Weighted isotonic regression under the L₁ norm

Isotonic Regression via Partitioning

“…For example, suppose the values and weights on a linear ordering are (4,1), (3,10), (1,1), (2,1). The first partitioning involves values 2 and 3, and results in all vertices being assigned to the interval [3,4].…”

“…A d-dimensional grid has Θ(dn) grid edges, and thus the general algorithm in [2] shows that the L 1 isotonic regression can be found in Θ(n 2 log n) time. However for vertices in arbitrary positions Θ(n 2 ) edges may be needed, as is shown in Figure 2, which would result in Θ(n 3 ) time.…”

“…by placing arbitrary values with weight 0 on V ′ \ V and applying the general algorithm in [2], noting that the number of stages is linear in the number of vertices.…”

“…For example, one may believe that average weight is an increasing function of height and of S < M < L < XL shirt size. That is, if the shirt size is held constant the average weight ordered metric set L 1 L 2 L ∞ linear Θ(n log n) Θ(n) Θ(n log n) [1,22] PAV * tree Θ(n 2 ) Θ(n log n) Θ(n log n) [5] [15] * 2-dim Θ(n 2 log n) Θ(n 2 ) Θ(n log n) grid * [21] * 2-dim Θ(n 3 ) Θ(n 3 ) Θ(n log 2 n) arbitrary * [21] [24] d ≥ 3 dim Θ(n 2 log n) Θ(n 4 ) Θ(n log n) grid * * * d ≥ 3 dim Θ(n 3 ) Θ(n 4 ) Θ(n log d n) arbitrary * * [24] arbitrary Θ(nm + n 2 log n) Θ(n 4 ) Θ(m log n) [2] [14] [24] * indicates that it is implied by the result for arbitrary ordered sets Table 1: Times of fastest previously known isotonic regression algorithms increases with height, and similarly the weight increases if the height is fixed and the shirt size increases. However, there are no assumptions of what happens if height increases but shirt size decreases.…”

“…For arbitrary DAGs, Maxwell and Muck-stadt [14], with a small correction by Spouge et al [21], gave a Θ(n 4 ) time algorithm for L 2 isotonic regression; Angelov et al [2] gave a Θ(nm + n 2 log n) time algorithm for L 1 ; and Stout [24] gave a Θ(m log n) time algorithm for L ∞ , based on a small modification of an algorithm of Kaufman and Tamir [12]. For 1 < p < ∞ the L p mean of a set is unique (see the Appendix), as is the isotonic regression, but this is not always true for L 1 .…”