{"id":"b1a2be96-a102-41d9-95a7-19a30ae6b348","arxiv_id":"2411.15265","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":5,"one_line_summary":"FreeMCG estimates on-manifold classifier gradients from black-box outputs using diffusion particles and an ensemble Kalman filter, and uses the same estimate for feature attribution and counterfactual explanation.","lead":"This paper introduces FreeMCG, a method that estimates a neural network's gradient direction using only its outputs, by generating on-manifold image variations with a diffusion model and an ensemble Kalman filter. It applies the same estimated gradient to both feature attribution and counterfactual explanation, aiming to make black-box image classifiers explainable.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"At t=400-700 the Tweedie-denoised particles are not in a small delta-neighborhood of x, so Theorem 2's premise fails and the central claim that FreeMCG is a derivative-free manifold-projected gradient at x is unsupported.","rationale":"The paper's selling point is a principled derivation: Eqs. (8)-(11) purport to convert black-box outputs into Cxx\\nabla log p. For that to be true, the particles in Alg. 2/3 must satisfy the hypotheses of Theorem 2. I checked the operating regime: Alg. 2 draws t up to 700, and Alg. 3 starts at t'=400; at such noise levels the denoiser output is dominated by the prior mean. This makes delta and the mean displacement large, so the O(delta^3) Taylor error and the missing gradient-at-\\bar{x} term are not controlled. This is not a disagreement with the community about what counts as state of the art; it is an internal mismatch between the claimed derivation and the implemented algorithm. The experiments may still show useful empirical behavior, and the human study is a genuine attempt to measure perceived change, but they do not rescue the theoretical claim that gFree is the on-manifold gradient at x. The reader's weakest-assumption analysis identifies the same load-bearing issue, and the recommended conditional verdict remains appropriate: accept only if the particle neighborhood assumption is quantitatively validated or the algorithm is restricted to a regime where the theorem applies.","tokens_in":15942,"tokens_out":6241,"duration_ms":76666,"concrete_test":"On ImageNet with the paper's ResNet-50 and diffusion model, run Algorithm 2 for t=700 (and t=400) with K=100 and compute: d_t = ||\\bar{x}_{0|t} - x||, delta_t = max_k ||x_{0|t}^{(k)} - \\bar{x}_{0|t}||, and the linearity residual R_t = max_k ||f(x_{0|t}^{(k)}) - f(\\bar{x}_{0|t}) - \\nabla f(\\bar{x}_{0|t})^T (x_{0|t}^{(k)} - \\bar{x}_{0|t})||, using the white-box classifier for this diagnostic only. If d_t is large relative to the local feature scale or R_t is comparable to ||Cxf(ec-p)||, Theorem 2's premise is contradicted; then restrict Alg. 2 to t<=300 and re-run ROAD: if full-range results change materially, the high-t particles are supplying nonlocal signal outside the derived approximation, so the central claim must be revised.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (11) estimates gFree from Tweedie-denoised particles, and Theorem 2 (Appendix A) justifies it only if all particles lie within delta of \\bar{x} and the Hessian is uniformly bounded. In Alg. 2, t is sampled up to 700; in Alg. 3, reverse diffusion starts at t'=400. At these signal-to-noise levels, D_theta(x_t) is close to the unconditional mean E[x0] and is nearly independent of x, so the particles are not in any small neighborhood of x nor on the tangent space at x. Two failures follow. (1) delta is large: the Taylor expansion in the proof of Theorem 2 around \\bar{x} incurs an O(delta^3) remainder that is not small, so Cxf does not approximate Cxx\\nabla f(\\bar{x})^T. (2) Even if the Taylor expansion held around \\bar{x}, Eq. (9) yields the gradient at \\bar{x}, not at the input x; the missing Hessian x (\\bar{x}-x) term is uncontrolled and can be of the same order as the signal when \\bar{x} is far from x. Thus, in the actual operating regime, gFree is not demonstrated to be the manifold-constrained gradient of the classifier at x; any useful behavior must come from a different, nonlocal aggregation of the particles, not from Theorem 2. This directly undermines the paper's central claim, independent of whether the downstream XAI metrics are favorable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":16263,"tokens_out":4372,"duration_ms":47530,"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":[{"comment":"","section":""}],"minor_comments":[{"comment":"The abstract contains the typo “Manifold-Contrained”; the title uses “Manifold-Constrained”.","section":"Abstract and title"},{"comment":"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.","section":"Algorithm 3, lines 6–7"},{"comment":"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)}].","section":"Algorithm 3, line 6"},{"comment":"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.","section":"Appendix A, Theorem 2 proof"},{"comment":"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.","section":"Theorem 1"},{"comment":"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).","section":"Eq. (8) and Fig. 2"}],"recommendation":"major_revision","confidential_remarks":"The paper comes from a group with strong prior work on diffusion-based inverse problems, and the empirical results are presented with care. The main reason for major revision rather than rejection is that the central theoretical claim is currently unsupported in the actual operating regime, but it is plausibly fixable by restricting timesteps, adding a direct gradient-approximation experiment, and either revising the theorems or weakening the claims. The missing direct validation of the core derivative-free gradient estimate is the most important gap; downstream XAI metrics cannot substitute for it. I would also encourage the authors to state whether code will be released, as the method's practical value depends on reproducibility of the pseudocode details."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nThe 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.\n\nWho 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.","headline":"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.","tokens_in":16832,"tokens_out":2044,"would_cite":true,"duration_ms":22619,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["explainable AI","feature attribution","counterfactual explanation","gradient-free optimization","diffusion models","ensemble Kalman filter","manifold learning","black-box model explanation"],"falsifier":"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.","tokens_in":15720,"feed_emoji":"🔍","tokens_out":7366,"duration_ms":71029,"temperature":0.7,"pith_summary":"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.","feed_headline":"No weights needed: one derivative-free gradient powers both XAI tasks","feed_subtitle":"FreeMCG reads classifier outputs and a diffusion model into one estimate that serves feature attribution and counterfactual explanation.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the manifold-projection result that Tweedie denoising is locally an orthogonal projection onto the data manifold, justifying the on-manifold premise of Theorem 1.","marker":"[7]"},{"why":"Gives Tweedie's formula for the posterior mean, which the paper uses to denoise forward-diffused particles in one step.","marker":"[14]"},{"why":"Provides the ensemble Kalman filter covariance analysis that underlies the derivative-free gradient approximation in Theorem 2.","marker":"[37]"},{"why":"Demonstrates ensemble Kalman diffusion guidance for inverse problems, the immediate precursor that the paper adapts from guidance to explainability.","marker":"[55]"},{"why":"Serves as the white-box diffusion counterfactual baseline and supplies the target-class guidance setup that FreeMCG compares against.","marker":"[4]"},{"why":"Serves as the latent diffusion counterfactual baseline and provides the evaluation scheme for selecting the closest target class.","marker":"[15]"},{"why":"Supplies the ROAD removal-based evaluation metric used to judge feature attribution faithfulness.","marker":"[34]"},{"why":"Provides the SDEdit-style forward-diffuse and reverse-diffuse recipe used in the counterfactual generation variant.","marker":"[28]"},{"why":"Supplies the pretrained unconditional diffusion models used as the image manifold prior in the ImageNet experiments.","marker":"[12]"}],"fun_headline_variants":["No weights, one gradient: unifies attribution and counterfactuals","Outputs only: derivative-free gradient unifies XAI tasks","One gradient, zero Jacobians: unified XAI","FreeMCG: output-only gradient for better explanations","Derivative-free manifold gradient: two tasks, one estimator"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["No weights, one gradient: unifies attribution and counterfactuals","Outputs only: derivative-free gradient unifies XAI tasks","One gradient, zero Jacobians: unified XAI","FreeMCG: output-only gradient for better explanations","Derivative-free manifold gradient: two tasks, one estimator"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000551,"raw_usage":{"total_tokens":2656,"prompt_tokens":1000,"completion_tokens":1656,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":616,"completion_tokens_details":{"reasoning_tokens":1574}},"tokens_in":616,"tokens_out":1656,"duration_ms":12505,"temperature":1.0,"reasoning_tokens":1574,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T14:47:20.042226+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Improving diffusion models for inverse prob- lems using manifold constraints","cited_arxiv_id":null,"evidence_quote":"Supplies the manifold-projection result that Tweedie denoising is locally an orthogonal projection onto the data manifold, justifying the on-manifold premise of Theorem 1."},{"cited_title":"Tweedie’s formula and selection bias","cited_arxiv_id":null,"evidence_quote":"Gives Tweedie's formula for the posterior mean, which the paper uses to denoise forward-diffused particles in one step."},{"cited_title":"Analysis of the ensemble kalman filter for inverse problems","cited_arxiv_id":null,"evidence_quote":"Provides the ensemble Kalman filter covariance analysis that underlies the derivative-free gradient approximation in Theorem 2."},{"cited_title":"Diffusion visual counterfactual explana- tions, 2022","cited_arxiv_id":null,"evidence_quote":"Serves as the white-box diffusion counterfactual baseline and supplies the target-class guidance setup that FreeMCG compares against."},{"cited_title":"Diffusion models beat GANs on image synthesis","cited_arxiv_id":null,"evidence_quote":"Supplies the pretrained unconditional diffusion models used as the image manifold prior in the ImageNet experiments."}],"review_version":1}