I think that this gives you a measure of the error due to all of the missing eigenvectors/values, not the accuracy of the ones that are present.
Both questions are interesting ones and the answers are clearly somewhat related. On Sun, Nov 6, 2011 at 4:29 PM, Danny Bickson <[email protected]>wrote: > Regarding accuracy, you can also compute norm(T- V'*A*V) > where T is the tridiagonal matrix, V is a matrix with the intermediate > solution vectors v_i , and A is the original matrix you want to > approximate. (If A is not symmetric then replace it with A*A') > > On Sun, Nov 6, 2011 at 7:19 PM, Jake Mannix <[email protected]> wrote: > > > The idea is this: after k iterations, you have a k-by-k tri-diagonal > > matrix, the > > eigenvalues of *this* are *an approximation* to the top k eigenvalues of > > A*A'. The largest of these k will have the least error, the next will > have > > more, > > and so on down the line. I've typically assumed that the bottom 1/3rd or > > so of the k are not going to be very accurate, but it depends on the > > spectrum > > of eigenvalues of A*A', sometimes all but the last few are very > accurately > > computed (but sometimes it's even worse: if the spectrum is very flat, > the > > mixing between eigenvectors can be near complete) > > > > But you don't need a formula: look at EigenVerificationJob for the way to > > measure your accuracy, you can just run this on your data after you've > > extracted out your first k singular vector/value pairs, and see how far > > into > > this list you want to keep. The measurement I use is "how close to an > > eigenvector is this vector?". I.e. how big is 1 - (v/|v|) . > > ((AA'v)/|AA'v|) ? > > > > -jake > > > > > > On Sun, Nov 6, 2011 at 3:59 PM, Lance Norskog <[email protected]> wrote: > > > > > Is there a formula (somewhere easy to find) for "slop" and "accurate"? > > > > > > On Sun, Nov 6, 2011 at 3:57 PM, Ted Dunning <[email protected]> > > wrote: > > > > > > > Almost exactly. If you stop at k iterations, you don't get the top k > > > > eigenvalues precisely. You need some extra slop to make sure your > top > > > > eigenvalues are accurate. > > > > > > > > On Sun, Nov 6, 2011 at 3:46 PM, Danny Bickson < > [email protected] > > > > >wrote: > > > > > > > > > Exactly, in each iteration you extract one more eigenvalue from > > largest > > > > to > > > > > smallest. If you run more iterations then the matrix rank then you > > get > > > > > again all eigenvalues. In that case you may encounter each > > eigenvalue a > > > > few > > > > > times. > > > > > > > > > > On Sun, Nov 6, 2011 at 6:42 PM, Sean Owen <[email protected]> > wrote: > > > > > > > > > > > True though I thought it was intended to run through all m steps > - > > is > > > > it > > > > > > simply that you stop after some number k and that still leads you > > to > > > > > > compute the k largest eigenvectors of the original? > > > > > > On Nov 6, 2011 11:36 PM, "Danny Bickson" < > [email protected] > > > > > > > > wrote: > > > > > > > > > > > > > To be exact, the size of the tridiagonal matrix > > > > > > > is number of iterations + 1. See description of the matrix T_mm > > > > > > > in Wikipedia: http://en.wikipedia.org/wiki/Lanczos_algorithm > > > > > > > > > > > > > > Best, > > > > > > > > > > > > > > DB > > > > > > > > > > > > > > On Sun, Nov 6, 2011 at 6:30 PM, Sean Owen <[email protected]> > > > wrote: > > > > > > > > > > > > > > > Oh do you only need the top k x k bit of the tridiagonal to > > find > > > > the > > > > > > > > top k eigenvalues? > > > > > > > > > > > > > > > > I really don't want to write a QR decomposer, phew. > > > > > > > > > > > > > > > > On Sun, Nov 6, 2011 at 11:26 PM, Ted Dunning < > > > > [email protected]> > > > > > > > > wrote: > > > > > > > > > The tridiagonal is much smaller than you would need if you > > > wanted > > > > > all > > > > > > > the > > > > > > > > > eigenvalues. Since you only want a small number, you only > > > have a > > > > > > > > > tri-diagonal matrix that is some multiple of that size. > > > > In-memory > > > > > > > > > decomposition makes total sense. > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > > -- > > > Lance Norskog > > > [email protected] > > > > > >
