ספי השחזור המדויקים לבחירת נתונים מועדפת ברגרסיה ליניארית ערכי-וקטור

תקציר מקורי באנגליתarXiv:2608.30254v2 Announce Type: replace Abstract: 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. We study vector-valued linear regression with square loss $\ell_{(x,y)}(W)=\lVert Wx-y\rVert_2^2$, where $x\in\mathbb{R}^d$ and $y\in\mathbb{R}^m$. The learner returns the minimum-Frobenius-norm empirical risk minimizer. We prove that the minimal budget of weighted examples for recovering the full-data loss on every finite dataset is exactly $n^{\star}(d,m)=(m+1)d$. We determine the weighted selection profile $F_{\mathrm{weighted}}(d,m,n)$ at the near-threshold budget: $F_{\mathrm{weighted}}(d,m,(m+1)d-1)=1+1/(dm^2)$. We recover the known spanning-budget value $F_{\mathrm{weighted}}(