Unconstrained support points can leave a convex support
Abstract
Support points provide a practical and effective way to approximate a probability distribution by a small set of equally weighted representatives through energy-distance minimization. In many applications, the variables satisfy convex constraints that the representatives should also respect. We investigate whether unconstrained support points automatically satisfy such constraints and show that this is not guaranteed. We construct counterexamples using three planar observations, a density with full triangular support, and positive-definite matrix data admitting an indefinite optimal representative under the Frobenius norm. We also identify cases in which containment holds, establish stability under distributional perturbations, and quantify the objective cost of imposing a convex-hull constraint. For a fixed number of representatives and a positive population constraint gap, empirical configurations with vanishing global objective error eventually leave the sample convex hull almost surely.
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BibTeX
@misc{li2026unconstrained,
title = {{Unconstrained support points can leave a convex support}},
author = {Zhaohui Li},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00514},
eprint = {2611.00514},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00514},
}
RIS
TY - UNPB
TI - Unconstrained support points can leave a convex support
AU - Zhaohui Li
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00514
UR - https://arxiv.org/abs/2611.00514
AB - Support points provide a practical and effective way to approximate a probability distribution by a small set of equally weighted representatives through energy-distance minimization. In many applications, the variables satisfy convex constraints that the representatives should also respect. We investigate whether unconstrained support points automatically satisfy such constraints and show that this is not guaranteed. We construct counterexamples using three planar observations, a density with full triangular support, and positive-definite matrix data admitting an indefinite optimal representative under the Frobenius norm. We also identify cases in which containment holds, establish stability under distributional perturbations, and quantify the objective cost of imposing a convex-hull constraint. For a fixed number of representatives and a positive population constraint gap, empirical configurations with vanishing global objective error eventually leave the sample convex hull almost surely.
ER -