Multi-dimensional Persistent Sheaf Laplacians for Image Analysis

TL;DR

Experimental results show that the proposed MPSL framework provides more stable performance across a wide range of reduced dimensions and achieves consistent improvements to PCA-based baselines in moderate dimensional regimes.

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 (6)

• 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.
• 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.
• 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.
• 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.

Theorems & Definitions (5)

• Definition 2.1: Simplicial Complex
• Definition 2.2: Cellular Sheaf
• Remark 2.1
• Remark 3.1
• Remark 3.2: Choice of the kernel scale parameter