Fortunately, if one applies the sequence of Givens transformations to the Identity matrix
, the structure of (12) evolves slowly from the top left-hand corner and from the bottom right-hand corner.
Canalization is possible so long as E cannot be diagonalized to produce the identity matrix
. In other words, so long as [Lambda] [not equal to] I.
Notice that we have used [I.sub.1:i - 1] to represent an identity matrix
of size [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [I.sub.i + 1:N] to represent one of size [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
where e1 is the first column of the identity matrix
of order k.
which corresponds to the power gain of an identity matrix
, shown in (11).
[B.sup.-1.sub.i] is fixed as the identity matrix
for the first iteration ([B.sup.-1.sub.0] = I).
where [I.sub.n-6] denotes a (n - 6) x (n - 6) identity matrix
. Note that concept extraction can map each image [x.sub.i] to [y.sub.i] from the original feature space to the new concept space.
From (2), we know that if the packets from layer 3 to layer 4 are original packets, then there exists generator matrix G such that GT = I, where I is an identity matrix
. In the last section, we pointed out that the transfer matrix T in the five-layer network model must be a full-rank matrix.
Scheinberg and Tang establish a sublinear rate of convergence for this method when the Hessian approximations are suitably modified by adding a scaled identity matrix
and when the scaled proximal maps are evaluated with increasing accuracy.
Start with initializing the matrix T by setting it to T = I, where I is the identity matrix
. Then, apply the following steps:
According to assumption (4), there exist nonsingular matrices T and N such that TE + NC = [I.sub.n], where [I.sub.n] [member of] [R.sup.nxn] denotes an identity matrix
. The general solution for T and N is given as
Defining [[zeta].sub.h] as the hth row of the [n.sub.u]-dimensional identity matrix
, we have