Abstract
The analysis and processing of massive amount of multivariate data and high-dimensional functions have become a basic need in many areas of science 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 13thproblem 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 L2(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 we briefly report some new results of study in seeking 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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Kolmogorov A N. On the representation of continuous functions of many variables by superposition of continuous funcitons of one variable and addition. Dokl. Akad. Nauk. SSSR, 1957, 114, No.5, 953–956
Arnold V I. On functions of three variables. Dokl. Akad. Nauk. SSSR, 1957, 114, No.4, 679–681
Pontryagin L S. Continuous Groups. Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit., 1954. 92–93
Bourbaki N. Espaces Vectoriels Topologiques. Paris: Hermann & Cie, 1955. 307–311
Wainberg M M. Some problems of differential calculus in linear spaces. YMN, 1952, 4(50), 88–102
Ljusternik L A, Sobolev V I. Elements of Functional Analysis. Moscow: Nauka Press, 1965, 450–461, 489–492
Lang S. Real and Functional Analysis. New York: Springer-Verlag New York Inc., 1990. 70–72
Zeidler E. Nonlinear Functional Analysis and Its Applications. New York: Springer-Verlag New York Inc., 1990. II/A, 255–262; III, 603–605
Guan Zhao-Zhi. Lectures on Functional Analysis. Beijing: High Education Press, 1958. 143–149, 274–293
Kato T. Perturbation Theory for Linear Operators. New York: Springer-Verlag New York Inc., 1966. 275–276
Beauzamy B. Introduction to Banach Spaces and their Geometry. Amsterdam-New York: North-Holland Publishing Co., 1982. 175–204
Diestel J. Geometry of Banach Spaces-Selected Topics. Berlin-New York: Springer-Verlag, 1975. 29–94
Yosida K. Functional Analysis. Berlin: Springer-Verlag, 1965. 86–88, 119–125
Fihtengolts G M. Lectures on Differential and Integral Calculus, II. Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit., 1959. 292–294
Sobolev S L. Equations of Mathematical Physics. 1954
Gohberg I T, Krein M G. Introduction to Theory of Non-self-adjoint Linear Operators. Moscow: Nauka Press, 1965. 153–160
Fan K. Maximum properties and inequalities for the eigenvalues of completely continuous operators. USA: Proc. Nat. Acad. Sci., 1951, 760–766
Weyl H. Inequalities between the two kinds of eigenvalues of a linear transformation. USA: Proc. Nat. Acad. Sci., 1949, (35): 408–411
Stinespring W. A sufficient condition for an integral operator to have a trace. J. Reine Angew. Math., 1958, No.3–4, 200–207
Butkovski A G, Song J. On Construction of Functional Generator with Multiple Variables. Energetics and Automation, 1961 (2): 153–160
Kostiutsenko A G, Krein S G, Sobolev V I. Linear Operators in Hilbert Spaces. In Book: Functional Analysis, NAYKA, 1964
Krasnoselski M A. Topological Methods in Theory of Non- Linear Equations. Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit, 1956. 57–58
Song J. Design of Time-Optimal Controller. Automatica and Tele-mechanics. 1958, 20(3): 273–228
Song J. Optimal representation of multivariate functions or data in visualizable low-dimensional spaces, to appear in Chinese Science Bulletin, Vol 46, No. 16, 2001.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2003 Springer Science+Business Media New York
About this chapter
Cite this chapter
Song, J. (2003). Optimal Representation and Visualization of Multivariate Data and Functions in Low-Dimensional Spaces. In: Gong, W., Shi, L. (eds) Modeling, Control and Optimization of Complex Systems. The International Series on Discrete Event Dynamic Systems, vol 14. Springer, Boston, MA. https://doi.org/10.1007/978-1-4615-1139-7_1
Download citation
DOI: https://doi.org/10.1007/978-1-4615-1139-7_1
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4613-5411-6
Online ISBN: 978-1-4615-1139-7
eBook Packages: Springer Book Archive