how to solve weighted low rank approximation problem with diagonal weight matrix

Asked Apr 12 '15 at 08:34

Active Apr 12 '15 at 10:32

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Without weights the low rank approximation problem can be solved in terms of svd of the original matrix.But is there any way to solve the problem following problem

$$\min_C{\sum_{i=1}^n \sum_{j=1}^d (N_{ij}-C_{ij})^2\sigma_j^{2}}$$ in terms of svd of N?

N is the original matrix.C is the low rank approximation.I need to find the low rank matrix C such that the above function is minimized.here $\sigma$ s are column wise weights.This a special case of weighted low rank approximation.If all the weights are 1s the we can get a solution based on svd.

edited Apr 12 '15 at 10:32

cangrejo

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asked Apr 12 '15 at 08:34

biswpo

What are all these notations? – ttnphns Apr 12 '15 at 08:55
edited the post please have look – biswpo Apr 12 '15 at 09:11
I feel that you may be speaking of weighted (=generalized) SVD which is what is being done in weighted biplot, for example. I recommend you to read [this](http://stats.stackexchange.com/a/141755/3277) my answer. – ttnphns Apr 12 '15 at 10:35
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Would it be better to use "column weights" instead of "diagonal weights" in the title? – ttnphns Apr 12 '15 at 10:55

how to solve weighted low rank approximation problem with diagonal weight matrix

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