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How quickly minimizing $M*x-v$ (numerically) ?

asked 2013-10-21 13:03:37 -0500

updated 2013-10-22 04:51:20 -0500

Let $v$ in $R^m$ and let $M$ be a matrix from $R^n$ to $R^m$, with $m>n$ big numbers.
I want to compute a vector $x$ in $R^n$ such that the norm of $M*x-v$ is minimal.

One way is to compute the projection $w$ of $v$ on the image of $M$.
For so, we can compute the projection $p$ on the image of $M$, as follows :

MTGS=M.transpose().gram_schmidt()[0]  # it's orthogonalization, not orthonormalization
l=MTGS.rank()
U=[]
for i in range(l):
   v=MTGS[i]
   u=v/(v.norm())
   L=list(u)
   U.append(L)
N=matrix(m,l,U)  
p=N.transpose()*N

Then:

w=p*v  
x=M.solve_right(w)

This vector $x$ minimizes the norm of $M*x-v$, but this method is very expensive in time, because it computes $p$ and $w$, while I just need $x$.

Is there another method, less expensive in time, for computing $x$ ?

Remark : I'm ok with numerical methods.

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answered 2013-10-22 06:06:36 -0500

ppurka gravatar image

updated 2013-10-22 06:13:12 -0500

If I am understanding your question correctly - if you put the additional constraint of getting $x$ with the minimum norm, then you are asking for the Moore-Penrose pseudoinverse. In Sage you can explicitly call the pseudoinverse function of numpy

sage: MN = M.numpy()
sage: import numpy
sage: x = matrix(numpy.linalg.pinv(MN))*v

Edit: Even the command

sage: x = M \ v

works for a nonsquare matrix M. But I am not sure what this is actually doing for a nonsquare matrix. The documentation does not seem to explain. The solution obtained is not the same as obtained from the pseudoinverse.

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Thank you very much ppurka ! Do you know if we can compute x directly, without computing the pseudo-inverse before ?

Sébastien Palcoux gravatar imageSébastien Palcoux ( 2013-10-22 09:26:26 -0500 )edit

Well, that's what the Sage command `M \ v` does - it does not compute the inverse or pseudoinverse. Except, I am not sure exactly what it is computing for nonsquare matrices. Maybe you can put up a question in the sage-support mailing list.

ppurka gravatar imageppurka ( 2013-10-22 15:27:10 -0500 )edit

Thank you ppurka. I see a difference between "x = matrix(numpy.linalg.pinv(MN))*v" and "x = M \ v" : if v is not in the image of M then the first proposes a pseudo-solution and the second gives "ValueError: matrix equation has no solutions".

Sébastien Palcoux gravatar imageSébastien Palcoux ( 2013-10-24 00:00:44 -0500 )edit

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Asked: 2013-10-21 13:03:37 -0500

Seen: 407 times

Last updated: Oct 22 '13