We resolve the threshold part of Question 4 of the COLT 2025 open problem "Data Selection for Regression Tasks" of Hanneke, Moran, Shlimovich and Yehudayoff. In vector-valued linear regression with square loss
ℓ(x,y)(W)=∣Wx−y∣22, where
x∈Rd,
y∈Rm and the learner is the empirical risk minimizer of minimal Frobenius norm, we prove that the minimal budget of weighted examples that recovers the full-data loss on every finite dataset is exactly
n∗(d,m)=(m+1)d. We further determine two more values of the weighted selection profile
Fw(d,m,n): at the near-threshold budget,
Fw(d,m,(m+1)d−1)=1+dm21, and at the spanning budget,
Fw(d,m,d)=d+1 for every
m, while
Fw(d,m,n)=∞ for
n<d. For the smallest open intermediate cell
(d,m)=(2,2) we prove
Fw(2,2,3)∈[13/8,15/8] and
Fw(2,2,4)∈[5/4,3/2], reduce the conjectured exact values
13/8 and
5/4 to a finite moment problem on the circle with at most seven atoms, and establish strong structural evidence for the conjecture. The upper-bound techniques (a fixed-basis conic compression lemma, a determinant-facet rigidity theorem for maximal certificates, and sharp sparsification lemmas for zero-mean weighted point systems) are of independent interest. As a byproduct we correct an erroneous claim circulating in a recent unrefereed preprint, exhibiting an explicit dataset with
m=2 on which no weighted selection of
2d points recovers the optimal loss. All results are new only for
m≥2; the scalar case
m=1 is due to Hanneke et al.