parameter-space geometry analysis

This work addresses the lack of effective support for Kendall’s 3D shape space in existing Python libraries—such as Geomstats—which has hindered the practical application of Riemannian geometry in three-dimensional shape analysis. We present the first systematic implementation of an efficient and user-friendly Python toolkit tailored specifically to Kendall’s 3D shape space, enabling scale-, position-, and orientation-invariant shape modeling and statistical analysis. By providing a ready-to-use, open-source solution for shape statistics on manifolds, this contribution fills a critical software gap in advanced 3D shape analysis, substantially lowering the barrier to entry for researchers, improving computational efficiency, and enhancing the reproducibility of results.

Existing mathematical modeling lacks a rigorous, unambiguous ontological foundation, hindering a unified characterization of the mapping between models and real-world phenomena. This paper introduces, for the first time, an axiomatic definition of mathematical models grounded in Hilbert-space operator theory: a model is formalized as a computable operator acting on random variables, systematically unifying theoretical derivation, experimental implementation, and statistical identification. We further establish a geometric correspondence between the model manifold and the prediction surface, exposing intrinsic structural properties and the fundamental nature of model computability. This framework fills a critical gap in the formal ontology of modeling, providing a unified mathematical foundation for interdisciplinary model construction. It significantly enhances the logical rigor of theoretical inference and the reliability of empirical validation.

Traditional machine learning struggles to effectively model shape data with nonlinear geometric structures and their intrinsic variability. This work proposes a unified analytical framework that systematically integrates differential geometry, manifold statistics, and geometric deep learning to address the challenges posed by complex, unaligned shapes exhibiting nonlinear variation. The framework encompasses key components including shape representation, geodesic metrics, parametrization, and statistical inference. It has been successfully applied to multiscale biological geometric data—such as cellular morphologies and primate dental evolution—revealing structural patterns and evolutionary trajectories underlying shape variation. This approach establishes both a theoretical foundation and a practical paradigm for geometry-aware learning in shape analysis.

Why do deep neural networks (DNNs) generalize well despite residing in high-dimensional parameter spaces—a phenomenon unexplained by classical model selection theory? This paper proposes a novel theoretical framework integrating singular differential geometry and information geometry. Specifically, it defines the local effective dimension of the parameter manifold via spectral analysis of the Fisher information matrix and constructs a short-description-length measure aligned with the network’s intrinsic singularity. For the first time, this approach unifies singular semi-Riemannian geometry with locally varying-dimensional information theory, reconstructing model complexity from the geometric singularity of the parameter manifold and thereby providing a rigorous geometric foundation for Occam’s razor. Experiments demonstrate that the proposed measure quantitatively captures the low generalization error of DNNs even under high parametric complexity, substantially enhancing the interpretability of deep learning generalization mechanisms.

Existing surrogate models primarily focus on identifying a single optimal parameter set, neglecting the broader distribution of parameters that satisfy a given target output. Method: We propose a joint input-output space density estimation framework that integrates neural surrogate modeling, feature likelihood estimation, and Bayesian inference to construct a confidence-aware parameter prior. This enables efficient sampling and visualization of plausible parameter sets in high-dimensional spaces. Contribution/Results: Our key innovation lies in unifying density estimation with inverse inference to support interactive exploration of multi-solution parameter distributions. Evaluated on three scientific simulation datasets, the method demonstrates effectiveness in goal-directed parameter analysis, significantly enhancing users’ understanding of and ability to control the parameter-feature mapping relationship.

Geospatial impact evaluations often grapple with ambiguity in defining the exposure units, timing, and intensity of interventions, particularly when multiple plausible exposure definitions exist. This study introduces the concept of “treatment geometry” as a foundational framework to systematically characterize the spatiotemporal footprint of interventions derived from Earth observation data. Centered on key trade-offs—including spatial resolution, temporal alignment, spillover effects, and boundary uncertainty—the framework provides diagnostic tools that enable researchers to identify which geometric definition choices are most critical for causal identification, rather than defaulting to a single methodological approach. Empirical applications to air pollution, wildfires, and forest policy demonstrate that this approach substantially enhances the credibility of causal inference and the rigor of empirical design.

This work addresses the limitations of traditional volumetric parameterization methods, which rely on fixed canonical domains—such as the unit ball—and often neglect the global geometric structure of the input 3-manifold, leading to significant distortion. To overcome this, the authors propose an adaptive volumetric parameterization framework that dynamically adjusts the target domain and jointly optimizes local shape preservation and quality distortion for high-fidelity parameterization of simply connected 3-manifolds. The approach innovatively introduces three progressively refined target domains: a prescribed ellipsoid, a volume-normalized adaptive ellipsoid, and a free-boundary domain embedded in hyperbolic space. It integrates 3D quasi-conformal shape updates, diffusion-driven density equalization, and fold-over-free geometric correction, enabling multi-resolution and locally adaptive remeshing. Experiments demonstrate substantial reductions in geometric distortion, enhanced parameterization quality, and successful applications in volumetric registration, deformation, and adaptive remeshing.

Traditional multiverse analyses struggle to characterize the structure of inferential uncertainty across different analytical specifications. This work proposes a distribution–geometry framework that models each plausible analysis specification as a probability distribution over the target outcome space and constructs a geometric structure based on distances between these distributions, thereby introducing a geometric perspective to dissect effect heterogeneity and uncertainty variation for the first time. The framework leverages geometric tools—such as neighborhoods, diameters, Fréchet means, and dispersion measures—to effectively preserve and reveal variability inherent in the multiverse of analyses. Numerical experiments and real-world case studies demonstrate that this approach substantially complements existing summarization strategies, enhancing analytical depth while retaining fine-grained details of both effect estimates and their associated uncertainties.

This study addresses the long-standing question of whether multivariate Kriging outperforms single-output modeling under heterotopic observations. The work establishes, for the first time, a theoretical link between output-specific design geometry and the estimability of cross-output dependencies, yielding a model-free diagnostic criterion. It derives an exact expression for predictive gain along with its geometric bounds, leading to a practical first-order net benefit criterion. Theoretical analysis reveals the statistical nonequivalence between collocated and heterotopic designs. Extensive experiments—conducted using separable multi-output Gaussian processes, radial basis functions, and linear models of coregionalization—validate the proposed criterion on synthetic datasets, an M/M/1 queueing system, and a multi-pollutant monitoring network, demonstrating both its effectiveness and practical utility.

This study addresses the challenge of smooth modeling and geometric feature extraction for parameterized curves in $\mathbb{R}^p$ subject to discrete measurement errors. The proposed methodology leverages separable Hilbert spaces and Sobolev frameworks, employing penalized least squares to achieve smooth curve fitting. It further extends functional principal component analysis to $\mathbb{R}^3$, utilizing variational methods to solve for the eigenfunctions of the covariance operator in order to decompose spatial variance. By integrating the Euler–Lagrange theorem, regularization techniques, and differential geometry, this work effectively captures key differential features such as velocity and curvature. The resulting approach demonstrates significant advantages over conventional multivariate analysis methods.