Regularized least squares locality preserving projections with applications to image recognition

Area of Science:

• Machine Learning
• Data Science
• Dimensionality Reduction

Background:

• Locality Preserving Projection (LPP) is a dimensionality reduction technique preserving local data structure.
• LPP faces challenges with undersampled data (high feature dimension relative to samples), leading to ill-posed problems.
• The generalized eigenvalue problem in LPP becomes unstable under these conditions.

Purpose of the Study:

• To address the undersampled problem in Locality Preserving Projection (LPP).
• To establish a novel connection between LPP and multivariate linear regression.
• To develop robust methods for solving LPP in small-sample scenarios.

Main Methods:

• Demonstrating the equivalence of LPP to a multivariate linear regression under specific conditions.
• Connecting LPP to a multi-column least squares problem.
• Proposing two novel regularized least squares methods for LPP.

Main Results:

• The proposed methods effectively solve LPP even with undersampled data.
• Experimental results validate the performance of the new LPP approaches on real-world datasets.
• The connection to least squares provides a stable framework for LPP.

Conclusions:

• The study successfully reformulates LPP as a solvable least squares problem.
• The proposed regularized methods offer a robust solution to LPP's small-sample-size limitations.
• This work enhances the applicability of LPP in practical, data-constrained scenarios.