ndscape: a structural map of the nested dichotomy space
Abstract
Nested dichotomies (NDs) reduce multiclass classification to a tree of binary classifiers. Tree structure can change accuracy, bias, and variance, but the full space of structures has not been mapped in aggregate. We introduce ndscape, a two-dimensional mapping of ND structures based on depth-weighted structural tokens. We use this embedding to conduct what is, to our knowledge, the first bias-variance analysis across the ND structure space. In simulations with 6 to 10 classes, NDs with similar tree structures exhibit similar squared bias, variance, and squared probability loss, regardless of whether the binary classifier is logistic regression or a decision tree. Variance, however, follows mean root-to-leaf depth when logistic regression is used at each node, so flatter, more balanced NDs are more stable, whereas with decision-tree node models, variance is not associated with depth and differs little across ND structures. On six datasets with 6 to 19 classes, held-out accuracy, log loss, and prediction variance remain spatially organised on the same topology-only map. None of the six standard construction heuristics tested selects a tree from the best-performing cluster on any dataset, although some select high-performing individual trees. The ndscape embedding turns ND selection from one opaque construction step into a readable comparison of class groupings, exposing the upper-level splits that distinguish high-performing and stable regions.
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BibTeX
@misc{dixmatthews2026ndscape,
title = {{ndscape: a structural map of the nested dichotomy space}},
author = {Maxwell Dix-Matthews and Melinda Hodkiewicz and Adriano Polpo},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18050},
eprint = {2610.18050},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18050},
}
RIS
TY - UNPB
TI - ndscape: a structural map of the nested dichotomy space
AU - Maxwell Dix-Matthews
AU - Melinda Hodkiewicz
AU - Adriano Polpo
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18050
UR - https://arxiv.org/abs/2610.18050
AB - Nested dichotomies (NDs) reduce multiclass classification to a tree of binary classifiers. Tree structure can change accuracy, bias, and variance, but the full space of structures has not been mapped in aggregate. We introduce ndscape, a two-dimensional mapping of ND structures based on depth-weighted structural tokens. We use this embedding to conduct what is, to our knowledge, the first bias-variance analysis across the ND structure space. In simulations with 6 to 10 classes, NDs with similar tree structures exhibit similar squared bias, variance, and squared probability loss, regardless of whether the binary classifier is logistic regression or a decision tree. Variance, however, follows mean root-to-leaf depth when logistic regression is used at each node, so flatter, more balanced NDs are more stable, whereas with decision-tree node models, variance is not associated with depth and differs little across ND structures. On six datasets with 6 to 19 classes, held-out accuracy, log loss, and prediction variance remain spatially organised on the same topology-only map. None of the six standard construction heuristics tested selects a tree from the best-performing cluster on any dataset, although some select high-performing individual trees. The ndscape embedding turns ND selection from one opaque construction step into a readable comparison of class groupings, exposing the upper-level splits that distinguish high-performing and stable regions.
ER -