After arguing that choosing the right probability distance is critical for achieving the elusive goals of unsupervised learning, we compare the geometric properties of the two currently most promising distances: (1) the earth-mover distance, and (2) the energy distance, also known as maximum mean discrepancy. These insights allow us to give a fresh viewpoint on reported experimental results and to risk a couple predictions. Joint work with Leon Bottou, Martin Arjovsky, David Lopez-Paz, and Maxime Oquab.