Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold-valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator.

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.

Structured prediction provides a general framework to deal with supervised problems where the outputs have semantically rich structure. While classical approaches consider finite, albeit potentially huge, output spaces, in this paper we discuss how structured prediction can be extended to a continuous scenario. Specifically, we study a structured prediction approach to manifold-valued regression. We characterize a class of problems for which the considered approach is statistically consistent and study how geometric optimization can be used to compute the corresponding estimator. Papers published at the Neural Information Processing Systems Conference.