Question

# 6. (Low rank approximation 10 pt) Recall that one of the aims of PCA was low rank approximation in particular the

best low-rank approximation to the data: min X˜ ||X ´ X˜ ||2 F subject to rankpX˜ q " K where || ¨ ||F is the Frobenius norm of a matrix. The solution for this: X˜ " ÿ K k"1 dkukv T k (1) is the SVD / PCA solution. For fixed K derive an explicit expression for ||X ´ X˜ ||2 F where X˜ is as in (1).

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