analyze weight-space geometry

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.

This paper addresses the challenges of evaluating dimensionality reduction (DR) effectiveness and estimating intrinsic data dimensionality. We propose a geometric profiling method based on sectional curvature in discrete metric spaces, which characterizes large-scale data geometry via metric relationships among point triplets. For the first time, this approach systematically introduces differential-geometric curvature into quantitative DR quality assessment and intrinsic dimension estimation—without requiring embedded coordinates or manifold assumptions, thus ensuring both theoretical rigor and computational feasibility. Experiments across diverse synthetic and real-world datasets demonstrate that our method robustly discriminates DR algorithm performance, achieves significantly lower intrinsic dimension estimation error than state-of-the-art methods (e.g., MDS- and PCA-based estimators), and successfully uncovers latent negative curvature in empirical networks—including social and biological networks.

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.

This paper addresses the lack of geometric characterization for the implicit spatial partitioning induced by machine learning models. We propose a modeling framework based on Riemannian simplicial complexes: model decision regions are represented as metric-bearing simplicial complexes, enabling systematic quantification of geometric features—including volumes, facet areas, and dihedral angles. To track geometric evolution across neural network layers, we introduce pullbacks of differential forms and an extended Laplacian operator. Furthermore, we define vertex-wise discrete curvature and edge-wise statistical Ricci curvature to explicitly link model geometry with underlying data distributions. The resulting geometric regularization method directly constrains spatial configurations, enhancing both generalization and interpretability. Empirically and theoretically, it demonstrates consistency in regularizer design and efficacy in diagnosing learning dynamics, offering a computationally tractable and principled approach to geometric deep learning.

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.

Existing neural representational similarity measures focus solely on the extrinsic geometry of state space, limiting their ability to reveal the essential intrinsic differences among neural network solutions. This work proposes Metric Similarity Analysis (MSA), which introduces Riemannian geometry into representational similarity research for the first time. Grounded in the manifold hypothesis, MSA characterizes the geometric structure of neural representations through intrinsic metrics defined on statistical manifolds. The method effectively distinguishes computational mechanisms of deep networks trained under different learning paradigms, enables precise comparison of nonlinear dynamical behaviors, and successfully extends to the analysis of diffusion models. Empirical validation demonstrates its broad applicability across diverse settings and its mathematical rigor.

Traditional linear dimensionality reduction methods often fail to effectively uncover the intrinsic low-dimensional manifold structure embedded in high-dimensional data. This work systematically traces the historical development of manifold fitting and, for the first time, categorizes it into three distinct phases: nonparametric statistics, mathematically inspired analysis, and modern practical statistics. It clarifies manifold fitting’s role as an independent geometric data analysis tool and delineates its conceptual boundaries from related techniques such as manifold embedding and denoising. By integrating nonparametric methods, differential geometry, and contemporary statistical learning approaches, the paper explores cutting-edge applications of manifold fitting in neural networks and bioinformatics, offering a comprehensive reference framework that elucidates both its theoretical limits and practical utility.

This work addresses the gap in deep learning theory caused by the manifold hypothesis’s lack of precise characterization and reliable benchmarks for the geometric properties of data manifolds. We introduce a controllable benchmark framework that extends the dSprites and COIL-20 datasets through dense, axis-aligned sampling and employs finite-difference estimators to recover geometric quantities—such as curvature, reach, and volume—with high accuracy. This framework provides the first testbed combining known ground-truth geometry with the complexity of real-world data, enabling near-ground-truth geometric estimation even in regimes where generic estimators fail. Leveraging this platform, we calibrate and evaluate the scaling behavior of generalization bounds proposed by Genovese, Fefferman, and colleagues, thereby exposing fundamental limitations in current theoretical analyses.

This study addresses the limitations of conventional fMRI functional connectivity analyses, which overlook the non-Euclidean geometric structure of correlation matrices, thereby constraining statistical sensitivity and scalability. To overcome this, the authors propose a scalable geometric framework that maps correlation matrices to symmetric zero-diagonal matrices via the Off-log metric, enabling closed-form statistical modeling. Furthermore, they introduce a comparison of eigenspaces using principal angles on the Grassmann manifold, effectively resolving ambiguities arising from eigenvector sign and basis indeterminacy. Evaluated on Parkinson’s disease, psychiatric disorders, and three aging-related fMRI datasets, the method achieves classification performance comparable to or better than existing approaches, significantly enhances sensitivity in permutation testing, and accurately identifies disease-relevant brain networks.