Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 205 / 5 / c / Solution

Put $Z=[X,W]$. Its ridgeless minimum-norm fit is $Z^T(Z{Z}^T{)}^{+}Y$, so its first $p$ coordinates areThe strong law of large numbers applied entrywise gives $W{W}^T\to \lambda I_n$ almost surely as $d\to \infty$. Continuity of inversion then yieldsalmost surely, where the middle equality is the push-through identity.