Given a real-valued weighted function $f$ on a finite dag, the $L_p$ isotonic regression of $f$, $p \in [0,\infty]$, is unique except when $p \in [0,1] \cup \{\infty\}$. We are interested in determining a ``best'' isotonic regression for $p \in \{0, 1, \infty\}$, 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 $L_\infty$ the strict and lex regressions are unique and the same. For $L_1$, strict $q \scriptstyle\searrow 1$ is unique, but we show that $q \scriptstyle\nearrow 1$ 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 $p{ \scriptstyle \nearrow} 0$ is the same as lex regression. We also give algorithms for computing the best $L_p$ isotonic regression in certain situations.

翻译：给定有限有向无环图上的实值加权函数 $f$，其 $L_p$ 保序回归（$p \in [0,\infty]$）在 $p \in [0,1] \cup \{\infty\}$ 时并非唯一。我们感兴趣的是确定 $p \in \{0, 1, \infty\}$ 时的“最优”保序回归，其中“最优”指回归结果不仅满足最小范数，还具有更强的性质。一种方法是采用严格 $L_p$ 回归，即当 $q$ 趋近于 $p$ 时最优 $L_q$ 逼近的极限；另一种是词典回归，基于回归误差的词典序。对于 $L_\infty$，严格回归和词典回归唯一且相同。对于 $L_1$，$q \scriptstyle\searrow 1$ 的严格回归唯一，但我们证明 $q \scriptstyle\nearrow 1$ 可能不唯一，且即使唯一，两个极限也可能不同。对于 $L_0$，严格回归和词典回归通常既不唯一，也不总具有相同的最优回归集合，但通过将 $L_p$ 优化的目标扩展到 $p < 0$，我们证明 $p{ \scriptstyle \nearrow} 0$ 与词典回归等价。我们还给出了在某些情况下计算最优 $L_p$ 保序回归的算法。