Abstract:
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A sequential method for approximating vectors in Hilbert spaces,
called Sequential Approximation with Optimal Coefficients and
Interacting Frequencies (SAOCIF), is presented. SAOCIF combines two
key ideas. The first one is the optimization of the coefficients (the
linear part of the approximation). The second one is the flexibility
to choose the frequencies (the non-linear part). The only relation
with the previous residue has to do with its approximation capability
of the target vector f. The approximations defined by SAOCIF always
exist, and maintain orthogonal-like properties. The theoretical
results obtained prove that, under reasonable conditions, the residue
of the approximation obtained with SAOCIF (in the limit) is the best
one that can be obtained with any subset of the given set of vectors.
In the particular case of L^2, it can be applied to approximations
by algebraic polynomials, Fourier series, wavelets and feed-forward
neural networks, among others. Also, a particular algorithm with
neural networks is presented. The resulting method combines the
locality of sequential approximations, where only one frequency is
found at every step, with the globality of non-sequential methods,
such as Backpropagation, where every frequency interacts with the
others. Experimental results show a very satisfactory performance of
this new method and several suggesting ideas for future experiments. |