Derivative-Free Diffusion Manifold-Constrained Gradient for Unified XAI

Byeongsu Sim; Hyungjin Chung; Jaemin Kim; Jong Chul Ye; Sangmin Lee; Won Jun Kim

REVIEW 1 major objections 6 minor 1 cited by

Derivative-Free Diffusion Manifold-Constrained Gradient for Unified XAI

T0 review · 1 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper claims that the gradient of a black-box classifier can be approximated on the data manifold from output scores alone, and that the same estimate powers both feature attribution and counterfactual generation.

desk verdict A useful black-box XAI package built on known EnKF machinery, but the central manifold-gradient theorem is not supported under the paper's own operating conditions. read the letter →

arxiv 2411.15265 v2 pith:ZSW3VQ5S submitted 2024-11-22 cs.CV cs.LG

Won Jun Kim , Hyungjin Chung , Jaemin Kim , Sangmin Lee , Byeongsu Sim , Jong Chul Ye This is my paper · ORCID

classification cs.CVcs.LG

keywords explainableAIfeatureattributioncounterfactualexplanationgradient-freeoptimizationdiffusionmodelsensembleKalmanfiltermanifoldlearningblack-boxmodel

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper sets out to fix three failures of gradient-based explainability: it usually needs white-box access to model weights, it is easily fooled into adversarial directions, and it produces attributions that leave the image manifold. The proposed method, FreeMCG, approximates a model's input gradient projected onto the data manifold using only the model's output logits and a pretrained diffusion model. It does this by generating an ensemble of on-manifold image perturbations and reading the gradient off the covariance between those perturbations and the classifier's outputs. The paper then shows that this single derivative-free estimate can serve both major XAI tasks: feature attribution maps and counterfactual explanations. If correct, it means a fully black-box service can be explained as faithfully as a white-box one, and the two traditionally separate explanation tasks collapse into one operation.

What carries the argument

The load-bearing object is the FreeMCG estimator $g_{\mathrm{Free}} = \mathbb{E}\big[(x_{0|t}^{(k)}-\bar{x}_{0|t})(f(x_{0|t}^{(k)})-\bar{f})^\top\big](e_c-p)$, an ensemble Kalman-filter-style cross-covariance between Tweedie-denoised image particles and the classifier's logit outputs, multiplied by the softmax residual $e_c-p$. It carries the argument in three roles: Theorem 2 shows the cross-covariance equals $C_{xx}\nabla_x f(x)^\top$ plus a cubic error, so the formula is a genuine gradient surrogate; Theorem 1 shows the $C_{xx}$ preconditioner keeps the resulting direction tangent to the data manifold; and the diffusion denoiser provides the particles that satisfy the on-manifold premise in one step. This single object is what lets the paper treat feature attribution and counterfactual generation as two uses of the same black-box gradient.

What would settle it

Compute the exact preconditioned gradient $C_{xx}\nabla_x f(x)^\top$ by backpropagation for a set of ImageNet images and compare it with the FreeMCG estimate at increasing timesteps $t\in\{100,200,300,400,500,600,700\}$; if the relative error does not stay small as the particle spread $\delta$ grows, or if replacing the diffusion model with one trained on a different domain destroys the ROAD ranking, then the $O(\delta^3)$ approximation is not what carries the method.

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Extended reading notes

Core claim

The central discovery is that the preconditioned gradient $C_{xx}\nabla_x \log p(y=c|x)$—the gradient of the classifier's class probability multiplied by the empirical covariance of nearby on-manifold particles—can be computed without the Jacobian $\partial f/\partial x$. Because $C_{xf} = \frac{1}{K}\sum_k (x^{(k)}-\bar{x})(f(x^{(k)})-\bar{f})^\top$ approximates $C_{xx}\nabla_x f(x)^\top$ up to $O(\delta^3)$ when the particles stay within distance $\delta$ of their mean, the paper defines $g_{\mathrm{Free}} = C_{xf}(e_c - p)$ as a derivative-free estimate of the manifold-constrained gradient. Diffusion models supply the particles: forward-diffuse the input, apply Tweedie's formula to denoise, and the resulting particles lie approximately on the manifold, so the covariance preconditioner expands tangent directions and contracts normal ones. The same $g_{\mathrm{Free}}$ is then used as the attribution map and as the ascent direction for counterfactuals, with only the target class vector $e_c-p$ changed.

Load-bearing premise

The entire estimate works only if the diffusion-denoised copies of the input stay within a tiny neighborhood of the input and on a locally flat part of the image manifold; the paper uses them at noise levels where this has not been checked, so if those copies drift off, the 'gradient' is no longer the model's gradient.

Editorial extensions

If this is right

An image classifier that exposes only its softmax outputs can be explained with attribution maps competitive with white-box methods such as Integrated Gradients.
Counterfactual explanation can be generated without access to classifier weights or adversarial robustness training, and the resulting images are perceived as more semantically changed and more realistic than gradient-based baselines.
A single gradient estimate now serves both feature attribution and counterfactual generation, so the two XAI pipelines no longer need separate machinery.
Because the estimated gradient lies on the image manifold, explanations should be more robust to the off-manifold adversarial perturbations that plague vanilla gradients.
The method also works in the latent space of Stable Diffusion and on chest X-rays with a domain-trained diffusion model, indicating the framework transfers across image domains.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not explore it, but the same covariance trick may extend beyond images: any black-box function evaluated on on-manifold perturbations could be locally explained in the same way, which would open XAI to API-only models in medicine, finance, or robotics.
The heavy reliance on a pretrained diffusion model suggests a testable boundary condition: when the diffusion model is trained on a different distribution than the classifier, the 'manifold' it imposes may not be the classifier's decision manifold, so explanations could be systematically biased by the generative prior.
One reading of the results is that FreeMCG counterfactuals target a human-aligned boundary on the data manifold rather than the classifier's true adversarial boundary, which would reframe what a counterfactual explanation should be optimizing.
The method's success on a small six-class chest X-ray classifier with a custom diffusion model suggests a practical path for deploying black-box XAI in clinical settings, but this is an extrapolation from the paper's two datasets.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Two things to know. First, this is a serious, readable attempt to unify feature attribution and counterfactual explanation under one black-box gradient estimate, with real experiments on ImageNet and MIMIC-CXR, including a human study. Second, the theoretical justification is weaker than it looks: the theorem that makes FreeMCG a manifold-constrained gradient at x assumes particles lie in a tiny delta-ball around their mean, and the operating regime (t up to 700 for attribution, t'=400 for counterfactuals) likely violates that. I side with the stress-test note here. At high noise levels, Tweedie-denoised particles sit near the unconditional mean, not near x, so Theorem 2's Taylor remainder is not small and Eq. (9) is not demonstrated at x. The useful behavior of FreeMCG probably comes from a nonlocal, ensemble-level aggregation, not from the theorem as stated.

What is actually new: the EnKF derivative-free covariance approximation and diffusion guidance both appear in prior work [55], but applying this machinery to classification XAI and packaging it for both attribution and counterfactuals is a legitimate extension. The black-box framing is genuinely practical, and the experiments include white-box baselines plus a CXR transfer. The authors also deserve credit for reporting hyperparameter details and using only 100 particles, far fewer than the EnKF literature typically uses.

The soft spots are real but not fatal if framed honestly. Theorem 1 is close to a tautology: if particles lie on the manifold, their deviations span the tangent direction. Theorem 2 is a standard Taylor argument, but its assumptions are unverified in the regime the paper actually uses. Algorithm 3 introduces per-particle probabilities p(k), while the derivation in Eq. (10) uses a fixed p(x); that mismatch is never addressed. The counterfactual evaluation is also weakened by per-example beta selection and no error bars, so calling the results state-of-the-art overclaims when the flip rate is 51.5 versus 85.8 for DVCE, and the CXR table shows worse L2 and FID. The higher perceived change scores are meaningful, but they do not alone establish superiority.

Who is this for? Researchers working on black-box XAI or diffusion-based counterfactuals will get value, and the paper is a good reading-group starting point for discussing why ensemble methods work even when their stated assumptions fail. My recommendation: send it to serious peer review, not desk reject, but require major revision—verify or soften the theorems, fix the Algorithm 3 mismatch, report error bars with fixed hyperparameters, and release code.

Referee Report

1 major / 6 minor

Summary. The paper proposes FreeMCG, a black-box explainability method that approximates a classifier's input gradient preconditioned by the empirical covariance of diffusion-denoised image particles. The central derivation combines Tweedie's formula, an ensemble Kalman filter-style covariance estimate, and a Taylor expansion (Theorem 2) to replace the unavailable Jacobian-gradient product C_xx ∇f(x)^T with a derivative-free cross-covariance C_xf computed from classifier outputs only. The same gradient estimate is then applied to two downstream tasks: feature attribution (via a weighted average over denoised particles) and counterfactual explanation (via reverse-diffusion guided sampling). Experiments on ImageNet and MIMIC-CXR compare FreeMCG against DVCE and LDCE for counterfactuals and against gradient-based attributions plus SHAP under ROAD evaluation.

Significance. If the theoretical derivation were solid, FreeMCG would be a valuable unified, derivative-free XAI framework: it would offer a single gradient-like object for both feature attribution and counterfactual explanation without model-weight access, and the empirical results include human-subject evaluation and cross-architecture counterfactual demonstrations. The use of EnKF-style covariance from diffusion particles is a promising idea, and the derivative-free property is practically important. However, the paper's central claim that gFree is a manifold-constrained gradient at the input x is currently supported only by theorems whose assumptions are not met in the operating regime, so the significance is conditional on a substantial revision of either the theory or the experimental protocol.

major comments (1)

minor comments (6)

[Abstract and title] The abstract contains the typo “Manifold-Contrained”; the title uses “Manifold-Constrained”.
[Algorithm 3, lines 6–7] Algorithm 3 computes a per-particle probability p^(k) = softmax(f([x_k])) and passes it to FREE MCG, whereas the derivation in Eq. (10) uses a fixed p = p(x). This makes the implemented estimator different from the derived one; the pseudocode should be aligned with the derivation, or a per-particle variant should be derived explicitly.
[Algorithm 3, line 6] The notation [x_k] is ambiguous; it should presumably be the Tweedie-denoised estimate [x_{0|t}^{(k)}] rather than the noisy particle [x_t^{(k)}].
[Appendix A, Theorem 2 proof] The theorem statement writes ∇_x f(x), but the proof expands around x̄ and the final line conflates the two centers; this notation should be corrected to avoid ambiguity about which point the Jacobian is evaluated at.
[Theorem 1] The proof of Theorem 1 is essentially a restatement of the construction: if all particles lie on the manifold, then C_xx b lies in the span of tangent vectors by definition. This should be presented as intuition or a direct consequence, not as a substantive theorem, and the claim that C_xx “contracts” normal components should be quantified for finite sample size.
[Eq. (8) and Fig. 2] The notation gFree := C_xx ∇_x log p(c|x) is introduced before C_xx is defined; consider reordering or adding a one-line definition. Also, the equation in Fig. 2 appears to omit the argument of ∇f; it should read something like C_xx (∂f/∂x)^T(e_c − p).

Circularity Check

2 steps flagged · score 3.0 of 10

The derivative-free EnKF/Taylor approximation is genuinely independent, but the advertised 'on-manifold' property of FreeMCG is built into the estimator's definition and the on-manifold premise is imported from the authors' own prior work.

self definitional [Section 3.1, Eq. (8) and Theorem 1 (proof in Appendix A)]
"gFree := Cxx∇x log p(c|x) = Cxx(∂f(x)/∂x)^⊤(ec − p(x)), (8) ... Theorem 1. Suppose that M is locally linear at x ... Suppose further, that the particles x(k) ∈ M. Then, gFree approximately lies within M. Precisely, Cxx acts as a linear transformation, expanding along TxM and contracting the vectors normal to TxM."

The proof of Theorem 1 in Appendix A shows Cxx·b = (1/K)Σ_k (x(k)−x̄)((x(k)−x̄)^⊤·b), so gFree is, by its definition in (8), an affine combination of the particle deviations x(k)−x̄. Once the premise 'x(k) ∈ M' is granted, 'gFree approximately lies in M' is a restatement of that premise plus the algebraic form of the estimator. The advertised manifold-constrained character is therefore an input (on-manifold particles) presented as an output theorem, not an independent first-principles result.
self citation load bearing [Section 3.1, immediately after Eq. (11); operating points in Alg. 2 and Appendix E]
"Notice that under our assumption of local linearity, we can use Proposition 2 of [7], which states that Tweedie’s formula [14] is locally an orthogonal projection to the manifold M, satisfying the assumption of Thm. 1."

The premise that makes Theorem 1 applicable—that Tweedie-denoised particles x(k)_{0|t} lie on M—is not established here; it is imported from Proposition 2 of the authors' own prior work [7]. The paper supplies no proof or independent check of that proposition, and the practical operating regime (Alg. 2 draws t up to 700; Alg. 3 uses t′ = 400) is far outside the small-δ local-linearity regime stated in Theorem 2. Thus the central 'on-manifold gradient' claim rests on a load-bearing self-citation whose assumptions are not verified at the points where FreeMCG is actually run.

full rationale

The core derivative-free approximation Cxf ≈ Cxx∇xf(x)^⊤ (Theorem 2) is a genuine Taylor/EnKF result and the XAI benchmarks (ROAD, flip rate, human study) are external, so the paper is not wholly circular. However, the headline 'manifold-constrained' property is partially circular: gFree is defined as a covariance-weighted combination of particles, so its lying on M is a direct consequence of the on-manifold-particle premise, and that premise is supplied by a self-citation to Proposition 2 of [7] rather than by verification in the operating regime. These issues undermine the strength of the central claim but do not reduce the entire derivation to a fit, so a moderate score is appropriate.

Assumptions & free parameters 5 free parameters · 4 assumptions · 0 invented entities

The method introduces no new physical or abstract entities. Its central claim rests on manifold geometry assumptions, a Taylor expansion through a black-box classifier, and several experimentally chosen hyperparameters.

free parameters (5)

Diffusion timestep range for feature attribution = t in [100, 200, 300, 400, 500, 600, 700], or normalized [0.1, 0.7]
Chosen by the authors to capture coarse and fine attributions; no sensitivity analysis is provided.
Number of ensemble particles K = 100
Reported as much smaller than the 10,000+ particles used in EnKF diffusion guidance [55]; no study of K's effect.
Counterfactual reverse diffusion timestep t' = 400 for ImageNet, 300 for MIMIC-CXR
Set experimentally; affects generation fidelity and is not derived.
Gradient ascent step size alpha = 0.2
Set experimentally for counterfactual generation.
Proximal regularization beta = 0.01 or 0.02, selected per example for the 'most informative' result
Hyperparameter selection during evaluation is a selection risk and inflates perceived quality.

assumptions (4)

domain assumption The data manifold M is locally linear at x and particles x(k) lie on M near x.
Invoked in Section 3.1 and Theorem 1 to conclude Cxx projects gradients onto the tangent space; not verified for Tweedie-denoised particles.
domain assumption The classifier f has uniformly bounded Hessian and all particles are within delta of the mean, so the O(delta^3) Taylor error is small.
Needed for Theorem 2 and Eq. (9); neural network Hessians are not bounded in this sense and delta is not controlled.
domain assumption Tweedie denoising is locally an orthogonal projection onto the data manifold, as per Proposition 2 of [7].
Used to justify that x_{0|t} particles are on-manifold; this relies on the authors' prior work.
domain assumption The diffusion model D_theta accurately estimates E[x0|xt] well enough for particle generation.
Standard denoising objective in Eq. (3), but model error and the choice of timestep range affect particle quality.

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Cite this review

Pith. "Pith review of Derivative-Free Diffusion Manifold-Constrained Gradient for Unified XAI." pith.science (2026). https://pith.science/paper/ZSW3VQ5S

@misc{pith2026241115265,
  author       = {Pith},
  title        = {Pith review of: Derivative-Free Diffusion Manifold-Constrained Gradient for Unified XAI},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZSW3VQ5S}},
  note         = {Machine review of arXiv:2411.15265}
}

read the original abstract

Gradient-based methods are a prototypical family of explainability techniques, especially for image-based models. Nonetheless, they have several shortcomings in that they (1) require white-box access to models, (2) are vulnerable to adversarial attacks, and (3) produce attributions that lie off the image manifold, leading to explanations that are not actually faithful to the model and do not align well with human perception. To overcome these challenges, we introduce Derivative-Free Diffusion Manifold-Constrainted Gradients (FreeMCG), a novel method that serves as an improved basis for explainability of a given neural network than the traditional gradient. Specifically, by leveraging ensemble Kalman filters and diffusion models, we derive a derivative-free approximation of the model's gradient projected onto the data manifold, requiring access only to the model's outputs. We demonstrate the effectiveness of FreeMCG by applying it to both counterfactual generation and feature attribution, which have traditionally been treated as distinct tasks. Through comprehensive evaluation on both tasks, counterfactual explanation and feature attribution, we show that our method yields state-of-the-art results while preserving the essential properties expected of XAI tools.

Figures

Figures reproduced from arXiv: 2411.15265 by the authors.

**Figure 1.** Representative results of FreeMCG, operating as a universal framework for black box XAI. Among various XAI methods, the gradient of the target (e.g. classifier) model is the prototypical method of understanding the local behavior of a function around a given input data point. It has thus been used since the beginning of the rise of neural networks to explain their decisions [43]. There have been many subsequent w… view at source ↗

**Figure 2.** Unified framework of XAI for feature attribution and counterfactual explanation using our method, Derivative-Free Diffusion [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Toy example of FreeMCG, illustrating the gradient (red arrow) and the gradient aligned with the tangent of manifold (gray dashed) approximated by (9) (green arrow), using neighbor data points (blue dots). Theorem 1 states that when the particles x (k) lies on the manifold surrounding the original data point x, Cxx captures the geometric properties of the tangent space TxM. Specifically, the covariance matrix refine… view at source ↗

Figures from the paper (8 more)

**Figure 4.** Figure 4: Counterfactual Generation. Images on the leftmost column are given as input, and the counterfactual for the target class (shown in blue) are generated using each method. Baseline methods generate counterfactuals by perturbing the image towards the direction of a given …

**Figure 5.** Figure 5: (a) Feature attribution maps obtained using each method, (b) Corresponding ROAD [ [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

**Figure 6.** Figure 6: Disease→Normal class counterfactuals produced using FreeMCG for each type of disease lesion. In Disease→Normal counterfactuals ( [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

**Figure 7.** Figure 7: Normal→Disease counterfactuals for each type of disease lesion. From Normal→Disease counterfactuals ( [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]

**Figure 8.** Figure 8: Disease→Normal counterfactuals for each lesion using DVCE vs. FreeMCG. Both DVCE and FreeMCG produce counterfactuals that reduce the lesion features and move the CXR image towards the normal class, but FreeMCG produces more sparse changes than DVCE ( [PITH_FULL_IMAGE:…

**Figure 9.** Figure 9: Illustration of decision boundaries of different architectures using FreeMCG-based counterfactual generation. All models are [PITH_FULL_IMAGE:figures/full_fig_p016_9.png]

**Figure 10.** Figure 10: FreeMCG-counterfactual generation using Stable Diffusion [ [PITH_FULL_IMAGE:figures/full_fig_p017_10.png]

**Figure 11.** Figure 11: Interface for the human user study. Questions are repeated for each input image and corresponding generated counterfactuals. [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

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