loss landscape analysis

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.

Traditional low-dimensional approaches struggle to capture the complex topological structure of neural network loss landscapes, limiting our understanding of optimization and generalization mechanisms. This work proposes Landscaper, an open-source tool that integrates Hessian-guided subspace sampling with topological data analysis (TDA) to enable geometric characterization of loss landscapes in arbitrary dimensions, revealing the hierarchy and connectivity of energy basins. The study introduces the Saddle-Minimum Average Distance (SMAD) metric, a novel indicator that, for the first time, leverages multidimensional topological analysis to detect landscape simplification during training. SMAD demonstrates sensitivity to training dynamics across diverse neural architectures and pretrained language models. Furthermore, in a molecular property prediction task, SMAD proves effective as a diagnostic indicator for out-of-distribution generalization.

This work addresses the limitations of conventional Gaussian-noise perturbations by systematically uncovering previously overlooked one-dimensional (1D) and two-dimensional (2D) local geometric structures in deep neural network (DNN) loss landscapes. Methodologically, we introduce a progressive taxonomy of five types of 1D loss curves (e.g., *v-basin*, *vvv-basin*) and design a perturbation-direction mining algorithm that integrates low-dimensional subspace projection with Hessian spectral analysis to automatically extract and visualize complex geometric structures. Our contributions include: (i) the first empirical observation and visualization of canonical 2D loss structures—including saddle surfaces and “bottle-bottom” geometries—in real DNNs; (ii) a theoretical characterization linking the geometric properties of perturbation directions to the eigenvalue distribution of the Hessian; and (iii) a novel geometric perspective for understanding generalization behavior and optimization dynamics.

This work addresses the challenge of accurately characterizing geometric singularities—such as corners, edges, and self-intersections—in the underlying manifold of point cloud data. We propose a singularity modeling and estimation framework grounded in the graph Laplacian operator. For the first time, we establish explicit functional bounds for the graph Laplacian within singular neighborhoods, thereby bridging spectral methods with local geometric inference. Based on this theoretical foundation, we derive a principled existence test for singularities and develop robust, nonparametric estimators for intrinsic geometric quantities—including manifold dimension and curvature. The method provides rigorous theoretical guarantees while maintaining interpretability. Extensive experiments on both synthetic and real-world point clouds demonstrate high estimation accuracy and strong consistency with theoretical predictions. Our approach establishes a novel paradigm for singularity-aware manifold learning, enabling geometrically faithful analysis of complex, nonsmooth structures in point cloud data.

This work addresses the challenge of characterizing the distribution of critical points in the empirical risk landscape and its impact on optimization dynamics under high-dimensional Gaussian single-index models. By leveraging the Kac–Rice formula within a proportional asymptotic regime, the authors reduce the original problem to a tractable finite-dimensional scalar variational problem. This approach yields the first complete topological phase diagram of the loss landscape for phase retrieval, along with a precise characterization of a BBP-type spectral instability transition in the Hessian along the signal direction. Integrating high-dimensional asymptotic analysis, Hessian spectral theory, and gradient flow simulations, the framework accurately predicts the distribution of critical points, their joint label statistics, and the dynamical behavior of local optimization algorithms, with theoretical predictions showing excellent agreement with finite-dimensional numerical experiments.

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.

This study addresses the lack of computable and interpretable spatial entropy metrics in pathological point pattern analysis by proposing Dense Basin Entropy and a simplified Brownian motion Markov chain. Methodologically, novel spatial entropy metrics are defined based on Voronoi/Delaunay tessellations, and a Markov chain approximating Brownian motion is constructed. Combined with spectral gap analysis, this framework enables the generalization of multi-set hypothesis testing and feature extraction. In histopathological experiments, the proposed metric demonstrates high sensitivity to spatial clustering, providing analytical insights complementary to conventional methods. Ultimately, this work establishes a new paradigm for quantifying the spatial structure of complex point patterns.

本文提出一种直接在连续隐式模型中跟踪拓扑特征的方法，通过查询模型及其导数来追踪临界点的演变，避免了离散化带来的伪影。

本文提出一种基于几何的船体形式可制造性早期筛选框架，通过无量纲总偏差、曲率类面积分数等描述符来评估船体表面特征。

This work addresses the sensitivity of the Mapper algorithm to lens functions, cover parameters, and clustering strategies, for which no systematic evaluation framework previously existed. The authors propose the first triaxial assessment framework that comprehensively evaluates Mapper variants across three complementary dimensions: stability, cluster quality, and topological shape preservation. Experiments on synthetic data and the UCI handwritten digits dataset reveal inherent trade-offs among these dimensions, demonstrating that no single configuration achieves optimal performance across all metrics simultaneously. The study further identifies a “topological explosion” phenomenon at high resolutions, offering practical guidance for parameter selection in real-world applications and highlighting key challenges for future research in Mapper-based topological data analysis.