МАТЕМАТИКА
J i a n S O N G
(Chinese Academy of Engineering, Beijing 100038)
OPTIMAL REPRESENTATION OF MULTIVARIATE
FUNCTIONS OR DATA IN VISUALIZABLE
LOW-DIMENSIONAL SPACES
1
It is intended to find the best representation of high-dimensional
functions or multivariate data in the
L
2
(Ω)
space with the fewest
number of terms, each of them is a combination of one-variable function.
A system of non-linear integral equations has been derived as an
eigenvalue problem of gradient operator in the above-said space. It is
proved that the complete set of eigenfunctions generated by the gradient
operator constitutes an orthonormal system, and any function of
L
2
(Ω)
can be expanded with the fewest terms and exponential rapidity of
convergence. It is also proved as a Corollary, all eigenvalues of the
integral operators has multiplicity equal to 1 if the dimension of the
underlying space
R
n
is
n
= 2
,
4
and 6.
The analysis and processing of massive amount of multivariate data
or high-dimensional functions have become a basic need in many areas
of scientific exploration and engineering. To reduce the dimensionality for
compact representation and visualization of high-dimensional information
appear imperative in exploratory research and engineering modeling. Since
D. Hilbert raised the 13
th
problem in 1900, the study on possibility to
express high-dimensional functions via composition of lower-dimensional
functions has gained considerable success [1, 2]. Nonetheless, no methods
of realization are ever indicated, and not even all integrable functions
can be treated this way,
a fortiori
functions in
L
2
(Ω)
. The common
practice is to expand high-dimensional functions into a convergent series
in terms of a chosen orthonormal basis with lower dimensional ones.
However, the length and rapidity of convergence of the expansion heavily
depend upon the choice of basis. In this paper an attempt is made to
seek an optimal basis for a given function provided with fewest terms and
rapidest convergence. All elements of the optimal basis turned out to be
products of single-variable functions taken from the unit balls of ingredient
spaces. The proposed theorems and schemes may find wide applications
in data processing, visualization, computing, engineering simulation and
decoupling of nonlinear control systems. The facts established in the
theorems may have their own theoretical interests.
The paper is published without any redaction.
ISSN 1812-3368. Вестник МГТУ им. Н.Э. Баумана. Сер. “Естественные науки”. 2005. № 4
33
