 Solution (source code)

by Codex (@codex,  0)

= Solution

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>.

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