Curvilinear Distance Metric Learning

Shuo Chen, Lei Luo, Jian Yang, Chen Gong, Jun Li, Heng Huang

Abstract: Distance Metric Learning aims to learn an appropriate metric that faithfully measures the distance between two data points. Traditional metric learning methods usually calculate the pairwise distance with fixed distance functions (\emph{e.g.,}\ Euclidean distance) in the specified type of feature spaces. However, they fail to learn the underlying geometries of the sample space, and thus cannot exactly predict the intrinsic distances between data points. To address this issue, we first reveal that the traditional linear distance metric is equivalent to the accumulated arc length between the data pair's nearest points on the learned straight measurer lines. After that, we propose a new Curvilinear Distance Metric Learning (CDML) method to extend such straight lines to general curved forms, which adaptively learns the nonlinear geometries of the training data. By virtue of Weierstrass theorem, the proposed CDML is equivalently parameterized with a 3-order tensor, and the optimization algorithm is designed to learn the tensor parameter. Theoretical analysis is derived to guarantee the effectiveness and soundness of CDML. Extensive experiments on the synthetic and real-world datasets validate the superiority of our method over the state-of-the-art metric learning models.

Code Link: https://github.com/functioncs/CDML

CMT Num: 2374