cross-layer and cross-subject geometry comparison

This study addresses the challenge of comparing high-dimensional neural representations across neuroscience and artificial intelligence: specifically, how to select similarity measures that best reveal functional correspondences and divergences. We systematically evaluate eight mainstream representational similarity metrics—including linear CKA, Procrustes distance, CCA, inner-product kernel, and nearest-neighbor alignment—against behavioral functional alignment (e.g., recognition accuracy, generalization, robustness) as a ground-truth benchmark. Our evaluation spans both biological neural data and artificial neural network models. Results show that geometry-sensitive metrics—particularly linear CKA and Procrustes distance—consistently outperform predictive metrics, achieving superior alignment with human behavioral performance and effectively distinguishing trained versus untrained models. In contrast, linear predictivity exhibits only moderate behavioral alignment. This work establishes the first behavior-driven representational similarity benchmark, providing a principled, cross-domain methodology for mechanistic interpretation and comparative analysis of neural computation.

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.

Existing activation alignment methods struggle to capture differences in the sensitivity of neural representations to local stimulus perturbations and thus fail to reflect how systems leverage local evidence for discrimination. This work proposes a novel analytical framework based on locally decodable information, integrating Fisher information, pullback metrics, and log-spectral distances on the SPD manifold to construct the Spectral Riemannian Alignment Score (S-RAS). S-RAS provides, for the first time, a minimal, dataset-level summary of neural representational sensitivity from the perspective of local discriminative tasks, with guaranteed multiplicative consistency. The method successfully aligns corresponding layers across independently trained networks, enables transferable class-conditional probing, reveals representational differences between standard and robustly trained models, and uncovers stimulus coordinate family effects in mouse visual cortex.

This study addresses the limitations of traditional anomaly detection methods, which focus solely on threshold-exceeding signals and fail to capture structural stress preceding critical events. The authors propose a latent-geometry-based structural monitoring framework that models large-scale behavioral ensembles as geometric energy landscapes, enabling early warning through the detection of structural deformations—embodying the core principle that “structure precedes geometry.” Applied to the Tor network, the method identifies a stable nine-dimensional support subspace and reveals, for the first time, a novel detectable failure mode: connectivity degradation without topological change. By integrating a dual-observer pipeline, subspace alignment, Monte Carlo simulation, and high-dimensional geometric analysis, the approach achieves a 0.0% false positive rate across 24 stable observation windows and attains a 16.8σ significance level in retrospective analysis of the infrastructure event on February 20, 2026.

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.

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 approaches to measuring functional similarity between models rely on the true data distribution, making it difficult to characterize alignment of decision boundaries across the entire input space. This work proposes Rashomon Alignment (RA), a novel framework that, for the first time, evaluates functional similarity between models from a geometric perspective over the full input space without dependence on any specific data distribution. By uniformly sampling the input space and employing geometric similarity metrics, RA enables a global analysis of decision boundary alignment. Experiments across more than 90 datasets demonstrate that geometric alignment provides a complementary perspective to distribution-based alignment, and that RA effectively supports model selection, ensemble construction, and enhanced interpretability.

This work addresses the limitations of existing representation alignment methods, which predominantly rely on geometric properties and struggle to capture the global structural organization of model representations. To overcome this, the study introduces topological data analysis into the field for the first time, proposing a Mapper-based visual analytics framework. By integrating force-directed layout, Bubble Sets, motif querying, and membrane-inspired heuristics, the framework enables a unified analytical pipeline spanning global structure alignment, local region matching, and fine-grained pattern exploration. Case studies on language and multimodal models, complemented by expert evaluations, demonstrate that the approach effectively reveals and compares the topological organization of representations across different models or layers, offering deep structural insights.

This work addresses the limited generalization performance in cross-subject brain functional decoding caused by inter-individual variability in neural responses. To overcome this challenge, the authors propose SpectralOT, a novel method that, for the first time, integrates spectral features of the Laplace–Beltrami operator into functional data and leverages optimal transport theory to construct a geometry-aware whole-brain alignment framework. By explicitly incorporating cortical geometric structure during functional alignment, the approach enhances computational efficiency while preserving anatomical consistency. Experimental results demonstrate that SpectralOT significantly improves the generalization capability of cross-subject decoding models, offering a new paradigm for high-precision brain functional analysis.