Isotonic Median Regression via Scaling

This paper gives algorithms for determining isotonic median regressions (i.e., isotonic regression using the L1 metric) satisfying order constraints given by various ordered sets. For a rooted tree the regression can be found in Θ(n log n) time, while for a star it can be found in Θ(n) time, where n is the number of vertices. For bivariate data, when the set is a grid the regression can be found in Θ(n log n) time, while for general sets it can be found in Θ(n log n) time. For vertices in d-dimensional index space, d ≥ 3, the regression can be found in Θ(n log n) time for general placement. When there are multiple data values per point, with N total values, the regression for tree and bivariate grid orders can be determined in Θ(n log n +N log logN) time. Most of the algorithms are based on a scaling approach which exploits the fact that L1 regression values can always be chosen to be data values.