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.

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.

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.

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 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.