The paper brings new insights into the connection between metric learning and Riemannian statistics. Having parametric and generative models that can go beyond the training data is very useful in many scenarios. However, what I am mainly concerned is its applicability to real world data which is both large scale and high-dimensional. LAND requires the computation of a (here, diagonal) covariance matrix for each data point and of its inverse, which in high dimensions are known to be hard problems. The authors do mention high-dimensionality as a possible issue (lines 265-266), but the paper would need an analysis of the complexity and limitations of the metric introduced. How high is here high-dimensional?