Strict L∞ Isotonic Regression

Abstract

Given a function f and weights w on the vertices of a directed acyclic graph G, an isotonic regression of (f, w) is an order-preserving real-valued function that minimizes the weighted distance to f among all order-preserving functions. When the distance is given via the supremum norm there may be many isotonic regressions. One of special interest is the strict isotonic regression, which is the limit of p-norm isotonic regression as p approaches infinity. Algorithms for determining it are given. We also examine previous isotonic regression algorithms in terms of their behavior as mappings from weighted functions over G to isotonic functions over G, showing that the fastest algorithms are not monotonic mappings. In contrast, the strict isotonic regression is monotonic.

Isotonic Regression via Partitioning

“…A similar situation occurs for L ∞ isotonic regression. Algorithms have been developed for strict L ∞ isotonic regression [23,25], which is the limit, as p → ∞, of L p isotonic regression. This is also known as "best best" L ∞ regression [13].…”

Isotonic Regression for Multiple Independent Variables

“…It has been called the "best best" L ∞ isotonic regression [21], and the limit process is known as the "Polya algorithm" [28]. For arbitrary dags, given the transitive closure Strict can be determined in Θ(n 2 log n) time [37].…”

“…Strict minimizes the number of large errors [37], in that, for any dag G and data (f , w), if g = Strict(f, w) is an isotonic function on G, then there is a C > 0 such that g has more vertices with regression error ≥ C than does Strict(f, w), and for any D > C, g and Strict(f, w) have the same number of vertices with regression error ≥ D (there might be a d < C where g has fewer vertices with regression error ≥ d than does Strict, but the emphasis is on large errors). For example, for the function with values 3, 1, 2.5 and weights 2, 2, 1, Strict is 2, 2, 2.5, as are all L p isotonic regressions for 1 < p < ∞, but all of the other L ∞ regression mappings considered here have a nonzero error for the third value.…”

“…A proof of correctness, and the fact that there are only n iterations of the loop, appears in [37]. The algorithm in [37], applicable to all dags, first uses the transitive closure to find all violating pairs. Here rendezvous vertices are used to find important violating pairs incrementally.…”

“…The L ∞ Prefix and Strict algorithms for arbitrary dags utilize the transitive closure [37,38], and the rendezvous vertices in R were used to perform operations on the transitive closure more efficiently than can be done on the closure itself. For weighted data and d ≥ 2 these algorithms are faster for points in general position then the fastest previous results for points in a grid.…”