Supplementary material: Weston-Watkins Hinge Loss and Ordered Partitions

References to contents in the supplementary material are prefixed with an "S", e.g., Lemma S3.20, References to contents in the main article do not have any prefix, e.g., Theorem 1.3 and Section 1.4. We introduce notations in addition to those already defined in in the main article's Section 1.3. The exception is the last section Section S6.2, where the explicit All vectors are column vectors unless stated otherwise. If j = 1, then the result is trivial. Now, for the first case, suppose that y null { 1,j }.