Best L_p Isotonic Regressions, p ∈{0, 1, ∞}
share
Given a real-valued weighted function f on a finite dag, the L_p isotonic regression of f, p ∈ [0,∞], is unique except when p ∈ [0,1] ∪{∞}. We are interested in determining a “best” isotonic regression for p ∈{0, 1, ∞}, where by best we mean a regression satisfying stronger properties than merely having minimal norm. One approach is to use strict L_p regression, which is the limit of the best L_q approximation as q approaches p, and another is lex regression, which is based on lexical ordering of regression errors. For L_∞ the strict and lex regressions are unique and the same. For L_1, strict q ↘ 1 is unique, but we show that q ↗ 1 may not be, and even when it is unique the two limits may not be the same. For L_0, in general neither of the strict and lex regressions are unique, nor do they always have the same set of optimal regressions, but by expanding the objectives of L_p optimization to p < 0 we show p↗ 0 is the same as lex regression. We also give algorithms for computing the best L_p isotonic regression in certain situations.
READ FULL TEXT
