For $\phi(x)=x$ and centered data, the objective is the <Rayleigh quotient>
$$
u^TSu,
\qquad
S=\frac1n\sum_{i=1}^nX_iX_i^T,
\qquad \lVert u\rVert_2=1.
$$
This is the first-direction optimization in <principal component analysis>. Therefore $\widehat u$ is any unit <eigenvector> of the sample <covariance matrix> $S$ corresponding to its largest <eigenvalue>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 218 / 6 / c / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-218/6/c/i/solution