Hamzi - Eurandom

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$Continuous-time random walks, fractional calculus and ... - Eurandom$

Reproducing Kernel Hilbert Spaces • RKHS play an important in learning theory whose objective is to find an unknown function f : X → Y from random samples (xi, yi)| m i=1 . • For instance, assume that the random probability measure that governs the random samples is ρ and is defined on Z := X × Y . Let X be a compact subset of R n and Y = R. If we define the least square error of f as E = ∫ X×Y (f(x) − y)2dρ, then the function that minimzes the error is the regression function fρ fρ(x) = ∫ R ydρ(y|x), x ∈ X, where ρ(y|x) is the conditional probability measure on R. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 24 / 55
Reproducing Kernel Hilbert Spaces • Since ρ is unknown, neither fρ nor E is computable. We only have the samples s := (xi, yi)| m i=1 . The error fρ is approximated by the empirical error Es(f) by Es(f) = 1 m m∑ i=1 (f(xi) − yi) 2 + λ||f|| 2 H, for λ ≥ 0, λ plays the role of a regularizing parameter. . . . . . . Boumediene <strong>Hamzi</strong> (Imperial College) On Control and RDS in RKHS June 4th, 2012 25 / 55
Page 1 and 2: Preliminary Results on Control and
Page 3 and 4: Outline • Review of Some Concepts
Page 5 and 6: Review of Some Concepts for Linear
Page 11 and 12: Controllability and Observability E
Page 13 and 14: Balancing of Linear Control Systems
Page 15 and 16: Balancing of Nonlinear Systems •
Page 17 and 18: Empirical Estimates of the Gramians
Page 19 and 20: Reproducing Kernel Hilbert Spaces
Page 21 and 22: Reproducing Kernel Hilbert Spaces T
Page 23: Reproducing Kernel Hilbert Spaces
Page 33 and 34: Balanced Reduction of Nonlinear Con
Page 35 and 36: Balancing in RKHS • “Balancing
Page 37 and 38: Experiment: Inputs ◮ Excite with
Page 39 and 40: Experiment diag(TKcT T ) 10 −1 10
Page 45 and 46: Nonlinear SDEs in RKHSes • Assump
Page 47 and 48: Nonlinear SDEs in RKHSes • The st
Page 49 and 50: Parameter Estimation for SDEs • I
Page 51 and 52: Parameter Estimation for SDEs • I
Page 53 and 54: Example (A Fokker-Planck Integrable
Page 55: Open Questions • Derivation of da

Hamzi - Eurandom

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