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ε-superposition and truncation dimensions in average and probabilistic settings for ∞-variate linear problems
- Source :
- Journal of Complexity. 57:101439
- Publication Year :
- 2020
- Publisher :
- Elsevier BV, 2020.
-
Abstract
- The paper deals with linear problems defined on γ -weighted Hilbert spaces of functions with infinitely many variables. The spaces are endowed with zero-mean Gaussian measures which allows to define and study e -truncation and e -superposition dimensions in the average case and probabilistic settings. Roughly speaking, these e -dimensions quantify the smallest number k = k ( e ) of variables that allow to approximate the ∞ -variate functions by special ones that depend on at most k -variables with the average error bounded by e . In the probabilistic setting, given δ ∈ ( 0 , 1 ) , we want the error ≤ e with probability ≥ 1 − δ . We show that the e -dimensions are surprisingly small which, for anchored spaces, leads to very efficient algorithms, including the Multivariate Decomposition Methods.
- Subjects :
- Statistics and Probability
Numerical Analysis
Multivariate statistics
Pure mathematics
Control and Optimization
Algebra and Number Theory
Truncation
Applied Mathematics
General Mathematics
Gaussian
010102 general mathematics
Probabilistic logic
Hilbert space
010103 numerical & computational mathematics
01 natural sciences
symbols.namesake
Superposition principle
Random variate
Bounded function
symbols
0101 mathematics
Mathematics
Subjects
Details
- ISSN :
- 0885064X
- Volume :
- 57
- Database :
- OpenAIRE
- Journal :
- Journal of Complexity
- Accession number :
- edsair.doi...........2700055747fef6984a53f3419b3469de