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In matrix completion, the starting point is often stated as: the optimization problem for matrix completion:

min(X): (1/2) ||X-M||^2

s.t. rank(X)<= r

Where X is the reconstructed matrix and M are the starting samples (the sparse matrix), and r is some rank boundary.

My question is: where does the 1/2 come from? Is it an error in the paper I'm reading or am I missing some basic step? ref:

Thank you

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up vote 0 down vote accepted

The $\frac12$ is just a multiple to make things seem nicer, as it cancels out the square power if you differentiate the cost function $\|X-M\|^2$ with respective to the entries of $X$. Since $\|X-M\|^2$ and $\frac12\|X-M\|^2$ share the same minimizers, it's just a matter of taste to multiply by one-half.

By the way, I would call $\min_{\textrm{rank}(X)\le r}\|X-M\|^2$ a low-rank approximation problem rather than a matrix completion problem. To my understanding, a matrix completion problem is one that you need to fill in some missing entries of $M$, or modify some zero entries of $M$, so that the resulting matrix satisfies some requirement. In $\min_{\textrm{rank}(X)\le r}\|X-M\|^2$, the requirements are placed on $X$ rather than $M$, and existing nonzero entries of $M$ may change. So I wouldn't call it a "matrix completion problem". Yet different research communities have different usages of the term, so this is just my own opinion.

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ok - thanks for clarifying my understanding! – val Nov 29 '12 at 17:22

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