Multi-dimensional Persistent Sheaf Laplacians for Image Analysis

Xiang Xiang Wang; Guo-Wei Wei

cs.CVFeb 16, 2026

Multi-dimensional Persistent Sheaf Laplacians for Image Analysis

Authors: Xiang Xiang Wang, Guo-Wei Wei

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

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.

Figures & tables

Symbol	Description
$\mathcal{I}=\{I_{i}\}_{i=1}^{m}$	Image dataset consisting of $m$ images
$I_{i}$	The $i$ -th image in the dataset
$p\times q$	Spatial resolution of an image
$n=pq$	Dimension of the vectorized image
$\mathbb{R}^{n}$	Vector with dimension n
$\mathbb{R}^{m\times n}$	Matrix with size $m\times n$

Table 1: Summary of main notations used throughout the paper.

Figure 1: Overview of the proposed multi-dimensional persistent sheaf Laplacian framework. Given an image dataset, each image is first vectorized to form a feature matrix. Dimensionality reduction is then performed to obtain multiple reduced feature matrices corresponding to different dimensions $d\in\mathcal{D}$ . For each reduced dimension, a dimension-dependent distance matrix is computed. For each image and each pair $(d,k)$ with neighborhood size $k\in\mathcal{K}$ , a local simplicial complex $\Sigma_{i}^{(d,k)}$ is constructed from the $k$ -nearest neighborhood. A sheaf structure is defined on each local complex, leading to dimension- and scale-dependent sheaf Laplacian matrices $L_{i,h}^{(d,k)}$ for $h\in\{0,1\}$ . Their spectra $\Lambda_{i,h}^{(d,k)}$ are tracked across filtration scales to obtain persistent spectral information. Statistical descriptors extracted from the eigenvalue sets are denoted by $\mathbf{f}_{i,h}^{(d,k)}$ and aggregated across all dimensions and scales to form the unified feature vector $\mathbf{z}_{i}$ . The resulting representation is used for downstream tasks, such as classification.

Statistic	Definition	Interpretation
Zero eigenvalue count	$\phi_{\mathrm{zero}}(\Lambda)=\#\{\lambda\in\Lambda:\lambda=0\}$	Kernel dimension
Smallest nonzero eigenvalue	$\phi_{\min^{+}}(\Lambda)=\min\{\lambda\in\Lambda:\lambda>0\}$	Spectral gap
Maximum eigenvalue	$\phi_{\max}(\Lambda)=\max\Lambda$	Largest spectral value
Sum of eigenvalues	$\phi_{\mathrm{sum}}(\Lambda)=\sum_{\lambda\in\Lambda}\lambda$	Total spectral magnitude
Mean	$\phi_{\mathrm{mean}}(\Lambda)=\frac{1}{\|\Lambda\|}\sum_{\lambda\in\Lambda}\lambda$	Average value
Median	$\phi_{\mathrm{med}}(\Lambda)=\mathrm{median}(\Lambda)$	Central tendency

Table 2: Statistical descriptors computed from each persistent sheaf Laplacian spectrum

\Lambda_{i,h}^{(d,k)}

Dataset	Classes	Images	Image type	Resolution	Characteristics
COIL20 [ 29 ]	20	1440	Grayscale	$128\times 128$	Single object per class
ETH80 [ 18 ]	8	3280	Grayscale (converted)	$128\times 128$	Multiple object instances per class

Table 3: Summary of image datasets used in the experiments.

PCA dim	PCA			MPSL
	Acc	MR	Macro F1	Acc	MR	Macro F1
200	0.7604	0.7604	0.7692	0.9049	0.9050	0.9039
300	0.5632	0.5632	0.5956	0.9000	0.9003	0.8985
400	0.4396	0.4396	0.4689	0.9146	0.9149	0.9133
500	0.3854	0.3854	0.4019	0.9125	0.9130	0.9108
600	0.3410	0.3410	0.3443	0.9118	0.9119	0.9098

Table 4: COIL20 results for PCA and MPSL (single dimension plus multi-

k

) over PCA dimensions 200–1000.

Figure 2: COIL20 performance comparison between PCA and MPSL (single dimension

+

multi-

k

) over PCA dimensions 200–1000, evaluated using Accuracy (Acc), Macro Recall (MR), and Macro F1.

Figure 3: Qualitative UMAP visualizations of PCA-based and MPSL-based representations on the COIL20 dataset under different reduced dimensions. All embeddings are produced using the default UMAP parameters and are intended solely for qualitative visualization. The top row shows embeddings of PCA-based features at different reduced dimensions, while the bottom row shows MPSL representations constructed by aggregating single-dimension features across multiple neighborhood scales. UMAP is used here to highlight the geometric organization and structural consistency of the learned representations rather than to optimize separability.

Method	Acc	MR	Macro F1
PCA (average)	0.3675	0.3675	0.3682
MPSL (single dimension + multi- $k$ , average)	0.9091	0.9093	0.9074
MPSL (multi dimension, multi- $k$ aggregation)	0.9167	0.9169	0.9149

Table 5: Summary performance on COIL20 over PCA dimensions 200–1000. Average results are reported for PCA and MPSL (single dimension

+

multi-

k

), while MPSL (multi-dimension, multi-

k

aggregation) corresponds to a single aggregated representation.

Figure 4: Comparison between MPSL (single dimension

+

multi-

k

) and MPSL (multi-dimension, multi-

k

aggregation) on COIL20. Dashed lines denote the average performance of single-dimension MPSL, while solid lines indicate the aggregated multi-dimension MPSL result.

PCA dim	PCA			MPSL
	Acc	MR	Macro F1	Acc	MR	Macro F1
200	0.6061	0.6061	0.5876	0.6162	0.6162	0.6052
300	0.4817	0.4817	0.4370	0.6256	0.6256	0.6122
400	0.4149	0.4149	0.3520	0.6305	0.6305	0.6218
500	0.3713	0.3713	0.3106	0.6256	0.6256	0.6160
600	0.3326	0.3326	0.2866	0.6345	0.6345	0.6242

Table 6: ETH80 results for PCA and MPSL (single dimension

+

multi-

k

) over PCA dimensions 200 to 1000. Metrics are Accuracy (Acc), Macro Recall (MR), and Macro F1.

Figure 5: ETH80 performance comparison between PCA and MPSL (single dimension

+

multi-

k

) over PCA dimensions 200 to 1000, evaluated using Accuracy (Acc), Macro Recall (MR), and Macro F1.

Method	Acc	MR	Macro F1
PCA (average)	0.3576	0.3576	0.3147
MPSL (single dimension $+$ multi- $k$ , average)	0.6306	0.6306	0.6204
MPSL (multi-dimension, multi- $k$ aggregation)	0.6622	0.6622	0.6536

Table 7: Summary performance on ETH80 over PCA dimensions 200–1000. Average results are reported for PCA and MPSL (single dimension

+

multi-

k

), while MPSL (multi-dimension, multi-

k

aggregation) corresponds to a single aggregated representation.

Figure 6: ETH80 comparison between MPSL (single dimension

+

multi-

k

) and MPSL (multi-dimension, multi-

k

aggregation). Dashed lines indicate the average performance of MPSL (single dimension), while solid reference lines correspond to the aggregated MPSL representation.

Topological Representation Learning

Topological representation learning uses connectivity, holes, and higher-order structure to build or analyze machine-learning features.

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Analytic Torsion and Spectral Gap Capture Persistent-Laplacian Performance

Jernej Grlj, Aaron D. Lauda

Department of Mathematics University of Southern California Los Angeles, CA 90089 · Department of Mathematics Department of Physics University of Southern California Los Angeles, CA 90089

Dimensionality Reduction Spectral Methods Topological Representation Learning

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While persistent Laplacians (PL) offer a richer geometric representation of data than persistent homology, utilizing their full eigenspectrum for learning tasks is often hampered by high dimensionality and the ``varying length'' problem across different filtration scales. We propose a compact spectral representation that distills the persistent Laplacian into three mathematically grounded invariants: Betti numbers, the spectral gap, and analytic torsion. Across benchmark datasets including MNIST, QM-3D, and SKEMPI WT, we demonstrate that this reduced feature space captures the essential predictive signal of the full spectrum, and in some cases outperforms it, while significantly reducing computational overhead and preventing the noise introduced by higher-frequency eigenvalues. Our results suggest that these invariants provide a principled, fixed-length interface between spectral geometry and topological learning.

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