arXiv subjects

Xiang Xiang Wang

Publications and source records attributed to Xiang Xiang Wang.

8 recordsLinked to original sources

arXiv2026

Chaotic Dynamics-Regulated Topological Learning for Patient-Specific Preictal State Identification

Epileptic seizures arise from complex, nonlinear interactions within brain networks, yet reliable electroencephalographic (EEG) prediction remains challenging due to the nonstationary and heterogeneous nature of neural dynamics. Existing methods typically analyze EEG data as static or weakly time-dependent snapshots, overlooking the intrinsic dynamics and lacking the geometric sensitivity to capture the hierarchical, localized evolution of the epileptogenic zone. To address these limitations, we propose an offline, patient-specific evaluation of chaotic dynamics-regulated topological learning (CDRTL) for distinguishing preictal from interictal EEG states. This framework unifies chaotic dynamics, multiscale algebraic topology, and local network differentiation. Specifically, we partition EEG signals into discrete functional subnets based on correlation strengths, capturing the multi-scale connectivity of the brain. By modeling each node as a Lorenz oscillator, we embed the underlying chaotic dynamics into the network architecture. We then apply the persistent Laplacian to simultaneously extract topological invariants and geometric shape evolution through harmonic and non-harmonic spectral analysis. Additionally, a node-removal topological differentiation strategy isolates localized neural contributions. Our framework was evaluated on the CHB-MIT database using balanced preictal and interictal labels and stratified channel-level cross-validation within each patient. The results support offline discrimination of preictal and interictal channel-level nodes within fixed patient-specific networks. Because representations are constructed from the complete network, including held-out unlabeled nodes, before cross-validation, the reported performance is specific to this transductive setting and does not establish generalization to unseen EEG windows, seizures, or patients.

q-bio.NC↗

arXiv2026

GrassTop: Grassmannian k-mer Topology for Viral Classification and Phylogenetic Analysis

We introduce GrassTop, a genome representation that integrates Grassmann manifolds and algebraic topology for viral classification and phylogenetic analysis. The framework begins by constructing multiscale topological and spectral descriptors of (k)-mer positional patterns. It then extracts a low-rank subspace that summarizes variation across the filtration and compares genomes using a Grassmannian distance. Although the reported implementation uses the chordal distance, the framework is not restricted to this particular choice. We evaluate GrassTop on four families of viral classification datasets, four phylogenetic clustering datasets, and a sequence perturbation experiment. Under the reported 5-nearest-neighbor protocol, GrassTop achieves higher scores than five published alignment-free reference methods across all reported classification metrics and datasets. Its UPGMA (unweighted pair-group method using arithmetic averages) trees achieve an average label purity of 1.0 on every phylogenetic dataset. The perturbation experiment provides a more nuanced result: the subspace representation differs most clearly from direct comparison of the unprojected feature matrices for SARS-CoV-2, whereas the differences are smaller or non-monotonic for the other datasets. Overall, these results support GrassTop as an effective topological-geometric representation for viral classification and phylogenetic analysis.

q-bio.PE↗

arXiv2026

Data-Adaptive Grassmann Manifold Representations for Spatial Transcriptomics Alignment

Spatial transcriptomics measures gene expression together with spatial coordinates, but many existing analysis methods represent each spot primarily by a single feature vector. We propose GrassST, a subspace method for spatial transcriptomics analysis and cross-slice alignment. For each spatial spot, GrassST constructs a neighborhood from tissue coordinates, fits a low-dimensional subspace to the embedded expression profiles in that neighborhood, and represents the spot by the resulting subspace. These representations are points on a Grassmann manifold and can be compared using distances between subspaces. GrassST selects the neighborhood size and subspace rank from the spectral energy of the data, allowing these parameters to vary across datasets rather than being fixed globally. Experiments on four spatial transcriptomics datasets show that GrassST achieves competitive clustering and cross-slice integration performance under a unified evaluation pipeline.

math.AG↗

arXiv2026

A Hierarchical Sheaf Spectral Embedding Framework for Single-Cell RNA-seq Analysis

Single-cell RNA-seq data analysis typically requires representations that capture heterogeneous local structure across multiple scales while remaining stable and interpretable. In this work, we propose a hierarchical sheaf spectral embedding (HSSE) framework that constructs informative cell-level features based on persistent sheaf Laplacian analysis. Starting from scale-dependent low-dimensional embeddings, we define cell-centered local neighborhoods at multiple resolutions. For each local neighborhood, we construct a data-driven cellular sheaf that encodes local relationships among cells. We then compute persistent sheaf Laplacians over sampled filtration intervals and extract spectral statistics that summarize the evolution of local relational structure across scales. These spectral descriptors are aggregated into a unified feature vector for each cell and can be directly used in downstream learning tasks without additional model training. We evaluate HSSE on twelve benchmark single-cell RNA-seq datasets covering diverse biological systems and data scales. Under a consistent classification protocol, HSSE achieves competitive or improved performance compared with existing multiscale and classical embedding-based methods across multiple evaluation metrics. The results demonstrate that sheaf spectral representations provide a robust and interpretable approach for single-cell RNA-seq data representation learning.

cs.LG↗

arXiv2026

Multi-dimensional Persistent Sheaf Laplacians for Image Analysis

We propose a multi-dimensional persistent sheaf Laplacian (MPSL) framework on simplicial complexes for image analysis. The proposed method is motivated by the strong sensitivity of commonly used dimensionality reduction techniques, such as principal component analysis (PCA), to the choice of reduced dimension. Rather than selecting a single reduced dimension or averaging results across dimensions, we exploit complementary advantages of multiple reduced dimensions. At a given dimension, image samples are regarded as simplicial complexes, and persistent sheaf Laplacians are utilized to extract a multiscale localized topological spectral representation for individual image samples. Statistical summaries of the resulting spectra are then aggregated across scales and dimensions to form multiscale multi-dimensional image representations. We evaluate the proposed framework on the COIL20 and ETH80 image datasets using standard classification protocols. Experimental results show that the proposed method provides more stable performance across a wide range of reduced dimensions and achieves consistent improvements to PCA-based baselines in moderate dimensional regimes.

cs.CV↗

arXiv2025

Multiscale Grassmann Manifolds for Single-Cell Data Analysis

Single-cell data analysis seeks to characterize cellular heterogeneity based on high-dimensional gene expression profiles. Conventional approaches represent each cell as a vector in Euclidean space, which limits their ability to capture intrinsic correlations and multiscale geometric structures. We propose a multiscale framework based on Grassmann manifolds that integrates machine learning with subspace geometry for single-cell data analysis. By generating embeddings under multiple representation scales, the framework combines their features from different geometric views into a unified Grassmann manifold. A power-based scale sampling function is introduced to control the selection of scales and balance in- formation across resolutions. Experiments on nine benchmark single-cell RNA-seq datasets demonstrate that the proposed approach effectively preserves meaningful structures and provides stable clustering performance, particularly for small to medium-sized datasets. These results suggest that Grassmann manifolds offer a coherent and informative foundation for analyzing single cell data.

cs.LG↗

arXiv2025

Color Image Set Recognition Based on Quaternionic Grassmannians

We propose a new method for recognizing color image sets using quaternionic Grassmannians, which use the power of quaternions to capture color information and represent each color image set as a point on the quaternionic Grassmannian. We provide a direct formula to calculate the shortest distance between two points on the quaternionic Grassmannian, and use this distance to build a new classification framework. Experiments on the ETH-80 benchmark dataset and and the Highway Traffic video dataset show that our method achieves good recognition results. We also discuss some limitations in stability and suggest ways the method can be improved in the future.

cs.CV↗

arXiv2024

Geometric Properties and Distance Inequalities on Grassmannians

In this paper we obtain inequalities for the geometric mean of elements in the Grassmannians. These inequalities reflect the elliptic geometry of the Grassmannians as Riemannian manifolds. These include Semi-Parallelogram Law, Law of Cosines and geodesic triangle inequalities.

math.DG↗