subspace identification

In spatiotemporal data-driven system dynamics modeling, manifold learning algorithms suffer from heavy reliance on manual hyperparameter tuning and insufficient robustness in geometric representation. To address these issues, this paper proposes an automated manifold learning framework. First, it constructs a spatiotemporal proximity graph; then, it integrates classical methods (t-SNE, UMAP) with graph neural networks and introduces a meta-learning-driven subgraph sampling strategy to enable adaptive algorithm selection and joint hyperparameter optimization. Compared to conventional approaches, the framework significantly reduces human intervention while enhancing the fidelity of low-dimensional manifolds in capturing both local and global geometric structures of dynamical systems. Experiments across diverse dynamical systems demonstrate that the learned low-dimensional representations achieve superior accuracy and scalability, effectively supporting subsequent model order reduction and dynamic discovery.

This work addresses the challenges of modeling complex dependencies and mitigating redundancy in high-dimensional parameter spaces for discrete data generation. It introduces, for the first time, a Riemannian geometric structure with isometric properties into the exponential parameter space of product manifolds over categorical distributions, thereby constructing a low-dimensional latent subspace. By leveraging the Riemannian metric, geodesics within this subspace become straight lines, enabling consistent and efficient flow-matching training. The proposed approach substantially reduces the dimensionality of latent variables while preserving strong representational capacity for discrete data distributions. Experimental results demonstrate that the model achieves accurate and efficient discrete data generation using a significantly lower-dimensional latent space, effectively balancing computational efficiency with modeling performance.

Subspace clustering in high-dimensional data often yields multiple semantically distinct subspaces, yet existing methods require manual specification of both the number of subspaces and the number of clusters within each—rendering them parameter-sensitive and poorly interpretable. This paper proposes an automatic, non-redundant multi-subspace clustering framework. First, it introduces the Minimum Description Length (MDL) principle to non-redundant clustering, enabling joint, adaptive inference of both the optimal number of subspaces and the cluster count per subspace. Second, it designs a split-merge-based greedy search strategy coupled with a subspace-level outlier encoding mechanism, allowing simultaneous outlier detection. Evaluated on multiple benchmark datasets, the method achieves competitive accuracy against state-of-the-art approaches while significantly improving parameter robustness, model interpretability, and practical applicability.

This work addresses local structure modeling of point clouds in product spaces endowed with mixed Euclidean and directional metrics. We propose the first subspace-constrained mean shift algorithm tailored to such hybrid metric spaces, enabling joint estimation of density modes and density ridges. Theoretically, we establish convergence guarantees on product manifolds and provide practical implementation criteria. By integrating manifold gradient analysis with a customized product-space metric design, the method enhances interpretability and fidelity in capturing heterogeneous multi-source structures. Experiments on synthetic and real-world data—including 3D human poses and motion trajectories—demonstrate substantial improvements over state-of-the-art approaches in mode and ridge localization accuracy, robustness to noise, and structural interpretability. Our framework establishes a new paradigm for density-based topological modeling in complex geometric domains.

This work addresses the lack of a universal statistical interpretation for the manifold hypothesis—that high-dimensional data approximately reside on low-dimensional manifolds. We propose the Latent Metric Model (LMM), a generative framework grounded in fundamental statistical concepts: latent variables, variable dependence, and stationarity—providing the first unified statistical justification for the manifold assumption. Methodologically, LMM integrates neighborhood graph construction, spectral analysis, and an interpretable inference framework to enable unsupervised manifold discovery and geometric structure recovery under weak priors. Experiments demonstrate that complex manifold geometries naturally emerge from minimal statistical mechanisms; LMM significantly reduces reliance on hand-crafted priors on both synthetic and real-world datasets, while enabling interpretable reconstruction of manifold dimensionality, curvature, and coordinate systems.

This work addresses how to uncover interpretable concept manifolds embedded within the stacked representations of language models. The authors propose Manifold Probe, a method that generalizes traditional linear probing to manifold probing by integrating supervised manifold learning with linear predictability analysis. This approach identifies continuous geometric structures in representation space corresponding to high-level concepts—such as time or space—and determines their encoding directions. Beyond merely detecting the presence of such concepts, the method enables causal intervention: manipulating activations along discovered manifold directions directly alters model behavior. Experiments on Llama 2-7B demonstrate that perturbing representations along the extracted temporal manifold significantly shifts the model’s generated outputs regarding the release years of cultural works, thereby validating both the interpretability and causal efficacy of the recovered manifolds.

This work addresses the empirically observed phenomenon that weights in nonlinear multilayer perceptrons (MLPs) often evolve within a low-dimensional subspace during training, despite the lack of theoretical understanding. We provide the first precise characterization of this low-rank dynamics, proving that under smooth activation functions and gradient descent, the weight trajectories remain confined to an invariant low-dimensional subspace determined by initialization. By integrating dynamical systems theory, gradient flow analysis, and low-rank matrix approximation, we uncover the structure and emergence mechanism of this subspace and leverage these insights to design a low-rank parameterization scheme. Experiments across multiple classification tasks demonstrate that low-rank MLPs initialized within this subspace achieve performance comparable to their full-parameter counterparts, confirming the generalizability and practical relevance of our theoretical findings.

Modeling, classification, and forecasting of large-scale spatiotemporal data from high-dimensional nonlinear complex systems—such as brain activity, climate, and ecosystems—remain challenging due to the limited representational capacity of conventional dimensionality reduction and phase-space reconstruction methods. Method: We propose a geometric vector field analysis framework on discrete measure spaces, introducing for the first time a two-parameter family of vector field metrics applicable to spatiotemporal functions defined on graphs and simplicial complexes. This framework unifies representations of scalar fields, gradient fields, and multivalued fields, transcending classical attractor-geometric limitations. By integrating vector field representation theory, discrete differential geometry, and multidimensional scaling (MDS), it enables model-free, efficient dimensionality reduction, modal decomposition, phase-space reconstruction, and attractor characterization. Results: Extensive validation on biological and physical simulation datasets demonstrates substantial improvements in dynamical system analysis capability, particularly in capturing nonlinear, multiscale spatiotemporal structures.

This work addresses the problem of classifying trajectories generated by distinct nonlinear dynamical systems, where each class corresponds to a unique system. The authors propose Dynafit, a novel method that, for the first time, integrates the Koopman operator framework with kernel methods to achieve global linearization of dynamics in a reproducing kernel Hilbert space. By leveraging the kernel trick, Dynafit efficiently computes dynamical distances between trajectories while allowing incorporation of prior knowledge. The approach demonstrates significant performance gains over baseline methods across three diverse tasks: detecting chaos in logistic maps, recognizing handwritten dynamics, and classifying visual dynamic textures. These results validate Dynafit’s effectiveness and generality in multi-class classification of nonlinear dynamical systems.

This work addresses the limitation of existing neural classifiers that rely on linear readouts and struggle to capture the geometric structure of class representations, particularly under few-shot conditions where unilateral affine separability cannot be properly assessed. The authors propose a directional Linear Separability Metric (LSM) that quantifies the minimal proportion of competing-class samples intruding into an affine half-space containing all samples of a target class. LSM exhibits asymmetry, class-level granularity, target normalization, and invariance under full-rank linear transformations, thereby distinguishing the effects of linear reparameterizations from those of information loss or nonlinear distortions. An efficient penalty-based affine search algorithm is introduced to estimate LSM in high-dimensional feature spaces while preserving the original discrete constraints. Experiments demonstrate that LSM effectively reveals class intrusion phenomena induced by components such as coordinate gating, offering a novel tool for analyzing the geometry of neural representations.