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).

Image transcriptions

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 — Xlﬂ subject to rank(X) = K x where H - "F is the Hobenius norm of a matrix. The solution for this: ~ K X = Z dkukvf (1) 19:1 is the SVD / PCA solution. For ﬁxed K derive an explicit expression for "X — DUE; where X is as in (1).

(Low rank approximation 10 pt) Recall that one of the aims of PCA was low rank approximation in particular the best lowrank approximation to the...

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