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I am trying to figure out why do we need a low rank approximation of a matrix. Why is it used and where? Any insights?

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For example, if your matrix represents a graphic image, you could use a low rank approximation of it as a form of lossy compression. –  A.P. Apr 17 '13 at 7:20

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Please, read http://arxiv.org/pdf/0912.3599.pdf, page 3. This might be helpful.

I have also reformulated your question a bit here Applications of low-rank matrix approximation

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