Reviews: Manifold Structured Prediction

Summary: This paper is an extension of the results presented in "A Consistent Regularization Approach for Structured Prediction" by Ciliberto et al. It focuses on the specific case where the output space is a Riemannian manifold, and describes/proves sufficient conditions for loss functions defined over manifolds to have the properties of what is called a "Structure Encoding Loss Function" (SELF). Ciliberto et al presents an estimator that, when used with a SELF, has provable universal consistency and learning rates; this paper extends this estimator and these prior theoretical results to be used also with the aforementioned class of loss functions defined over manifolds, with a specific focus placed on the squared geodesic distance. After describing how inference can be achieved using the previously defined estimator for the specific output spaces defined here, experiments are run on a synthetic dataset with the goal of learning the inverse function over the set of positive-definite matrices and a real dataset consisting of fingerprint reconstruction. Comments: This work is well-written and well-organized, and it is easy to follow all of the concepts being presented.