Behavior of a functional in learning theory
Let \(A:H\to H\) be a continuous linear operator on a Hilbert space \(H\), \(R(A)\) the range of \(A\), and let \(y\in H\) satisfy \(y\in\overline{R(A)}\) and \(y\notin R(A)\). The author studies the functional \(F(\tau)\) defined as \(\inf\{\|y-Ax\|^2:x\in H,\|x\|\leq\tau\}\), \(\tau\geq 0\). Assuming that \(y\in R((AA^*)^{t/2})\) for some \(0<t<1\) he proves that \(F(\tau)\leq\tau^{-\frac{2t}{1-t}}\|(AA^*)^{-\frac 12}y\|^{\frac{2}{1-t}}\), \(\forall\tau>0\).
scientific article; zbMATH DE number 5305334