The CLEVER project
The Numerical Analysis Viewpoint
We promised that this paper has something to do with matrix decompositions. The basic "power iteration" described in the article can be viewed as "one unit" of the QR iteration (which forms the basis for many factorizations):
= M 
=
This iteration is guaranteed to converge to the principal eigenvector of M, if (i) the corresponding eigenvalue
is dominant, i.e., if
> 
.......... 
and (ii) the original starting vector V has a component in the direction of this eigen value. (To ensure that this is satisfied, the authors start with a non-degenerate choice of the eigen vector, which will have such an entry in all components). To analyze the convergence properties of this iteration notice that any vector V can be expressed as:
V =
+ 
+ ......... + 
V = 
+ 
+ ......... + 
+ 
+ ......... + 
[ 
+ ....... + 
The authors indicate that in their application, convergence is achieved within
a few iterations.
Expanding on this theme, we can make V to be a matrix of two columns, to obtain the
top two eigenvectors. The convergence in this case can be obtained similarly;
except
the ratio of the third and second eigenvalues is taken (instead of the first two).
The full-blown QR iteration obtains all the eigenvectors. The first column of Q would correspond
to the first eigenvector. The second column of Q would correspond to a linear combination of
the first two, and so on.
If M is symmetric (as is the case with the HITS matrix), then we know that
the eigenvalues are real and the eigenvectors can be chosen
orthonormal. In this case, the R matrix of the QR iteration would be diagonal and each of the
columns in Q would be an eigenvector.
