\beta_t}\bigl(c_t(z) - z\bigr),$$9

This construction implements hard constraints rather than soft regularization: moments are satisfied at each step, not merely asymptotically (Yao et al., 14 May 2026).

4. Algorithmic realization and limiting law

The implemented discretization is the Leimkuhler–Matthews (LM) Langevin integrator with projection. The algorithm computes empirical $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$0; initializes $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$1 i.i.d. from $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$2; maps to $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$3-coordinates; samples Gaussian noise matrices; projects both drift and noise onto the tangent space of $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$4; performs the LM update; retracts onto $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$5; and finally maps back to $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$6-space (Yao et al., 14 May 2026).

If $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$7 stacks the gradients $$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$8 rowwise, then the gradient in whitened coordinates is

$$c_t(z) = \sum_{i=1}^N w_i^t(z)\,e^{-\alpha t}x_i,\quad w_i^t(z) = \operatorname{softmax}_i\left(-\frac{\|z-e^{-\alpha t}x_j\|^2}{2\beta_t}\right).$$9

The projected LM step is

$t$0

followed by

$t$1

By construction, if $t$2, then after projection and retraction $t$3, hence in $t$4-coordinates each iterate has empirical mean $t$5 and covariance $t$6 (Yao et al., 14 May 2026).

The main theoretical result is the large-particle characterization of the stationary one-particle marginal. Let

$t$7

be the free energy functional, and let

$t$8

Then, for each $t$9, if $\beta_t$0 is the one-particle marginal in $\beta_t$1-space of the constrained Gibbs equilibrium, one has

$\beta_t$2

where

$\beta_t$3

Equivalently,

$\beta_t$4

for some $\beta_t$5 and symmetric $\beta_t$6 (Yao et al., 14 May 2026).

This density is a linear and quadratic exponential tilt of the naive score-smoothed target. The paper interprets it as the closest distribution in KL to $\beta_t$7 among all distributions with mean $\beta_t$8 and covariance $\beta_t$9. The tilting parameters satisfy

$w_i^t(z)$0

and

$w_i^t(z)$1

The proof is described as an equivalence of ensembles argument using whitened coordinates, conditioning on empirical mean and covariance, and applying a local CLT and coarea formula (Yao et al., 14 May 2026).

5. Empirical behavior, complexity, and scope

The computational bottleneck is evaluation of $w_i^t(z)$2 for all particles. Each evaluation requires computing $w_i^t(z)$3 at $w_i^t(z)$4 perturbed points, and direct evaluation would cost $w_i^t(z)$5. To reduce cost, the authors use a nearest-neighbor score estimator: for each query $w_i^t(z)$6, they find $w_i^t(z)$7 nearest neighbors and sample $w_i^t(z)$8 additional training points as a debiasing correction, then evaluate the GMM score using only a subsample of size $w_i^t(z)$9. This reduces the per-iteration cost to $L^p$00, with $L^p$01 (Yao et al., 14 May 2026).

The paper reports experiments on 2D distributions and latent-space image generation. On “Checkerboard” and “Two Spirals,” with 500 training points, $L^p$02 particles, 3000 iterations, step size $L^p$03, and isotropic Gaussian smoothing, the baseline $L^p$04-CFDM is described as very sensitive to $L^p$05 and $L^p$06: small $L^p$07 cause samples to stick close to training points, larger $L^p$08 spreads samples toward the convex hull with many off-support samples, and larger $L^p$09 collapses samples onto local centroids. MM-SOLD is reported as robust to $L^p$10 and $L^p$11, producing diverse samples closely matching target distributions and avoiding off-support collapses (Yao et al., 14 May 2026).

On handwritten digits compressed into a 100D latent space by a pretrained Nuclear Norm-Regularized AutoEncoder, the paper considers both classification and generation. For the classification setting, the reported overall test accuracies are 93.77% for an MLP on original training latents, 97.90% for an MLP with MM-SOLD augmentation, and 98.00% for the moment-matched minimum ECM classifier (Yao et al., 14 May 2026).

For single-class generation of digit “8,” the reported best results over a grid of $L^p$12 are:

Method	KID	Recall	DupRate
MM-SOLD	$L^p$13	$L^p$14	$L^p$15
$L^p$16-CFDM	$L^p$17	$L^p$18	$L^p$19
latent DDPM	$L^p$20	$L^p$21	$L^p$22

The corresponding train times are 0 h for MM-SOLD and $L^p$23-CFDM, and 0.20 h for latent DDPM; sampling times per sample are 2.80 ms for MM-SOLD on CPU, 6.67 ms for $L^p$24-CFDM on CPU, and 9.40 ms for latent DDPM on V100 GPU (Yao et al., 14 May 2026).

For CelebA-HQ $L^p$25 faces encoded into a 700D latent space, the best reported metrics are:

Method	FID	KID	Recall
MM-SOLD	$L^p$26	$L^p$27	$L^p$28
$L^p$29-CFDM	$L^p$30	$L^p$31	$L^p$32
latent DDPM	$L^p$33	$L^p$34	$L^p$35

The corresponding DupRate values are $L^p$36 for MM-SOLD, $L^p$37 for $L^p$38-CFDM, and $L^p$39 for latent DDPM; train times are 0 h for MM-SOLD and $L^p$40-CFDM and 13.15 h for DDPM; sampling times per sample are 45.21 ms on CPU for MM-SOLD, 54.91 ms on CPU for $L^p$41-CFDM, and 16.23 ms on H100 for DDPM (Yao et al., 14 May 2026).

The paper also reports ablations. On the 2D checkerboard task, SW2 to target is relatively flat over $L^p$42 and $L^p$43, consistent with a stability scale $L^p$44. For digit “8,” KID improves as $L^p$45 increases, surpassing kinetic Langevin around a few hundred particles. For fixed $L^p$46, MM-SOLD outperforms kinetic Langevin for small $L^p$47 (Yao et al., 14 May 2026).

The stated limitations are that the main theory assumes Gaussian smoothing noise, full-rank empirical covariance, isotropic GMM with $L^p$48, and characterizes only the infinite-particle stationary marginal; uniform-in-$L^p$49 discretization error bounds for the LM scheme with projection are not provided. The quadratic tilt $L^p$50 has $L^p$51 parameters, and direct fitting and sampling from the explicit tilted target become challenging in high dimensions. The paper further states that while MM-SOLD significantly improves over naive score smoothing and $L^p$52-CFDM, it still does not match state-of-the-art image diffusion models on very complex image manifolds (Yao et al., 14 May 2026).

6. Relations to robustness theory and adjacent Langevin methodologies

MM-SOLD is built from a modified score field inside overdamped Langevin dynamics, and that places it in direct conceptual contact with recent work on the robustness of Langevin samplers under score error. “On the Robustness of Langevin Dynamics to Score Function Error” studies continuous- and discrete-time Langevin dynamics with an estimated score $L^p$53 replacing the true score, and proves that Langevin dynamics is not robust to $L^p$54 errors, more generally $L^p$55 errors, in the estimate of the score function (Cao et al., 11 Mar 2026).

For the main finite-time lower bounds, the target distribution is an isotropic Gaussian $L^p$56 or $L^p$57, and the paper constructs Lipschitz score fields with exponentially small $L^p$58 error such that the resulting Langevin dynamics remains far from the target in total variation for all time horizons $L^p$59. One theorem uses $L^p$60 and a modified score that is correct in the outer region but induces an inner Ornstein–Uhlenbeck well, yielding

$L^p$61

Another theorem uses data-based initialization from the training samples themselves and a score field that places strongly mean-reverting OU wells around those samples, again with exponentially small $L^p$62 error and total variation nearly one over any polynomial time horizon. A third theorem constructs piecewise Lipschitz score errors for general targets, with arbitrarily small $L^p$63 error, such that asymptotically the Langevin mass concentrates in a bad cone of arbitrarily small $L^p$64-mass (Cao et al., 11 Mar 2026).

These results do not analyze MM-SOLD explicitly, but the source text states that a method like MM-SOLD fits squarely into the class

$L^p$65

with $L^p$66 a smoothed, moment-matched, or corrected score derived from a learned network or other approximation. The stated implication is that small $L^p$67 error alone is insufficient for such methods: even Lipschitz, globally smooth, exponentially accurate score fields can create OU-like traps or attractors in low-probability but dynamically accessible regions (Cao et al., 11 Mar 2026). This suggests that the practical success of MM-SOLD should not be read as a generic positive guarantee for all approximate-score Langevin schemes, but as evidence that its particular smoothing-and-constraint design can be useful in the tested settings.

The broader Langevin literature in the provided sources offers several adjacent perspectives. “Simulation-based Inference via Langevin Dynamics with Score Matching” proposes a score-based SBI method in which a neural score model approximates a likelihood score and is regularized by mean-zero and curvature moment identities; the resulting sampler is explicitly overdamped Langevin Monte Carlo with an approximate score (Jiang et al., 4 Sep 2025). “Sampling by averaging: A multiscale approach to score estimation” replaces explicit score estimation along a diffusion path by stochastic averaging in a slow–fast SDE and develops MultALMC and MultCDiff as training-free samplers using multiscale annealed Langevin dynamics or multiscale controlled diffusions (Cordero-Encinar et al., 20 Aug 2025). “Optimizing the diffusion coefficient of overdamped Langevin dynamics” studies reversible overdamped Langevin dynamics with a state-dependent diffusion matrix $L^p$68, showing that suitable choices of $L^p$69 can substantially increase the spectral gap; in the 1D homogenized limit, the optimal diffusion is

$L^p$70

that is, proportional to the inverse of the target density (Lelièvre et al., 2024).

Taken together, these papers situate MM-SOLD at the intersection of training-free score construction, moment-constrained interacting particle systems, and overdamped Langevin methodology. A plausible implication is that MM-SOLD’s distinctive contribution is not merely the use of a smoothed score, but the coupling of that score to an exactly enforced empirical first- and second-moment geometry, which the paper identifies as the mechanism preventing the global barycentric distortion of naive score smoothing (Yao et al., 14 May 2026).