Kernel-Based Adaptive Image Retrieval Methods - Multimedia Database Retrieval: Technology and Applications - page 34

The advantage of this network's use in the current application is that it finds the
input-to-output map using local approximators; consequently, the underlying basis
function responds only to a small region of the input space where the function is
centered, e.g., a Gaussian response, K i = exp

2 , where:

d 2

−

t

˃ − 2
i

d ( x , z i , ˃ i )= ( x − z ) ( x − z i ) (2.41)

This allows local evaluation for image similarity matching.
The parameters to learn for the LMN are the set of linear weight ʻ i , the center z i ,

and the width N m . The linear weights are

usually estimated by the least-squared (LS) method [ 43 ]. When using the Gaussian
function as the nonlinearity of hidden nodes, it has been observed that the same
width of

˃ i for each local approximator K i , i = 1 ,...,

˃ i is sufficient for the RBF network to obtain universal approximation
[ 42 ]. However, more recent theoretical investigations and practical results indicate
that the choice of center z i is most significant in the performance of the RBF network
[ 44 ]. As we shall see, this suggestion plays a central role in overcoming the variation
in the performance of the network in the adaptive retrieval application.

2.4.2 Learning Methods for the RBF Network

Various learning strategies have been proposed to structure and parameterize the
RBF network [ 41 , 43 - 45 ]. This section will consider two of these beside the new
learning strategy for adaptive image retrieval. For a given training set

N
i 1 ,the

initial approaches [ 41 ], constructed the RBF network by associating all available
training samples to the hidden units, using one-to-one correspondence. A radial-
basis function centered at z i is defined as:

{ x i , y i }

=

exp

2

−

x − z i

K ( x , z i )= , i = 1 ,..., N m (2.42)

2
i

2 ˃

where

N m
i

N
i

{ z i } = { x i } , N m = N (2.43)

= 1 = 1

This solution may be expensive, in terms of computational complexity, when N is
large. Thus, we may arbitrarily choose some data points as centers [ 43 ]. This gives
an approximation to the original RBF network, while providing a more suitable
basis for practical applications. In this case, the approximated solution is expanded
on a finite basis:

N m
i = 1 ʻ i K ( x , z i )

f

( )=

x (2.44)