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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

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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

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Abstract

  • We consider numerical estimations of Perron-Frobenius (P-F)
  • perators in RKHS.
  • A P-F operator is a linear operator which describes the time evolution
  • f 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

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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

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Dynamical systems with random noise

(Ω, F) : A measurable space, (X, B) : A Borel measurable and locally compact Hausdorff vector space, Xt, ξt : random variables from Ω to X, {ξt} : An i.i.d. stochastic process corresponds to the random noise in X (ξt is also independent of Xt), h: X → X (nonlinear in general)

Dynamical system with random noise

Xt+1 = h(Xt) + ξt, (1) P : A probability measure on Ω, Xt

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

4 / 17

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Dynamical systems with random noise

(Ω, F) : A measurable space, (X, B) : A Borel measurable and locally compact Hausdorff vector space, Xt, ξt : random variables from Ω to X, {ξt} : An i.i.d. stochastic process corresponds to the random noise in X (ξt is also independent of Xt), h: X → X (nonlinear in general)

Dynamical system with random noise

Xt+1 = h(Xt) + ξt, (1) P : A probability measure on Ω, Xt Transform Xt∗P, where Xt∗P(B) = P(Xt−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

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Dynamical systems with random noise

(Ω, F) : A measurable space, (X, B) : A Borel measurable and locally compact Hausdorff vector space, Xt, ξt : random variables from Ω to X, {ξt} : An i.i.d. stochastic process corresponds to the random noise in X (ξt is also independent of Xt), h: X → X (nonlinear in general)

Dynamical system with random noise

Xt+1 = h(Xt) + ξt, (1) P : A probability measure on Ω, Xt Transform Xt∗P, where Xt∗P(B) = P(Xt−1(B)) for B ∈ B : The push forward measure Xt+1∗P = βt∗(Xt∗P ⊗ P), (2) where βt(x, ω) = h(x) + ξt(ω)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

4 / 17

linear

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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 Hk = {∑m−1

t=0 ctϕ(xt) | m ∈ N, xt ∈ X, ct ∈ C} : The RKHS

M(X) : The space of all the finite signed Borel measures on X Φ : M(X) → Hk µ → ∫

x∈X ϕ(x) dµ(x) : The kernel mean embedding

Hk (An infinite dimensional Hilbert sp.) X (Usually a finite dimensional sp.) Feature map ϕ Kernel mean embedding Φ x µ Φ(µ) ϕ(x) ⟨Φ(µ), Φ(ν)⟩ = ∫

y∈X

∫

x∈X k(x, y)dµ(x)dν(y)

: 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

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Perron-Frobenius operators in RKHSs

Linear relation between Xt∗P and Xt+1∗P Xt+1∗P = βt∗(Xt∗P ⊗ P) (2)

Definition 1 (Perron Frobenius operator in RKHS)

An operator K : Φ(M(X)) → Hk is called a Perron-Frobenius operator in Hk if it satisfies KΦ(µ) := Φ(βt∗(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

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Construction of a Krylov subspace of the P-F operator

{x0, x1, . . . , xT } ⊆ X : observed time-series data µS

t,N := 1/N ∑N−1 i=0 δxt+iS (t = 0, . . . , m) : empirical measures

(We will drop superscript S for simplicity) δx : Dirc measure of x ∈ X

Assumptions

  • 1. µt,N converge to a finite Borel measure µt weakly as N → ∞ for

t = 0, . . . m

  • 2. limN→∞ 1

N

∑N−1

i=0

∫

ω∈Ωf(h(xt+iS) + ξt(ω)) dP(ω)

(Space average) = limN→∞ 1

N

∑N−1

i=0 f(h(xt+iS) + ξt+iS(η))

(Time average) a.s. η ∈ Ω If K is bounded, KΦ(µt) = Φ(µt+1) (t = 0, . . . , m − 1) (4) Span{Φ(µ0), . . . , Φ(µm−1)} = Span{Φ(µ0), KΦ(µ0), . . . , Km−1Φ(µ0)} : Krylov subspace of K and Φ(µ0), denoted as Km(K, Φ(µ0))

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

7 / 17

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Numerical estimation for the P-F operator1

Km(K, Φ(µ0)) = Span{Φ(µ0), . . . , Φ(µm−1)} · · · constructed only with observed data [Φ(µ0), . . . , Φ(µm−1)] = QmRm : QR decomposition

Proposition 1 (Numerical estimation of K)

Let Km := Q∗

mKQm, the operator projected onto Km(K, Φ(µS 0 )). Then,

Km is represented only with observed data as: Km = Q∗[Φ(µ0), . . . , Φ(µm−1)]R−1

m

(5) For application, we want to know the time evolution of the dynamical system at some time t > T → Estimate Kv for v = ϕ(xt) (T : The number of observables in the time-series data)

Arnoldi approximation of Kv

Kv ≈ QT KmQ∗

T v

(6)

1Hashimoto et al., arXiv:1909.03634v3, 2019.

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

8 / 17

Observable at time t

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Unboundedness of the P-F operator

In fact, the P-F operators can be unbounded2. In this case, the Arnoldi approximation Km does not converge to K as m → ∞. K (Unbounded) Transform (γI − K)−1 (Bounded), where γ / ∈ Λ(K) ut,N = ∑t

i=0

(t

i

) (−1)iγt−iΦ(µi,N), limN→∞ ut,N = ut (γI − K)−1ut+1 = ut (7) Km((γI − K)−1, uS

m) = Span{u1, . . . , um}

[uS

1 , . . . , uS m] = QmRm : QR decomposition

Lm := Q∗

T (γI − K)−1QT = Q∗ T [u1, . . . , um]R−1 m

Shift-invert Arnoldi approximation of Kv

Kv ≈ QT KmQ∗

T v,

Km := γI − L−1

m

(8)

2Ikeda, Ishikawa and Sawano, arXiv:1911.11992, 2019.

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

9 / 17

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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. Our setting with a P-F operator K

  • Data driven approach
  • K is not given, instead,
  • bserved data are given
  • Kv for a vector v have to be

estimated by data Estimate Model Data The Classical setting with a linear

  • perator A

(A typical example of A : Laplace operator)

  • Model driven approach
  • Operators are given
  • Av for a vector v can be

computed Extract Model Data Behavior

  • f 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

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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 ≈ um → ∥v − K−1um∥ : residual of um 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)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

11 / 17

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof. QmQ∗

mv =: pm−1(K)Φ(µ0)

∈ Km(K, Φ(µ0)) = Span{Φ(µ0), KΦ(µ0), . . . , Km−1Φ(µ0)}

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

12 / 17

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof. QmQ∗

mv =: pm−1(K)Φ(µ0)

arg min

u∈Km(K,Φ(µ0))

∥v − u∥ = QmQ∗

mv

(10)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

12 / 17

Projection onto Km(K, Φ(µ0))

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof. QmQ∗

mv =: pm−1(K)Φ(µ0)

arg min

u∈Km(K,Φ(µ0))

∥v − u∥ = QmQ∗

mv

= pm−1(K)K−1Φ(µ1) = K−1pm−1(K)Φ(µ1). (10)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

12 / 17

Projection onto Km(K, Φ(µ0)) Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof. QmQ∗

mv =: pm−1(K)Φ(µ0)

arg min

u∈Km(K,Φ(µ0))

∥v − u∥ = QmQ∗

mv

= pm−1(K)K−1Φ(µ1) = K−1pm−1(K)Φ(µ1). (10)

  • arg minu∈Km(K,Φ(µ0)) ∥v − u∥ = K−1pm−1(K)Φ(µ1),

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

12 / 17

Projection onto Km(K, Φ(µ0)) Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof. QmQ∗

mv =: pm−1(K)Φ(µ0)

arg min

u∈Km(K,Φ(µ0))

∥v − u∥ = QmQ∗

mv

= pm−1(K)K−1Φ(µ1) = K−1pm−1(K)Φ(µ1). (10)

  • arg minu∈Km(K,Φ(µ0)) ∥v − u∥ = K−1pm−1(K)Φ(µ1),
  • For v ∈ Km(K, Φ(µ1)), K−1v ∈ Km(K, Φ(µ0)),

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

12 / 17

Projection onto Km(K, Φ(µ0)) Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof. QmQ∗

mv =: pm−1(K)Φ(µ0)

arg min

u∈Km(K,Φ(µ0))

∥v − u∥ = QmQ∗

mv

= pm−1(K)K−1Φ(µ1) = K−1pm−1(K)Φ(µ1). (10)

  • arg minu∈Km(K,Φ(µ0)) ∥v − u∥ = K−1pm−1(K)Φ(µ1),
  • For v ∈ Km(K, Φ(µ1)), K−1v ∈ Km(K, Φ(µ0)),

∴ pm−1(K)Φ(µ1) = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (11)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

12 / 17

Projection onto Km(K, Φ(µ0)) Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) (12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

(12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

(12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

= Qm+1Q∗

m+1Kpm−1(K)Φ(µ0)

(12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

= Qm+1Q∗

m+1Kpm−1(K)Φ(µ0) ∈ Km+1(K, Φ(µ0))

(12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

= Qm+1Q∗

m+1Kpm−1(K)Φ(µ0) ∈ Km+1(K, Φ(µ0))

= Qm+1Q∗

m+1KQm+1Q∗ m+1pm−1(K)Φ(µ0)

(12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

Φ(µ1) = KΦ(µ0) : Time evolution of µ0

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SLIDE 27

Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

= Qm+1Q∗

m+1Kpm−1(K)Φ(µ0) ∈ Km+1(K, Φ(µ0))

= Qm+1Q∗

m+1KQm+1Q∗ m+1pm−1(K)Φ(µ0)

(12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

Φ(µ1) = KΦ(µ0) : Time evolution of µ0 By the def. of Km+1 By the def. of pm−1

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SLIDE 28

Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

= Qm+1Q∗

m+1Kpm−1(K)Φ(µ0) ∈ Km+1(K, Φ(µ0))

= Qm+1Q∗

m+1KQm+1Q∗ m+1pm−1(K)Φ(µ0)

= Qm+1Km+1Q∗

m+1QmQ∗ mv

(12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

Φ(µ1) = KΦ(µ0) : Time evolution of µ0 By the def. of Km+1 By the def. of pm−1

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SLIDE 29

Residual of a Krylov approximation

Theorem 1 (Minimizer of the Residual)

Let ˜ um := Qm+1Km+1Q∗

m+1QmQ∗

  • mv. Then, ˜

um minimizes the residual in Km(K, Φ(µ1)), that is: ˜ um = arg min

u∈Km(K,Φ(µ1))

∥v − K−1u∥. (9) Proof pm−1(K)Φ(µ1) ∈ Km+1(K, Φ(µ0)) = Qm+1Q∗

m+1pm−1(K)Φ(µ1)

= Qm+1Q∗

m+1Kpm−1(K)Φ(µ0) ∈ Km+1(K, Φ(µ0))

= Qm+1Q∗

m+1KQm+1Q∗ m+1pm−1(K)Φ(µ0)

= Qm+1Km+1Q∗

m+1QmQ∗ mv

= ˜ um. (12)

Krylov subspace methods for P-F operators in RKHSs

  • Y. H. and T. N.

13 / 17

Φ(µ1) = KΦ(µ0) : Time evolution of µ0 By the def. of Km+1 By the def. of pm−1

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SLIDE 30

Residual of a Krylov approximation

˜ um = arg minu∈Km(K,Φ(µ1)) ∥v − K−1u∥ · · · A minimizer of the residual

Theorem 2 (Residual of an Arnoldi approximation)

let um := QmKmQ∗

mv ∈ Km(K, Φ(µS 0 )) be the Arnoldi approximation of

  • Kv. Then, there exists Cm > 0 such that

∥v − K−1um∥ ≤ (1 + Cm)∥v − K−1˜ um−1∥. (13) For ϵ > 0, if m is sufficiently large so that the Krylov subspace Km(K, Φ(µS

0 )) is sufficiently close to Hk, and if K is bounded, then

Cm ≤ 1 + ∥K−1∥∥K∥ ϵ. (14) We have also shown similar theorems about the Shift-invert Arnoldi approximations.

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SLIDE 31

Numerical experiments

Example 1 (Landau equation)

X = [0, ∞) dr dt = 0.5r − r3 (15) Discretizing and adding random noise Xt = Xt−1 + ∆t(0.5Xt−1 − X3

t−1 + ξt)

(16)

  • Fig. 1: The dimension of the Krylov subspace m versus the residual

Krylov subspace methods for P-F operators in RKHSs

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Numerical experiments

Example 2 (Real-world Internet traffic data)

xt : the amount of Internet traffic (gbps) that passed through a certain node (ID 12) in a network composed of 23 nodes and 227 links at time t.

  • Fig. 2: The amount of Internet traffic at

each t

  • Fig. 3: The dimension of the Krylov

subspace m versus the residual

Krylov subspace methods for P-F operators in RKHSs

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SLIDE 33

Conclusions

  • We considered Krylov subspace methods for P-F operators.
  • Since the setting with P-F operators is different from classical settings

in numerical linear algebra, new analyses for the Krylov subspace methods for P-F operators are required.

  • We have shown connections of the Krylov approximations with their

residuals.

Krylov subspace methods for P-F operators in RKHSs

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