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Overcoming Dimensionality Constraints: A Gershgorin Circle Theorem-Based Feature Extraction for Weighted Laplacian

Sahaj Anilbhai Patel¹, Abidin Yildirim¹

¹Department of Electrical and Computer, University of Alabama at Birmingham, Birmingham, AL 35205, USA.

Journal of Imaging

|May 24, 2024

Summary

This study introduces a new dimensionality reduction method for weighted Laplacian matrices in computer vision, called Gershgorin Circle Feature Extraction (GCFE). GCFE effectively reduces matrix size while maintaining high accuracy and computational efficiency.

Keywords:

Gershgorin circle theorem complex graph structure convolution neural network dimensionality reduction feature extraction weighted Laplacian matrix

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Area of Science:

Computer Vision
Graph Theory
Linear Algebra

Background:

Weighted Laplacian matrices are crucial for analyzing complex graph structures in computer vision.
Increasing graph complexity leads to high-dimensional Laplacian matrices, causing the "curse of dimensionality".

Purpose of the Study:

To introduce a novel method for reducing the dimensionality of weighted Laplacian matrices.
To address the computational challenges posed by high-dimensional matrices in computer vision applications.

Main Methods:

Utilized the Gershgorin circle theorem to transform the weighted Laplacian matrix.
Developed a feature extraction technique (GCFE) by estimating eigenvalue inclusions.
Transformed the matrix into a strictly diagonal domain for dimensionality reduction.

Main Results:

GCFE demonstrated superior performance with notable Z-scores compared to existing methods like I-PCA and kernel PCA.
GCFE maintained consistent accuracy across varying image patch sizes, unlike other methods.
Achieved high classification accuracy and computational efficiency, with low standard deviation on datasets like E_Balanced and E_MNSIT.

Conclusions:

The Gershgorin Circle Feature Extraction (GCFE) method offers an effective solution for weighted Laplacian matrix dimensionality reduction.
GCFE shows significant potential for efficient and accurate feature extraction in computer vision tasks.
Requires fewer training parameters for deep learning models compared to traditional methods.