MCA EXIN NEURON - Neural-based Orthogonal Data Fitting: The EXIN Neural Networks - page 64

MCA-EXIN (Solid), MCA-FENG1 (Dashed)

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

500 1000 1500 2000

Iterations

Figure 2.10 Smallest eigenvalue computation by MCA EXIN and FENG for a three-
dimensional well-conditioned autocorrelation matrix.

Proof. Consider the FENG ODE (2.30) and replace w( t ) with the expression
(2.85):

df i ( t )
dt

T

=− w ( t ) w ( t ) λ i f i ( t ) + f i ( t ) ∀ i = 1, ... , n (2.132)

Using the notation and the same analysis as in the proof of Theorem 16, it holds
that

d ϕ i ( t )
dt

2
2

= w ( t ) (λ n − λ i ) ϕ i ( t ) (2.133)

which is coincident with the formula for LUO [124, App.]. Recalling that, near
convergence,

2
2

w ( t ) → 1 /λ n ∀ i = 1, ... , n − 1, it holds that

w ( 0 ) − 2

convergence
−−−−−−−−−−→

λ n
λ n − 1

2

τ FENG = (2.134)

λ n − 1 − λ n − λ n

2.6.4 Weight Increments and RQ Minimal Residual Property

The MCA learning laws are iterative and so have the common form

w( t + 1 ) = w( t ) + α( t )w( t ) (2.135)