Supervised Quantile Normalization for Matrix Factorization using Optimal Transport

We present in this recent work (https://arxiv.org/pdf/2002.03229.pdf) a recent application of our framework to carry out "soft" sorting and ranking using regularized OT. We expand this framework to include "soft" quantile normalization operators that can be differentiated efficiently, and apply it to the problem of dimensionality reduction: we ask how features can be normalized with a target distribution of quantiles to recover "easy to factorize" matrices. We provide algorithms to do this, as well as empirical evidence of recovery.