Supremum for the coupled five-point positive-basis class

Determine whether $3/2$ is the exact supremum of the weighted four-point selection risk ratio over datasets consisting of exactly five points whose gradients form a coupled five-vector positive basis.

Background

The paper analyzes datasets of exactly five points in dimension three whose loss gradients form a coupled five-vector positive basis. It proves that the four-point selection ratio for every such dataset is strictly below $3/2$.

The authors report numerical evidence that a family of instances approaches ratio $3/2$ from below, but they do not establish a matching lower bound when all selections are allowed, including rank-deficient selections governed by the minimum-norm tie rule. Thus the exact supremum for this restricted class remains unresolved.

References

We do not know whether $\frac32$ is the exact supremum of $(4;)/$ over this class. On the path $c_1\downarrow0$, $c_2=c_3=c_4=c_5\uparrow\frac54$ the upper bound above tends to $\frac12$, and exact optimization over all full-rank supports approaches ratio $\frac32$ from below; but a matching lower bound over all selections, including rank-deficient supports under the min-norm rule, has not been proved.

— Exact Risk Ratios for Weighted Data Selection in Linear Regression  (2608.28007 - Zhang, 28 Aug 2026) in Section 4, Remark following Proposition 4.5, “The coupled class stays below $3/2$”