Neural Information Processing Systems http://nips.cc/

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 projected 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 cumulative arc length between the data pair's nearest points on the learned straight measurer lines. After that, by extending such straight lines to general curved forms, we propose a Curvilinear Distance Metric Learning (CDML) method, 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.

Neural Information Processing Systems http://nips.cc/

Originality: The method is new and provides a direct generalization of the Linear Distance Metric learning. Quality: Theorems are clearly interesting to validate the methodology. Fitting capacity result (Theorem 2) ensures that there exists a curvilinear metric that can well separate the data. The Generalization bound ensures empirical loss converges to the expected loss. However, it is unclear whether this ensures that the algorithm converges to the/a Distance introduced by Theorem 2 (the distance well separating the data).

This paper had somewhat mixed reviews, with two positive reviews and one negative review. After the rebuttal, I also took a look at the paper and initiated a discussion with the reviewers. The main concerns of the negative reviewer were in the comparison to existing non-linear metric learning approaches; I also had a similar concern when reading the paper. I think the rebuttal responded well to this (also, it's unrealistic that one could compare with many of the existing non-linear approaches, given how many exist); please include the new results in the final version. My other concern was that the writing is often not super clear---for example, terms like "measurer lines" are not ever precisely defined, and may be unfamiliar to readers who work on metric learning.

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 projected 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 cumulative arc length between the data pair's nearest points on the learned straight measurer lines. After that, by extending such straight lines to general curved forms, we propose a Curvilinear Distance Metric Learning (CDML) method, 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.

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 projected 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 cumulative arc length between the data pair's nearest points on the learned straight measurer lines. After that, by extending such straight lines to general curved forms, we propose a Curvilinear Distance Metric Learning (CDML) method, which adaptively learns the nonlinear geometries of the training data.