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

Xiang Xiang Wang; Guo-Wei We

cs.LGMar 27, 2026

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

Authors: Xiang Xiang Wang, Guo-Wei We

Organizations: Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA. · Department of Biochemistry and Molecular Biology, Michigan State University, East Lansing, MI 48824, USA. · Department of Electrical and Computer Engineering, Michigan State University, East Lansing, MI 48824, USA.

Abstract

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.

Figures & tables

Symbol	Description
$\mathbb{R}$	Real number field
$\mathbb{R}^{n}$	$n$ dimensional real vector space
$\mathbb{R}^{n\times n}$	Set of all real $n\times n$ matrices
$X$	Single cell data matrix with rows representing cells
$N$	Number of cells in the dataset
$d$	Ambient feature dimension of the data

Table 1: Notations used in the paper.

Figure 1: Overview of the proposed cell-centered persistent sheaf Laplacian framework for single-cell data analysis. Starting from the single-cell gene expression matrix, a family of scale-dependent low-dimensional representations is constructed to capture cellular structures at different resolutions. Based on these representations, scale-dependent distance matrices are computed to quantify pairwise relationships between cells. For each cell $i$ , a cell-centered neighborhood is identified from the distance matrices, and neighborhood-induced simplicial complexes are constructed to encode local topological structures around the cell. On these cell-centered simplicial complexes, persistent sheaf Laplacians are computed via filtration sampling and spectral computation, yielding spectral information that characterizes the underlying sheaf-enhanced topological structures. These spectral features are aggregated to form a feature vector $Z_{i}$ associated with cell $i$ . Collecting the feature vectors for all cells gives $\{Z_{i}\}_{i=1}^{m}$ , where $m$ denotes the total number of cells, which serves as input for downstream learning tasks, e.g., classification.

GEO accession	Reference	Cell types	Samples	Genes
GSE45719	Deng [ 14 ]	8	300	22 431
GSE67835	Dramanis [ 13 ]	8	420	22 084
GSE75748cell	Chu [ 7 ]	7	1 018	19 189
GSE75748time	Chu [ 7 ]	6	758	19 189
GSE82187	Gokce [ 19 ]	10	705	18 840
GSE84133human1	Baron [ 2 ]	14	1 895	20 125

Table 2: Summary of benchmark single-cell RNA-seq datasets used in the experiments. Each dataset includes its GEO accession number, reference, number of annotated cell types, and the numbers of samples (cells) and genes.

Dataset	HSSE	MDG [ 17 ]	PCA [ 17 ]	UMAP [ 17 ]	NMF [ 17 ]	tSNE [ 17 ]
GSE45719	0.935	0.959	0.770	0.842	0.739	0.526
GSE67835	0.949	0.926	0.879	0.929	0.799	0.707
GSE75748cell	0.990	0.969	0.885	0.936	0.846	0.814
GSE75748time	0.941	0.929	0.901	0.871	0.872	0.708
GSE82187	0.986	0.950	0.860	0.968	0.852	0.815
GSE84133human1	0.961	0.949	0.636	0.946	0.622	0.895

Table 3: Macro-F1 comparison across 12 benchmark single-cell RNA-seq datasets. HSSE denotes the proposed Hierarchical Sheaf Spectral Embedding. All baseline results (MDG, PCA, UMAP, NMF, and tSNE) are taken from Feng et al. [ 17 ] . For each dataset, the best-performing method is highlighted in bold.

Dataset	HSSE	MDG [ 17 ]	PCA [ 17 ]	UMAP [ 17 ]	NMF [ 17 ]	tSNE [ 17 ]
GSE45719	0.937	0.962	0.780	0.848	0.747	0.532
GSE67835	0.940	0.920	0.876	0.921	0.800	0.699
GSE75748cell	0.990	0.968	0.887	0.934	0.848	0.811
GSE75748time	0.941	0.930	0.901	0.870	0.873	0.708
GSE82187	0.985	0.955	0.863	0.966	0.857	0.807
GSE84133human1	0.955	0.945	0.644	0.938	0.626	0.883

Table 4: Macro-Recall comparison across 12 benchmark single-cell RNA-seq datasets. All baseline results (MDG, PCA, UMAP, NMF, and tSNE) are taken from Feng et al. [ 17 ] . Boldface indicates the highest Macro-Recall achieved for each dataset.

Dataset	HSSE	MDG [ 17 ]	PCA [ 17 ]	UMAP [ 17 ]	NMF [ 17 ]	tSNE [ 17 ]
GSE45719	0.991	0.994	0.950	0.965	0.944	0.831
GSE67835	0.986	0.986	0.984	0.992	0.975	0.920
GSE75748cell	0.998	0.996	0.983	0.992	0.974	0.965
GSE75748time	0.989	0.990	0.979	0.971	0.980	0.927
GSE82187	1.000	0.995	0.992	0.993	0.988	0.961
GSE84133human1	0.989	0.987	0.951	0.979	0.944	0.980

Table 5: Macro-AUC comparison across 12 benchmark single-cell RNA-seq datasets. All baseline results (MDG, PCA, UMAP, NMF, and tSNE) are taken from Feng et al. [ 17 ] . For each dataset, the best result is shown in bold.

Figure 2: Dataset-wise performance differences between HSSE and MDG across three evaluation metrics. Each panel shows paired results for HSSE and MDG on the same dataset using a dumbbell plot: (left) Macro-F1, (middle) Macro-Recall, and (right) Macro-AUC (OVR). Horizontal line segments connect MDG (circle) and HSSE (square), and the percentage annotation indicates the relative change of HSSE compared to MDG. Positive values denote improvements achieved by HSSE, while negative values indicate datasets where MDG performs slightly better.

Figure 3: Performance gains of HSSE over classical dimensionality reduction methods (PCA, UMAP, NMF, and t-SNE) across twelve benchmark single-cell RNA-seq datasets. Each panel reports the gain of HSSE in terms of (left) Macro-F1, (middle) Macro-Recall, and (right) Macro-AUC, respectively. Positive values indicate consistent performance improvements achieved by HSSE over the corresponding baseline method.

Figure 4: Sensitivity analysis with respect to the number of neighborhood sizes under a fixed scale configuration

S=\{5,14,25,37,50\}

. Results are shown for four representative datasets: GSE67835, GSE82187, GSE84133mouse1, and GSE94820. The horizontal axis denotes the number of neighborhood sizes used (

|\mathcal{K}|=1

10

). Six evaluation metrics are reported: Accuracy, Macro F1, Weighted F1, Macro Recall, MCC, and Macro AUC.

Figure 5: Effect of the scale configuration size (

|S|

) on classification performance under a fixed neighborhood-size set

\mathcal{K}=\{5,10,15,20,30,40,50,60,70,80\}

. Macro F1 and Accuracy are reported as

|S|

increases from 1 to 5 on four representative datasets: GSE67835, GSE82187, GSE84133mouse1, and GSE94820.

Single-Cell RNA Sequencing

Single-cell RNA sequencing measures gene expression in individual cells to map cell types, states, and responses across tissues and conditions.

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Single-cell RNA-seq representation learning is fundamentally label-free: cell identities, states, and contexts are not fixed training targets, so what constitutes signal or nuisance is analysis-dependent. A single representation must therefore preserve biological identity and state, remain robust to nuisance context, and retain the gene-level variation needed for expression analysis, three demands we call the representation trilemma. To tackle this problem, we introduce scTrilemma, a latent-bottleneck VAE that routes expression-derived variation to the embedding, the decoder, or the prior rather than forcing all of it through one embedding. It gates gene tokens by expression, routes the cell representation through the decoder, and conditions the prior on unlabeled pseudo-bulk context, under a single reconstruction objective and without target annotations or auxiliary representation losses. In release-based zero-shot evaluation on successive CZ CELLxGENE Census releases, scTrilemma leads all three demands at once and preserves biological-state, differential-expression, and pathway structure across multiple disease settings. Latent interventions further show that context can be removed at almost no cost to the other demands, leaving identity against fidelity as the remaining tension. Code is publicly available at https://github.com/yunhak0/scTrilemma.

Abstract

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