  Quantile normalisation is a popular normalisation method for data subject to
unwanted variations such as images, speech, or genomic data. It applies a
monotonic transformation to the feature values of each sample to ensure that
after normalisation, they follow the same target distribution for each sample.
Choosing a "good" target distribution remains however largely empirical and
heuristic, and is usually done independently of the subsequent analysis of
normalised data. We propose instead to couple the quantile normalisation step
with the subsequent analysis, and to optimise the target distribution jointly
with the other parameters in the analysis. We illustrate this principle on the
problem of estimating a linear model over normalised data, and show that it
leads to a particular low-rank matrix regression problem that can be solved
efficiently. We illustrate the potential of our method, which we term SUQUAN,
on simulated data, images and genomic data, where it outperforms standard
quantile normalisation.
