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Mahalanobis distance on extended Grassmann manifolds for variational pattern analysis.

Yoshikazu Washizawa, Seiji Hotta

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    |October 21, 2014
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    This study introduces novel methods to measure pattern similarity by extending the Mahalanobis distance on Grassmann manifolds. These advanced techniques improve classification accuracy by better accounting for pattern structure variations.

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    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.