$L_p$ Isotonic Regression Algorithms Using an $L_0$ Approach

Significant advances in maximum flow algorithms have changed the relative performance of various approaches to isotonic regression. If the transitive closure is given then the standard approach used for $L_0$ (Hamming distance) isotonic regression (finding anti-chains in the transitive closure of the violator graph), combined with new flow algorithms, gives an $L_1$ algorithm taking $\tilde{\Theta}(n^2+n^\frac{3}{2} \log U )$ time, where $U$ is the maximum vertex weight. The previous fastest was $\Theta(n^3)$. Similar results are obtained for $L_2$ and for $L_p$ approximations, $1 < p < \infty$. For weighted points in $d$-dimensional space with coordinate-wise ordering, $d \geq 3$, $L_0, L_1$ and $L_2$ regressions can be found in only $o(n^\frac{3}{2} \log^d n \log U)$ time, improving on the previous best of $\tilde{\Theta}(n^2 \log^d n)$, and for unweighted points the time is $O(n^{\frac{4}{3}+o(1)} \log^d n)$.