Mahalanobis distance on extended Grassmann manifolds for variational pattern analysis

Area of Science:

• Computer Science
• Machine Learning
• Pattern Recognition

Background:

• Pattern variations are often modeled as linear manifolds or low-dimensional subspaces.
• Conventional similarity measures are deterministic and fail with non-isotopic distributions, leading to unreliable distance measurements.
• Existing vector-based distance measurements in Euclidean space also face limitations with complex data distributions.

Purpose of the Study:

• To develop flexible methods for extending the Mahalanobis distance on extended Grassmann manifolds.
• To enable more reliable measurement of pattern (dis)similarity based on intrinsic pattern structure.
• To improve the performance of pattern classification algorithms.

Main Methods:

• Systematized representations of variational patterns using the Grassmann manifold.
• Introduced Mahalanobis distance on the Grassmann manifold as an extension of Euclidean distance.
• Developed two novel methods to flexibly extend Mahalanobis distance on extended Grassmann manifolds.

Main Results:

• The proposed methods effectively measure pattern (dis)similarity by considering the underlying pattern structure.
• Experimental evaluations demonstrated the efficacy of the extended Mahalanobis distance on Grassmann manifolds.
• The new methods resulted in a lower error classification rate compared to conventional approaches.

Conclusions:

• The extended Mahalanobis distance on Grassmann manifolds provides a robust framework for pattern similarity measurement.
• These methods offer a significant improvement over traditional techniques, particularly for non-isotopic data distributions.
• The developed approach enhances classification accuracy by leveraging the geometric structure of pattern variations.