Reviews: Learning Low-Dimensional Metrics

Summary of the paper: The paper considers the problem of learning a low-rank / sparse and low-rank Mahalanobis distance under relative constraints dist(x_i,x_j) dist(x_i,x_k) in the framework of regularized empirical risk minimization using trace norm / l_{1,2} norm regularization. The contributions are three theorems that provide 1) an upper bound on the estimation error of the empirical risk minimizer 2) a minimax lower bound on the error under the log loss function associated to the model, showing near-optimality of empirical risk minimization in this case 3) an upper bound on the deviation of the learned Mahalanobis distance from the true one (in terms of Frobenius norm) under the log loss function associated to the model. Quality: My main concern here is the close resemblance to Jain et al. (2016). Big parts of the present paper have a one-to-one correspondence to parts in that paper. Jain et al. study the problem of learning a Gram matrix G given relative constraints dist(x_i,x_j) dist(x_i,x_k), but without being given the coordinates of the points x_i (ordinal embedding problem).