For each training instance (x t , y t ), each aspect i ∈ 1...m, and each rank r ∈ 1...k, define an auxiliary variable r indicates whether the true rank y[i] t is to the right or left of a potential rank r. Now suppose that a training set (x 1 , y 1 ), ..., (x T , y T ) is perfectly rankable for each aspect independently.
for each aspect i ∈ 1...m, there exists some ideal model ) such that the signed distance from the prediction to the r th boundary: r has the same sign as the auxiliary variable y[i] t r .