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.

💡 Research Summary

The paper introduces a novel framework called Hierarchical Sheaf Spectral Embedding (HSSE) for representing single‑cell RNA‑seq data in a way that captures heterogeneous local structures across multiple scales. The authors start by generating a family of low‑dimensional embeddings of the gene‑expression matrix, each obtained under different neighborhood‑size or distance‑threshold settings, thereby providing representations at various resolutions. For every cell, and for each embedding scale, a cell‑centered k‑nearest‑neighbor patch is constructed, which is then turned into a simplicial complex (including 0‑, 1‑, and possibly 2‑simplices). On each complex a data‑driven cellular sheaf is defined: each simplex carries a vector space (e.g., the average expression of the cells it contains) and linear restriction maps encode how information is transferred between adjacent simplices, reflecting local gene‑co‑expression patterns.

To capture multi‑scale structure, the authors compute persistent sheaf Laplacians over sampled filtration intervals. For a filtration parameter ℓ, the coboundary operators δ are built from the restriction maps, and the sheaf Laplacian L⁽q⁾₍a,b₎ = δ₍a,b₎⁽q⁾(δ₍a,b₎⁽q⁾)ᵀ + δ₍a₎⁽q‑1⁾(δ₍a₎⁽q‑1⁾)ᵀ is formed for each degree q. The eigenvalue spectrum of L encodes both topological invariants (harmonic eigenvalues near zero) and geometric variations (non‑harmonic eigenvalues) as the filtration progresses. For each combination of embedding scale, neighborhood size, and filtration interval, the authors extract statistical descriptors of the eigenvalues (mean, variance, extrema, higher‑order moments) and concatenate them into a fixed‑length feature vector zᵢ for each cell.

These vectors are then fed directly into downstream classifiers (e.g., linear SVM, Random Forest) without any additional model training. The authors evaluate HSSE on twelve publicly available single‑cell RNA‑seq datasets spanning mouse, human, immune, neuronal, and pancreatic systems. Using a consistent 5‑fold cross‑validation protocol, HSSE achieves an average classification accuracy of 92.3 % and often outperforms state‑of‑the‑art multiscale embedding methods (PHATE, Diffusion Maps), persistent homology, and persistent Laplacian approaches by 2–4 % absolute gain. Notably, in datasets where cell types are finely intermingled (e.g., immune subpopulations), the sheaf‑spectral features provide more stable separation than distance‑based embeddings.

Ablation studies demonstrate that (1) the data‑driven design of restriction maps, (2) the number of sampled filtration intervals, and (3) the diversity of neighborhood sizes all contribute positively to performance, with the full combination yielding the best results. The paper also discusses computational scalability: the persistent sheaf Laplacian package used can handle large complexes efficiently, and the overall pipeline scales linearly with the number of cells after the initial embedding step.

In summary, HSSE integrates topological, geometric, and statistical information through the lens of cellular sheaves and persistent Laplacians, delivering a model‑free, interpretable, and scalable representation for single‑cell transcriptomics. The work opens avenues for applying sheaf‑theoretic tools to spatial transcriptomics, multimodal omics integration, and real‑time analysis of massive single‑cell atlases.