$L_0$ Isotonic Regression With Secondary Objectives

We provide algorithms for isotonic regression minimizing $L_0$ error (Hamming distance). This is also known as monotonic relabeling, and is applicable when labels have a linear ordering but not necessarily a metric. There may be exponentially many optimal relabelings, so we look at secondary criteria to determine which are best. For arbitrary ordinal labels the criterion is maximizing the number of labels which are only changed to an adjacent label (and recursively apply this). For real-valued labels we minimize the $L_p$ error. For linearly ordered sets we also give algorithms which minimize the sum of the $L_p$ and weighted $L_0$ errors, a form of penalized (regularized) regression. We also examine $L_0$ isotonic regression on multidimensional coordinate-wise orderings. Previous algorithms took $\Theta(n^3)$ time, but we reduce this to $o(n^{3/2})$.