معيارهاي جديد فاصله برمبناي هندسه داده‌ها و استفاده از هندسه‌هاي نااقليدسي

New Distance Metrics Based on the Geometry of Data an‎d Using Non-Euclidean Geometries

مهندسي هوش مصنوعي

Many machine learning algorithms use Euclidean distance as a common metric to calculate similarities between data. However, Euclidean distance is not valid when data lie on a manifold with non-zero curvatures. Therefore, we propose a new non-parametric approach that uses curvatures to calculate distances. Curvature is an appealing feature for this purpose since it is not altered by isometries. In this thesis, we propose two formulas for measuring distances on a manifold with constant curvature, an‎d their validities are proven using the theorems of differential geometry. Utilizing these formulas, an algorithm is developed to measure the distance between a point an‎d the center of a class. In the proposed algorithm geodesies are divided into equal linear segments, assuming that the curvature remains constant within each segment. This assumption is shown to be valid in many data spaces experimentally. Observed data near each segment is used to estimate the curvatures an‎d calculate the distance within each segment. Finally, the total distance is computed by summing up the non-Euclidean lengths of all segments. The proposed method is a supervised version of k-means, named non-Euclidean centers (NEK-centers). the correctness of the proposed method is validated using the Riemann tensor an‎d its related theorems in differential geometry. Furthermore, experimental results show that our method performs well in real-world data classification applications. Symmetric positive definite (SPD) matrices, which lie on manifolds, are used for input data representations.

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