Krylov subspace methods for Perron-Frobenius operators in RKHS Yuka - PowerPoint PPT Presentation

Krylov subspace methods for Perron-Frobenius operators in RKHS Yuka Hashimoto ∗ Takashi Nodera † ∗ NTT Network Technology Laboratories / Graduate School of Science and Technology, Keio University † Faculty of Science and Technology, Keio University 2020/2/11

Abstract • We consider numerical estimations of Perron-Frobenius (P-F) operators in RKHS. • A P-F operator is a linear operator which describes the time evolution of a dynamical system. • Recently, using P-F operators for time-series data analysis have been actively researched. • We investigate theoretical analyses of Krylov subspace methods for estimating P-F operators. Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 2 / 17

Contents 1. Background 2. Existing Krylov subspace methods for Perron-Frobenius operators 3. Difference from classical settings 4. New analyses of the Krylov subspace methods 5. Numerical experiments 6. Conclusion Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 3 / 17

Dynamical systems with random noise ( Ω, F ) : A measurable space, ( X , B ) : A Borel measurable and locally compact Hausdorff vector space, X t , ξ t : random variables from Ω to X , { ξ t } : An i.i.d. stochastic process corresponds to the random noise in X ( ξ t is also independent of X t ), h : X → X (nonlinear in general) Dynamical system with random noise X t +1 = h ( X t ) + ξ t , (1) P : A probability measure on Ω , X t Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 4 / 17

Dynamical systems with random noise ( Ω, F ) : A measurable space, ( X , B ) : A Borel measurable and locally compact Hausdorff vector space, X t , ξ t : random variables from Ω to X , { ξ t } : An i.i.d. stochastic process corresponds to the random noise in X ( ξ t is also independent of X t ), h : X → X (nonlinear in general) Dynamical system with random noise X t +1 = h ( X t ) + ξ t , (1) X t Transform X t ∗ P , P : A probability measure on Ω , where X t ∗ P ( B ) = P ( X t − 1 ( B )) for B ∈ B : The push forward measure Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 4 / 17

Dynamical systems with random noise ( Ω, F ) : A measurable space, ( X , B ) : A Borel measurable and locally compact Hausdorff vector space, X t , ξ t : random variables from Ω to X , { ξ t } : An i.i.d. stochastic process corresponds to the random noise in X ( ξ t is also independent of X t ), h : X → X (nonlinear in general) Dynamical system with random noise X t +1 = h ( X t ) + ξ t , (1) X t Transform X t ∗ P , P : A probability measure on Ω , where X t ∗ P ( B ) = P ( X t − 1 ( B )) for B ∈ B : The push forward measure X t +1 ∗ P = β t ∗ ( X t ∗ P ⊗ P ) , (2) where β t ( x, ω ) = h ( x ) + ξ t ( ω ) linear Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 4 / 17

RKHS and kernel mean embedding To define an inner product between measures, we employ the theory of RKHSs and kernel mean embeddings. k : A positive definite kernel, ϕ ( x ) = k ( x, · ) : The feature map H k = { ∑ m − 1 t =0 c t ϕ ( x t ) | m ∈ N , x t ∈ X , c t ∈ C } : The RKHS M ( X ) : The space of all the finite signed Borel measures on X Φ : M ( X ) → H k µ �→ ∫ x ∈X ϕ ( x ) dµ ( x ) : The kernel mean embedding Feature map ϕ x ϕ ( x ) µ Φ( µ ) Kernel mean embedding Φ X (Usually a finite dimensional sp.) H k (An infinite dimensional Hilbert sp.) ∫ ∫ ⟨ Φ( µ ) , Φ( ν ) ⟩ = x ∈X k ( x, y ) dµ ( x ) dν ( y ) y ∈X : The inner product between Φ( µ ) and Φ( ν ) Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 5 / 17

Perron-Frobenius operators in RKHSs Linear relation between X t ∗ P and X t +1 ∗ P X t +1 ∗ P = β t ∗ ( X t ∗ P ⊗ P ) (2) Definition 1 (Perron Frobenius operator in RKHS) An operator K : Φ( M ( X )) → H k is called a Perron-Frobenius operator in H k if it satisfies K Φ( µ ) := Φ( β t ∗ ( µ ⊗ P )) , (3) for µ ∈ M ( X ) . It can be shown that: • K is well-defined with some mild conditions of k . • K is linear. • K does not depend on time t . Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 6 / 17

Construction of a Krylov subspace of the P-F operator { x 0 , x 1 , . . . , x T } ⊆ X : observed time-series data t,N := 1 /N ∑ N − 1 µ S i =0 δ x t + iS ( t = 0 , . . . , m ) : empirical measures (We will drop superscript S for simplicity) δ x : Dirc measure of x ∈ X Assumptions 1. µ t,N converge to a finite Borel measure µ t weakly as N → ∞ for t = 0 , . . . m ∑ N − 1 2. lim N →∞ 1 ∫ ω ∈ Ω f ( h ( x t + iS ) + ξ t ( ω )) dP ( ω ) (Space average) i =0 N ∑ N − 1 = lim N →∞ 1 i =0 f ( h ( x t + iS ) + ξ t + iS ( η )) (Time average) N a.s. η ∈ Ω If K is bounded, K Φ( µ t ) = Φ( µ t +1 ) ( t = 0 , . . . , m − 1) (4) Span { Φ( µ 0 ) , . . . , Φ( µ m − 1 ) } = Span { Φ( µ 0 ) , K Φ( µ 0 ) , . . . , K m − 1 Φ( µ 0 ) } : Krylov subspace of K and Φ( µ 0 ) , denoted as K m ( K, Φ( µ 0 )) Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 7 / 17

Numerical estimation for the P-F operator 1 K m ( K, Φ( µ 0 )) = Span { Φ( µ 0 ) , . . . , Φ( µ m − 1 ) } · · · constructed only with observed data [Φ( µ 0 ) , . . . , Φ( µ m − 1 )] = Q m R m : QR decomposition Proposition 1 (Numerical estimation of K ) Let K m := Q ∗ m KQ m , the operator projected onto K m ( K, Φ( µ S 0 )) . Then, K m is represented only with observed data as: K m = Q ∗ [Φ( µ 0 ) , . . . , Φ( µ m − 1 )] R − 1 (5) m For application, we want to know the time evolution of the dynamical Observable system at some time t > T → Estimate Kv for v = ϕ ( x t ) at time t ( T : The number of observables in the time-series data) Arnoldi approximation of Kv Kv ≈ Q T K m Q ∗ T v (6) 1 Hashimoto et al., arXiv:1909.03634v3, 2019. Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 8 / 17

Unboundedness of the P-F operator In fact, the P-F operators can be unbounded 2 . In this case, the Arnoldi approximation K m does not converge to K as m → ∞ . K (Unbounded) Transform ( γI − K ) − 1 (Bounded), where γ / ∈ Λ( K ) u t,N = ∑ t ( t ) ( − 1) i γ t − i Φ( µ i,N ) , lim N →∞ u t,N = u t i =0 i ( γI − K ) − 1 u t +1 = u t (7) K m (( γI − K ) − 1 , u S m ) = Span { u 1 , . . . , u m } [ u S 1 , . . . , u S m ] = Q m R m : QR decomposition L m := Q ∗ T ( γI − K ) − 1 Q T = Q ∗ T [ u 1 , . . . , u m ] R − 1 m Shift-invert Arnoldi approximation of Kv K m := γ I − L − 1 Kv ≈ Q T K m Q ∗ T v, (8) m 2 Ikeda, Ishikawa and Sawano, arXiv:1911.11992, 2019. Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 9 / 17

Difference from classical settings Our setting with P-F operators is different from classical ones in numerical linear algebra. Although the analyses for classical settings have been actively investigated, those for P-F operators have not been investigated. The Classical setting with a linear operator A Our setting with a P-F operator K (A typical example of A • Data driven approach : Laplace operator) • K is not given , instead, • Model driven approach observed data are given • Operators are given • Kv for a vector v have to be • Av for a vector v can be estimated by data computed Data Data Model Estimate Model Behavior Extract of the model New analyses for the Krylov subspace methods for P-F operators are needed Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 10 / 17

Residual of a Krylov approximation Krylov approximations for classical settings have a strong connection with their residuals. → We also investigate a connection of the Krylov approximations of P-F operators with their residuals. Kv ≈ u m → ∥ v − K − 1 u m ∥ : residual of u m The steps of our analysis : 1. Find a minimizer of the residual 2. Derive the relation between the residual of the Arnoldi approximation and that of the minimizer Theorem 1 (Minimizer of the Residual) u m := Q m +1 K m +1 Q ∗ m +1 Q m Q ∗ Let ˜ m v . Then, ˜ u m minimizes the residual in K m ( K, Φ( µ 1 )) , that is: ∥ v − K − 1 u ∥ . u m = ˜ arg min (9) u ∈K m ( K, Φ( µ 1 )) Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 11 / 17

Residual of a Krylov approximation Theorem 1 (Minimizer of the Residual) u m := Q m +1 K m +1 Q ∗ m +1 Q m Q ∗ Let ˜ m v . Then, ˜ u m minimizes the residual in K m ( K, Φ( µ 1 )) , that is: ∥ v − K − 1 u ∥ . u m = ˜ arg min (9) u ∈K m ( K, Φ( µ 1 )) Q m Q ∗ Proof. m v =: p m − 1 ( K )Φ( µ 0 ) ∈ K m ( K, Φ( µ 0 )) = Span { Φ( µ 0 ) , K Φ( µ 0 ) , . . . , K m − 1 Φ( µ 0 ) } Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 12 / 17

Residual of a Krylov approximation Theorem 1 (Minimizer of the Residual) u m := Q m +1 K m +1 Q ∗ m +1 Q m Q ∗ Let ˜ m v . Then, ˜ u m minimizes the residual in K m ( K, Φ( µ 1 )) , that is: ∥ v − K − 1 u ∥ . u m = ˜ arg min (9) u ∈K m ( K, Φ( µ 1 )) Q m Q ∗ Proof. m v =: p m − 1 ( K )Φ( µ 0 ) Projection onto K m ( K, Φ( µ 0 )) ∥ v − u ∥ = Q m Q ∗ arg min m v u ∈K m ( K, Φ( µ 0 )) (10) Krylov subspace methods for P-F operators in RKHSs Y. H. and T. N. 12 / 17