Moment Guided Diffusion (MGD)

• MGD is a diffusion framework that explicitly matches prescribed moment constraints to sample from maximum-entropy distributions.
• It uses a predictor-corrector Euler–Maruyama scheme to balance drift and noise, ensuring efficient and robust convergence even in high dimensions.
• The method provides a computable entropy identity and negentropy estimator, making it practical for applications like structured data and overcoming energy barriers.

Moment Guided Diffusion (MGD) denotes a class of diffusion-based procedures in which moment information guides generation. In its most specific use, “MGD: Moment Guided Diffusion for Maximum Entropy Generation” defines a generative framework that produces samples from maximum-entropy distributions specified by a finite set of moment constraints, combining the non-equilibrium, finite-time transport of diffusion or flow models with Jaynes’ maximum-entropy principle (Lempereur et al., 19 Feb 2026). Related diffusion papers use closely adjacent notions of moment guidance—either through learned moment-like image features or through conditional moment matching in distillation—which suggests that the label is not yet fully standardized across subfields (Jung et al., 18 May 2025, Salimans et al., 2024).

1. Terminological scope and naming

The exact name “Moment Guided Diffusion” is used by “MGD: Moment Guided Diffusion for Maximum Entropy Generation” (Lempereur et al., 19 Feb 2026). In that work, MGD is a stochastic-process framework for sampling a maximum-entropy law under prescribed moment constraints.

A nearby but distinct use appears in “Guiding Diffusion with Deep Geometric Moments: Balancing Fidelity and Variation” (Jung et al., 18 May 2025). That paper does not use the term “MGD”; it uses “Deep Geometric Moments” (DGM) and refers to the method as DGM guidance. Its own clarification states that, if one refers to “a diffusion sampling procedure that is guided by moment-like features,” that corresponds to its training-free DGM-guided sampling, with DGM providing the learned “moment” prior.

A third usage appears in “Multistep Distillation of Diffusion Models via Moment Matching” (Salimans et al., 2024). That paper does not name its method MGD, but it explicitly presents the distillation procedure as “exactly ‘moment-guided’ in the sense of enforcing conditional moment constraints to guide the student diffusion model.” The relevant moment is the first conditional moment, namely the conditional mean of the clean data given a noisy observation.

By contrast, “Guided Motion Diffusion for Controllable Human Motion Synthesis” abbreviates to GMD, not MGD (Karunratanakul et al., 2023).

Paper	Term used in the paper	Guiding object
(Lempereur et al., 19 Feb 2026)	MGD	Finite set of prescribed moments
(Jung et al., 18 May 2025)	DGM guidance	Deep Geometric Moments feature prior
(Salimans et al., 2024)	Multistep distillation via moment matching	First conditional moment
(Karunratanakul et al., 2023)	GMD	Spatial constraints for motion

2. Maximum-entropy formulation

In the maximum-entropy formulation, the starting point is limited information about a random vector $X \in \mathbb{R}^d$ in the form of moment constraints

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$

where the observables are collected into

$\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$

and

$m = (m_1,\dots,m_k).$

Among all admissible distributions $p$ that satisfy the constraints, Jaynes’ principle selects the least informative one by maximizing Shannon’s differential entropy

$H(p) = -\int p(x)\log p(x)\,dx$

subject to normalization and moment matching (Lempereur et al., 19 Feb 2026).

The corresponding Lagrangian is

$L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$

with $\theta \in \mathbb{R}^k$. When a maximizer exists, the solution has exponential form

$$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$

and the optimal $\theta=\theta_*$ enforces $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$0.

Classical algorithms fit $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$1 by optimizing

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$2

with gradients

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$3

The difficulty is that sampling $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$4 and estimating $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$5 typically requires MCMC, such as Langevin or MALA, which mixes slowly in high dimension and can suffer exponentially in barrier height for non-convex energies. MGD is positioned precisely at this interface: it preserves a finite set of moments along a non-equilibrium transport, injects a tunable volatility that promotes exploration and entropy increase, and in the large-volatility limit converges to the maximum-entropy target while also providing an entropy estimator along the dynamics (Lempereur et al., 19 Feb 2026).

3. Stochastic interpolants and exact moment guidance

MGD adopts the stochastic interpolant framework to define a path of target moments $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$6 between an initial base distribution $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$7 and the desired constraints at terminal time $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$8. A convenient interpolant between Gaussian noise $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$9 and data $\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$0 is

$\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$1

with $\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$2 and $\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$3, so that $\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$4 and $\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$5. The target moment path is then

$\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$6

MGD constructs a diffusion $\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$7 whose moments match this path exactly:

$\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$8

The MGD SDE is

$\phi(x) = (f_1(x),\dots,f_k(x)) \in \mathbb{R}^k$9

where $m = (m_1,\dots,m_k).$0 is a tunable volatility and $m = (m_1,\dots,m_k).$1 are chosen by solving

$m = (m_1,\dots,m_k).$2

with Gram matrix

$m = (m_1,\dots,m_k).$3

Applying Itô’s lemma to $m = (m_1,\dots,m_k).$4 and taking expectations yields

$m = (m_1,\dots,m_k).$5

With the definitions above,

$m = (m_1,\dots,m_k).$6

and hence $m = (m_1,\dots,m_k).$7 for all $m = (m_1,\dots,m_k).$8 provided $m = (m_1,\dots,m_k).$9.

The associated Fokker–Planck equation for the PDF $p$0 is

$p$1

This decomposition gives the method its characteristic interpretation. The drift

$p$2

is a moment-guiding drift; $p$3 advances moments along the prescribed path $p$4, while the score-like component $p$5 compensates the moment changes induced by diffusion. Unlike a stochastic-interpolant SDE that matches the full density $p$6 of $p$7, MGD does not model the full score or density $p$8; it matches the moments $p$9 exactly while injecting noise. In the terminology of the paper, this makes it robust and data-efficient when only moment information is available or preferred (Lempereur et al., 19 Feb 2026).

4. Large-volatility limit, convergence, and entropy production

The core theoretical claim of MGD is that large volatility pushes the dynamics toward the maximum-entropy law consistent with the prescribed moments. Formally keeping leading-order terms in the Fokker–Planck equation as $H(p) = -\int p(x)\log p(x)\,dx$0 becomes large gives the stationary PDE

$H(p) = -\int p(x)\log p(x)\,dx$1

whose solutions are exponential families

$H(p) = -\int p(x)\log p(x)\,dx$2

with $H(p) = -\int p(x)\log p(x)\,dx$3 chosen so that $H(p) = -\int p(x)\log p(x)\,dx$4. In particular, at $H(p) = -\int p(x)\log p(x)\,dx$5 one obtains the maximum-entropy distribution $H(p) = -\int p(x)\log p(x)\,dx$6 with moments $H(p) = -\int p(x)\log p(x)\,dx$7 (Lempereur et al., 19 Feb 2026).

The paper’s main conjecture states that there exists $H(p) = -\int p(x)\log p(x)\,dx$8 such that

$H(p) = -\int p(x)\log p(x)\,dx$9

Because $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$0 and $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$1 share the same constrained moments, this is equivalently

$L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$2

Rigorous results are given in special cases and under confinement. For quadratic $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$3, namely $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$4, the MGD SDE becomes linear and can be solved explicitly:

$L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$5

where $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$6 is the covariance of the interpolant $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$7. If $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$8 is Gaussian, $L(p,\theta) = H(p) - \theta^T(E_p[\phi]-m),$9 is Gaussian with covariance $\theta \in \mathbb{R}^k$0 for any $\theta \in \mathbb{R}^k$1, so MGD exactly samples the maximum-entropy law for all $\theta \in \mathbb{R}^k$2. More generally, for non-Gaussian $\theta \in \mathbb{R}^k$3 but with commuting covariances, $\theta \in \mathbb{R}^k$4 as $\theta \in \mathbb{R}^k$5. The paper also proves existence and convergence for a regularized MGD with confinement, targeting

$\theta \in \mathbb{R}^k$6

and obtains a quantitative $\theta \in \mathbb{R}^k$7 rate under a Poincaré/log-Sobolev type condition and small moment deviations (Lempereur et al., 19 Feb 2026).

A second theoretical contribution is an entropy production identity. Writing

$\theta \in \mathbb{R}^k$8

the paper proves

$\theta \in \mathbb{R}^k$9

Integrating from $$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$0 to $$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$1 yields the computable lower bound

$$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$2

The gap is an integrated Fisher divergence:

$$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$3

In the large-$$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$4 limit, this gap vanishes if $$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$5 is itself exponential in $$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$6, implying $$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$7.

This entropy identity is also the basis for negentropy estimation. For a density $$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$8 with covariance $$p_\theta(x) = Z_\theta^{-1} e^{-\theta^T\phi(x)}, \qquad Z_\theta = \int e^{-\theta^T\phi(x)}\,dx,$$9, the negentropy is defined by

$\theta=\theta_*$0

where $\theta=\theta_*$1 is Gaussian with the same covariance. Since $\theta=\theta_*$2 is generally intractable in high dimension, MGD uses the lower bound $\theta=\theta_*$3 to form a tractable lower bound on negentropy per dimension:

$\theta=\theta_*$4

The paper presents this as a practical route to information-theoretic diagnostics directly from the dynamics, without evaluating the target density explicitly (Lempereur et al., 19 Feb 2026).

5. Numerical scheme, complexity, and design choices

The computational realization of MGD is a predictor-corrector Euler–Maruyama scheme that maintains the moment path exactly at each time step while injecting noise proportional to $\theta=\theta_*$5. The inputs are the moment functions $\theta=\theta_*$6, a target path $\theta=\theta_*$7, an initial law $\theta=\theta_*$8 that is often Gaussian, a volatility $\theta=\theta_*$9, a number of steps $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$00 so that $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$01, and a number of replicas $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$02 (Lempereur et al., 19 Feb 2026).

At time step $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$03 with $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$04, the predictor estimates the empirical Gram matrix

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$05

solves

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$06

and updates each particle by

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$07

The corrector recomputes the Gram matrix on predicted particles,

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$08

solves

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$09

and then projects particles back to the desired moment path:

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$10

The output is the final ensemble $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$11 together with the entropy bound

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$12

The complexity profile is explicit. The number of steps is $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$13. Per step, forming Gram matrices costs $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$14 and solving the $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$15 linear systems costs $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$16. The overall cost is therefore $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$17 in steps and depends polynomially on the number of moments $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$18. The paper emphasizes that, unlike MALA, this complexity does not blow up with barrier heights (Lempereur et al., 19 Feb 2026).

Several numerical design choices are singled out. Gram inversions can be stabilized by a small ridge $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$19 and normalization of rows of $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$20. One should use enough replicas, with examples in the range $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$21–$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$22, to stabilize empirical Gram estimates and moment matching. For non-smooth $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$23, such as $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$24 or scattering moduli, the predictor-corrector formulation avoids direct estimation of $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$25, which may be distributional. The paper also describes efficiency variants: precomputed transport, in which $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$26 is regressed offline from $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$27; offline coefficient learning for $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$28 on a time grid; and standard regularization devices for numerical robustness (Lempereur et al., 19 Feb 2026).

6. Applications and empirical behavior

The main empirical setting for high-dimensional structured data uses wavelet scattering moments. Here $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$29 is assembled from first-order amplitudes and energies,

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$30

cross-scale phase-sensitive correlations,

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$31

and cross-scale amplitude correlations,

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$32

The full vector $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$33 has dimension $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$34 and captures multiscale, orientation-dependent, and phase-asymmetric structures with lower variance than raw higher-order moments (Lempereur et al., 19 Feb 2026).

In one-dimensional synthetic tests, including bimodal and Laplacian targets, the paper reports that both $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$35 and $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$36 decay as $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$37. It also reports a barrier-insensitivity effect: for

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$38

MALA step counts to reach a fixed KL error grow exponentially in $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$39, whereas MGD requires an essentially constant number of steps because the homotopic transport places particles into correct modes early and maintains them as the landscape sharpens (Lempereur et al., 19 Feb 2026).

The financial application uses S&P 500 daily log-returns with dimension $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$40 and $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$41 scattering moments. The negentropy estimate $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$42 decreases with $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$43 and plateaus around $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$44, with estimated negentropy per dimension $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$45. Generated rolling volatility histograms match the data at converged $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$46 and differ markedly from the Gaussian with the same covariance.

For two-dimensional physical systems, the paper reports two benchmark regimes. In cosmic web dark-matter slices with $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$47 and $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$48, $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$49 plateaus by $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$50 and the estimated negentropy is $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$51. In two-dimensional turbulence with the same $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$52 and $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$53, $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$54, reflecting much stronger non-Gaussian structure; convergence is again reported as $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$55. The qualitative failure mode at small $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$56 is also consistent across applications: samples become under-entropic, with spiky modes, short tails, or too many near-zero fine-scale wavelet coefficients. Increasing $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$57 restores correct tails and spatial structures (Lempereur et al., 19 Feb 2026).

A deliberately misspecified scenario is also included. On CelebA faces with stationary scattering moments, where $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$58 is a poor model for the non-stationary data, MGD still converges quickly to $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$59 and the entropy bound again shows $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$60 convergence. The stated significance is that MGD can efficiently sample the maximum-entropy law defined by $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$61 even when the model error $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$62 is large (Lempereur et al., 19 Feb 2026).

7. Related diffusion usages, limitations, and future directions

Outside the maximum-entropy setting, two neighboring lines of work attach “moment guidance” to different objects. In DGM-guided text-to-image sampling, the guiding quantity is a learned, moment-inspired embedding rather than an explicit maximum-entropy constraint. The frozen DGM encoder $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$63 defines a feature-matching loss

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$64

with clean-image estimate

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$65

gradient

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$66

and guided noise prediction

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$67

The paper presents this as a training-free way to balance fidelity and diversity by matching a compact, geometry-aware feature vector of the subject rather than imposing segmentation or depth constraints. Its reported training-free quantitative block places DGM at CLIP-I $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$68 and I-DINO $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$69, compared with DINO guidance at CLIP-I $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$70, I-DINO $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$71, ResNet34 guidance at CLIP-I $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$72, I-DINO $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$73, CLIP guidance at CLIP-I $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$74, I-DINO $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$75, and segmentation maps at CLIP-I $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$76, I-DINO $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$77 (Jung et al., 18 May 2025).

In multistep distillation, the relevant moment is the first conditional moment of clean data. The central constraint is

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$78

where the teacher moment is

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$79

and the student moment is

$E[f_i(X)] = m_i,\qquad i=1,\dots,k,$80

The paper gives two practical implementations: alternating moment matching with an auxiliary denoiser $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$81, and instant parameter-space moment matching, formulated as an Efficient Method of Moments view. This multistep moment-guided perspective is used to distill many-step diffusion models into few-step models, with reported ImageNet results such as FID $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$82 and IS $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$83 for the alternating $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$84 model on class-conditional ImageNet $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$85, compared with teacher FID $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$86 and IS $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$87 (Salimans et al., 2024).

The maximum-entropy MGD framework also has explicit limitations. General global convergence to $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$88 at $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$89 is conjectured broadly but only proved under additional conditions. Existence and uniqueness of MGD solutions can fail if Gram matrices become singular. The quality of the maximum-entropy model depends on the choice of $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$90; if $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$91 is misspecified, the resulting $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$92 may be far from the true data distribution. Computational cost grows with $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$93 and with the number of moments $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$94, so careful engineering of $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$95 and solvers remains important (Lempereur et al., 19 Feb 2026).

The future directions stated for the maximum-entropy formulation are adaptive moment selection, learned volatility schedules $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$96, hybridization with data-driven score terms, higher-order integrators and variance-reduced estimators for $E[f_i(X)] = m_i,\qquad i=1,\dots,k,$97, and stronger theoretical guarantees, including global convergence without confinement, sharper rates, and finite-sample bounds for the entropy estimator. A plausible implication is that “Moment Guided Diffusion” will continue to denote more than one technical program unless the literature settles on a narrower naming convention: one centered on maximum-entropy moment-constrained sampling, and another centered on diffusion guidance or distillation by moment-like statistics.