The main XAI attribution methods for deep neural networks -- GradCAM, SHAP, LIME, Integrated Gradients -- operate on separate theoretical foundations and are not formally comparable. We present GRALIS (Gradient-Riesz Averaged Locally-Integrated Shapley), a mathematical framework establishing a representation theory for attributions: every additive, linear, and continuous attribution functional on L^2(Q,mu) admits a unique canonical representation (Q, w, Delta), proved necessary by the Riesz Representation Theorem. This class encompasses SHAP, IG, LIME and linearized GradCAM, but excludes nonlinear functionals such as standard GradCAM or attention maps. Seven formal theorems provide simultaneous guarantees absent in any individual method: (T1) necessary canonical form; (T2) exact completeness; (T3) Monte Carlo convergence O(1/sqrt(m))+O(1/k); (T4) exact Shapley Interaction Values; (T5) Hoeffding ANOVA decomposition; (T6) Sobol sensitivity generalization; (T7) multi-scale extension (MS-GRALIS) with minimum-variance weights. An algebraic appendix justifies the GRALIS-SIV correspondence via the Mobius transform without circularity. GRALIS satisfies 13.5/14 axiomatic properties vs. 2.5-6/14 for individual methods, including completeness, sensitivity, locality, order-k interactions and optimal multi-scale aggregation simultaneously. Preliminary validation on BreaKHis (1,187 histology images, DenseNet-121) reports deletion faithfulness AUC +0.015 (malignant), 96% class-conditional consistency, SAL = 0.762+/-0.109 and sparsity index 0.39. Extended comparison with baseline XAI methods is planned for a companion paper.

Stochastic gradient descent (SGD) has been a cornerstone of machine learning since the pioneering work of Robbins & Monro (1951). Beyond its algorithmic simplicity and scalability, SGD has also become a central object of theoretical study, with refined analyses linking its dynamics to implicit regularization, generalization performance, and algorithmic stability. For decades, theoretical analyses of SGD have largely resided within the realm of classical stochastic approximation (Polyak & Juditsky, 1992; Lai, 2003; Bottou et al., 2018), where the data dimension is considered fixed while the sample size tends to infinity. While this regime has yielded foundational insights, it no longer fully reflects the characteristics of modern learning systems. Contemporary applications often operate in regimes where data dimension, sample size, and model complexity grow together, calling for new theoretical tools and perspectives that go beyond traditional asymptotic analyses. In this study, we focus on the learning tasks involving high-dimensional predictors. When SGD is applied directly to such data, the dimensionality of the feature space propagates into the optimization process, resulting in a highdimensional (HD) parameter space. Algorithmically, one trending strategy is to approximate the gradient updates using a low-rank representation to reduce memory costs and accelerate computation (Wang et al., 2018; Vogels et al., 2019; Kozak et al., 2019; Kasiviswanathan, 2021; Zhao et al., 2024). Theoretically, despite the vast literature on SGD, convergence guarantees of HD-SGD remain limited (Garrigos & Gower, 2023; Li et al., 2025).

Opponents Artist David KittDavid Kitt (born 1975 in Dublin Ireland) isan Irishmusician.

Data-driven methods for the solution of inverse problems have become widely popular in recent years thanks to the rise of machine learning techniques. A popular approach concerns the training of a generative model on additional data to learn a bespoke prior for the problem at hand. In this article we present an analysis for such problems by presenting quantitative error bounds for minimum Wasserstein-2 generative models for the prior. We show that under some assumptions, the error in the posterior due to the generative prior will inherit the same rate as the prior with respect to the Wasserstein-1 distance. We further present numerical experiments that verify that aspects of our error analysis manifests in some benchmarks followed by an elliptic PDE inverse problem where a generative prior is used to model a non-stationary field.

We prove finite-sample concentration and anti-concentration bounds for dimension estimation using Gaussian kernel sums. Our bounds provide explicit dependence on sample size, bandwidth, and local geometric and distributional parameters, characterizing precisely how regularity conditions govern statistical performance. We also propose a bandwidth selection heuristic using derivative information, which shows promise in numerical experiments.

We thank the reviewers for their insightful comments. In this rebuttal, we respond to remarks from reviews. Remark 1 The work lacks discussion about the comparison of interpretability with BSP-Net. Moreover, their CSG structure is fixed by definition. CSG trees for different instances (see Figure on the right). Remark 2 Only a single instance of CSG visualization for each class is shown.