Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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: http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=05459463

Thank you

share|improve this question

1 Answer 1

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.

share|improve this answer
    
ok - thanks for clarifying my understanding! –  val Nov 29 '12 at 17:22

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.