Multi-dimensional Persistent Sheaf Laplacians for Image Analysis

2026-02-16

AI Summary

This work proposes a simplex-based multidimensional persistence Laplacian framework to address the sensitivity of dimensionality reduction methods like Principal Component Analysis (PCA) to the choice of target dimension and their resulting performance instability. The approach introduces, for the first time, the multidimensional persistence Laplacian into image analysis by integrating multiscale topological spectra across multiple reduced dimensions and employing statistical aggregation to produce robust feature representations. Experimental results on the COIL20 and ETH80 datasets demonstrate that the proposed method significantly outperforms PCA baselines at moderate dimensions and exhibits greater stability across varying target dimensions, thereby enhancing both the discriminability and robustness of image representations.

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Computer Vision: Representation Learning for Vision
• Intelligent Robots: Multimodal Perception & Sensor Fusion

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

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.

Problem

• dimensionality reduction
• PCA sensitivity
• image analysis
• reduced dimension selection

Innovation

• multi-dimensional persistent sheaf Laplacian
• simplicial complexes
• topological spectral representation
• dimensionality reduction
• image analysis