Optimal rank-two PCA

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Problem

A centered three-dimensional population has covariance eigenvalues $5$ , $2$ , and $1$ , with orthonormal eigenvectors $u_1$ , $u_2$ , and $u_3$ . For rank-two PCA, state the optimal subspace, retained variance, explained fraction, and minimum expected squared reconstruction error.

Reveal answer or reference solution

The optimal subspace is $\operatorname{span}(u_1,u_2)$ . It retains variance $5+2=7$ out of a total of $8$ , so the explained fraction is $\frac{7}{8}$ . The minimum expected squared reconstruction error is the discarded variance, $1$ .

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Optimal rank-two PCA

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