SEQKL Dynamic Rank Sequential Karhunen-Loeve
Description:
This function implements the dynamic-rank Sequential Karhunen-Loeve,
making one pass to approximate the dominant SVD U*S*V' and (optionally)
later passes improve on it.
Synopsis:
S = SEQKL(A) returns a vector S with approximations for the largest 6 singular
values of the matrix A. The elements of S are guaranteed to be non-negative
and sorted ascending.
[U,S] = SEQKL(A) returns a matrix U with orthonormal columns approximating
the left singular vectors corresponding to the approximate singular values in
S. S is a diagonal matrix with non-negative and non-decreasing elements.
[U,S,V] = SEQKL(A) returns a matrix V with orthonormal columns approximating
the right singular vectors.
[U,S,V,O] = SEQKL(A) returns a structure O with statstics from the computation
of U, S and V.
[...] = SEQKL(A,K) computes approximations for the largest K singular values
of A.
[...] = SEQKL(A,KMAX,THRESH) computes approximations for up to KMAX of the
largest singular values satisfying the relative threshhold THRESH, as follows:
If OPTS.ttype == 'rel', preserve all singular values such that
sigma >= thresh*min(sigma) if whch == 'L'
sigma <= thresh*min(sigma) if whch == 'S'
If OPTS.ttype == 'abs', preserve all singular values such that
sigma >= thresh if whch == 'L'
sigma <= thresh if whch == 'S'
[...] = SEQKL(A,K,...,'S') computes approximations for the smallest
singular values.
[...] = SEQKL(A,K,...,OPTS) allows the specification of other options, as
listed below.
Optional input:
opts.numpasses - number of times to pass through A
opts.mode - Update mode, default='auto2'
'restart' - compute SVD of A*W, W=[V Vperp], using last V
'sda' - IncSVD-based, implicit steepest descent on A'*A, variant A
'sdb' - IncSVD-based, implicit steepest descent on A'*A, variant B
opts.thresh - Tolerance for determining rank at each step. See above.
opts.ttype - Rank threshhold type: 'abs' or 'rel'. See above.
Default='rel'
opts.reortho - perform two-steps of classical Gram-Schmidt during SVD update
Default='yes'
opts.snaps - array of snapshot requests, by number columns processed
opts.kstart - number of columns used to initial decomp (lmax)
opts.kmin - minimum rank tracked by method, default=1
opts.extrak - extra dimension tracked.
applied after thresh-chosen k, before accounting for kmax.
opts.finalttype - final pass: same as opts.ttype, default=opts.ttype
opts.finalthresh - final pass: same as opts.fthresh, default=opts.fthresh
opts.finalkmin - final pass: same as opts.kmin, default=opts.kmin
opts.finalkmax - final pass: same as opts.kmax, default=kmax
opts.lmin - minimum value for l at each step, default=1
opts.lmax - maximum value for l at each step, default=kmax/sqrt(2)
opts.disp - print while running or not, default=0
0 - silent
1 - one-liners
2 - chatty
opts.debug: debugging checks and output [{0} | 1]. Expensive!!!
opts.paramtest: parameter test [{0} | 1]
do not actually run the algorithm, but instead test parameters,
set the defaults, and return them in a struct
Example: params = seqkl(A,kmax,thresh,opts);
opts.[USV] - initial factorization for starting multipass method:
.U - initial left vectors (m by o)
.S - initial left vectors (o by 1)
.V - initial left vectors (n by o)
Example:
data = randn(1000,100);
[U,S,V,OPS] = seqkl(data,10,.20,'S');
See also EIG, SVD, RSVD
