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 Θ̃(n^2+n^3/2log U ) time, where U is the maximum vertex weight. The previous fastest was Θ(n^3). Similar results are obtained for L_2 and for L_p approximations, 1 < p < ∞. For weighted points in d-dimensional space with coordinate-wise ordering, d ≥ 3, L_0, L_1 and L_2 regressions can be found in only o(n^3/2log^d n log U) time, improving on the previous best of Θ̃(n^2 log^d n), and for unweighted points the time is O(n^4/3+o(1)log^d n).
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