Regressive Virtual Metric Learning
We are interested in supervised metric learning of Mahalanobis like distances. Existing approaches mainly focus on learning a new distance using similarity and dissimilarity constraints between examples. In this paper, instead of bringing closer examples of the same class and pushing far away examples of different classes we propose to move the examples with respect to virtual points. Hence, each example is brought closer to a a priori defined virtual point reducing the number of constraints to satisfy. We show that our approach admits a closed form solution which can be kernelized. We provide a theoretical analysis showing the consistency of the approach and establishing some links with other classical metric learning methods. Furthermore we propose an efficient solution to the difficult problem of selecting virtual points based in part on recent works in optimal transport. Lastly, we evaluate our approach on several state of the art datasets.
Summary: This paper proposed a novel approach for efficient metric learning. The objective of classic metric learning is to solve the positive semi-definite matrix $M$ such that the distance between examples from the same class get closer while the distance between examples from different classes get further in the distance define by M. Unlike these methods, this paper proposed a idea of virtual point-based regression formulation. The points from the same class are pulled toward the virtual points, thus their distance to each other become closer and their distance to points in other classes become further. The paper describes a closed form solution to compute the regression matrix and two different ways to discover the virtual points. The method is evaluated in 13 different metric learning datasets and compared with several standard baselines.  Pros: - The experiment result is complete and supports the proposed method. It is evaluated in 13 different datasets, linear and RBF kernel, and compared with several baselines. - When learning the metric, the number of constraints only grows with number of virtual points, which is linear to the training size. For existing metric learning, the number of constraints is quadratic to the training size. - The paper is well written and easy to follow. The introduction and related work clearly explains the novelty and key contribution of the proposed method clearly. Supplementary material provides detail explanation of theorem.  Cons: - If I understand it correctly, the selection process described in in Algorithm 1 is quadratic to the size of S because of two argmax over x? - The performance of the proposed method doesn't work well with linear kernel. Suggestion: - Since the proposed method requires less constraints, the learning process should be more efficient than other methods. It would be helpful if we could also see some results for the training time comparison against other methods.- Compare to LMNN, could the proposed method provide additional advantage in adaptive(online) metric learning? - To further demonstrate the benefit of OT on learning set, the author could consider adding the baseline by randomly selecting an example from each class as virtual point.