analyze score functions

This paper addresses the modeling challenge of score functions for high-dimensional data under variable transformations. It establishes, for the first time, a rigorous differential-geometric theory for score functions under smooth invertible transformations and derives a general transformation formula. Methodologically: (1) it enables cross-space diffusion sampling via a reverse Itô lemma, decoupling spatial dependencies between forward and reverse processes; (2) it generalizes sliced score matching to arbitrary smooth transformations, yielding generalized sliced score matching. Key contributions include: the first principled separation of training in the original space from efficient sampling in the transformed space; stable diffusion modeling on constrained manifolds such as the probability simplex; and empirical results demonstrating that generalized sliced score matching significantly outperforms conventional linear-projection-based methods in high-dimensional density estimation.

This paper addresses the lack of a rigorous theoretical foundation for consistency of scoring functions under variable transformations, specifically examining conditions for consistency and identifiability when predictions and observations undergo one-sided or bijective transformations. Method: We establish formal necessary and sufficient conditions for (strict) consistency and identifiability under general transformations, integrating scoring function theory, Bregman divergence analysis, and techniques from elicitation and identification function characterization for expectation-like functionals. We introduce novel identifiable functionals—including the *g-transformed expectation* and *g-transformed quantile*—and analyze their elicitation properties. Contribution/Results: Our framework provides the first unified theoretical justification for transformed scoring functions in empirical modeling. It enables principled construction of interpretable and verifiable functionals, with broad applicability to probabilistic forecasting and robust regression. The results bridge theoretical statistics and practical model evaluation, ensuring that transformation-based scoring remains both statistically sound and operationally meaningful.

Traditional neural networks suffer from limited interpretability and weak theoretical foundations. Method: This paper proposes a novel machine learning paradigm grounded in infinite-dimensional Hilbert spaces, centering on linear operators. It integrates reproducing kernel Hilbert spaces (RKHS), spectral operator learning, wavelet representations, scattering transforms, and Koopman operator theory to formulate learning tasks as sampling, approximation, and dynamical inference in infinite-dimensional function spaces. Contribution/Results: We establish the first unified Hilbert-space-theoretic framework bridging spectral learning and symbolic reasoning. The approach significantly enhances mathematical rigor and model interpretability by grounding learning in well-defined functional-analytic principles. Moreover, it provides a rigorous mathematical foundation and new methodological pathways for deep interdisciplinary integration between signal processing and machine learning—enabling principled analysis of structured data, hierarchical feature extraction, and nonlinear dynamical system modeling.

This paper addresses the challenge of analytically solving the score function in nonlinear diffusion generative models. We propose the first theoretical framework grounded in Malliavin calculus, introducing a novel Bismut-type formula that integrates the Skorokhod integral with first- and second-order variational processes. This yields an exact closed-form expression for the score function applicable to general nonlinear diffusion processes. Crucially, our approach completely eliminates reliance on conventional Malliavin derivatives, substantially enhancing both theoretical tractability and computational feasibility. The derived closed-form solution provides a rigorous foundation for efficient sampling algorithms under complex data distributions and naturally extends to broader classes of nonlinear stochastic differential equation (SDE) modeling scenarios.

This work addresses operator learning for partial differential equation (PDE) solution operators and black-box physical systems, formalizing it as a regression problem between function spaces. Method: We propose the first systematic statistical learning framework for operators, integrating PDE-based physical priors with neural operator architectures and introducing a constraint-aware training paradigm. Contribution/Results: Our framework unifies and characterizes the statistical foundations of mainstream operator learning methods, significantly enhancing model interpretability and generalization. It is the first to explicitly incorporate active learning and uncertainty quantification as core components of operator learning. The resulting methodology provides a new paradigm for scientific machine learning—rigorous in theory and practical in implementation—that enables high-fidelity, data-efficient surrogate modeling of complex physical systems.

This work addresses the challenge of accurately estimating the score function of a probability measure from limited samples to enhance the generation quality of score-based generative models while mitigating overfitting. To this end, it introduces Sobolev space regularization into score function estimation for the first time, establishing a learning framework based on empirical risk minimization. The proposed method achieves minimax-optimal estimation rates on the flat torus and provides theoretical guarantees for score-based generative models, effectively balancing estimation accuracy with generalization capability.

This work addresses a critical limitation in the evaluation of neural operators for elliptic partial differential equations (PDEs), where performance deficiencies are often obscured by standard datasets and average error metrics. The authors propose ELADO—a systematic benchmark suite centered on variable-coefficient Poisson and Helmholtz equations—that precisely characterizes five key challenges: heavy-tailed solution distributions, input spectral shifts, heavy-tailed behavior in the solution frequency domain, operator sensitivity to input perturbations, and input complexity. These challenges are engineered through controlled random coefficient fields, spectral analysis, and local Lipschitz estimates. Experiments demonstrate that these factors substantially degrade model accuracy, yet remain undetected under conventional evaluation protocols. By moving beyond aggregate performance measures, this study establishes a new paradigm for systematically identifying and isolating the fundamental sources of difficulty in learning elliptic PDE operators.

本文介绍了一个开源Python包，用于解决几何函数理论中的极值问题，通过计算泰勒系数、Fekete-Szegő常数等，并提供不同级别的验证方法。

This work addresses the challenges posed by higher-order functions in mathematical optimization modeling, which often lead to unnatural LaTeX output and inefficient constraint verification. To overcome these issues, the authors propose an egglog-based optimization approach that performs desugaring reconstruction on the λ-calculus intermediate representation of JijModeling 2, thereby recovering comprehension-like syntactic structures. By integrating Henkin-style constants with Datalog-inspired rules, the method enables declarative, multi-step constraint checking. A custom cost model and equality saturation techniques are introduced to enhance performance significantly: the generated LaTeX aligns more closely with conventional mathematical notation, and complex constraint validation—previously requiring minutes or failing to terminate—now completes within seconds.

This work addresses the efficient algebraic representation and factorization of linear ordinary differential operators over compatible derivation modules. By implementing differential operators as first-class objects in Scratchpad II, the approach supports standard notation and provides a unified treatment of left and right module structures. For operators with coefficients in a field or polynomial ring, it integrates Ore localization, pseudo-division, and construction of right fraction fields to enable left and right division, computation of greatest common divisors, least common multiples, and extended Euclidean algorithms. Furthermore, by combining Riccati equations with Newton polygon analysis, the method effectively characterizes the singularities of factors. This framework facilitates constructive factorization and algebraic manipulation of operators with constant, elementary, rational, and even matrix-valued coefficients.