Weston

Sections S3 to S5 contain the proofs for all results stated in the matching sections from the main article. L always denotes the WW-hinge loss (Definition 1.4) and` always denotes the ordered partition loss (Definition 2.2). The main article sometimes works with generic lossesL and`. The exception is the last section Section S6.2, where the explicit For each i,j [k], σ(i,j) Rk k is the permutation matrix that switches thei-th and j-th index. By convention, ifi = j, then σ(i,j) is the identity.