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import OAI.Probability.LogConcave.Numerics.KernelActionPicardRms
import OAI.Probability.LogConcave.Sampling.ProbabilityEndpoint
namespace OAI
section
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory ProbabilityTheory RMSIntegral Quadrature
open scoped Classical BigOperators NNReal
lemma kernelActionPicard_continuous {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}
(hF : Primitive F lam) (x : Point d) {r T h : ℝ} (hr : 0≤r)
(hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) (hh : 0<h)
(ψ : ℝ) (n : ℕ) (hn : 0<n) (m N : ℕ) (s e : ProbabilityNode T h n) :
Continuous (kernelActionPicard F x r T h ψ n m N s e) := by
change Continuous (fun p => ∑i,(derivativeWeight (angleNodes m) i/ψ) •
kernelCorrectionPicard F x r T h (ψ*angleNodes m i) n N s e p)
apply continuous_finsetSum
intro i _
have hc : Continuous (fun v : Point d => (derivativeWeight (angleNodes m) i/ψ) • v) := continuous_const_smul _
exact hc.comp (kernelCorrectionPicard_continuous hF x hr hl hT0 hT1 hh (ψ*angleNodes m i) n hn N s e)
def kernelActionRmsBudget {d : ℕ} (F : Point d → ℝ) (x : Point d)
(lam Ap Ah Lp : ℝ≥0) (r R ρ h ψ C J : ℝ) (k n m N : ℕ) : ℝ :=
2*((∑i,|derivativeWeight (angleNodes m) i/ψ|)*
(∑i,|derivativeWeight (angleNodes m) i/ψ| *
kernelValueRmsBudget F x lam Ap Ah Lp r R ρ h (ψ*angleNodes m i) C J k n N))+
2*(((∑i,|derivativeWeight (angleNodes m) i|)/ψ)^2*
(((ψ*m)^(m+1)/(m.factorial:ℝ))^2*
((d*((lam:ℝ)*r^2)^2)*(R⁻¹)^(4*(m+1))*harmonicCorrectionBudget (m+1))))
theorem kernelActionPicard_terminal_rms (n : ℕ) (hn : 0<n) :
∃Ap Ah Lp Lh : ℝ≥0,∃C J : ℝ,0≤C ∧ 1≤J ∧ ∃k : ℕ,
∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam,
∀(x : Point d) {r : ℝ},∀hr : 0<r,0<lam → ∀hl : (lam:ℝ)*r^2≤1/2,1≤d →
∀{T h : ℝ},∀hT0 : 0<T,∀hT1 : T<1,∀hh : 0<h,h≤Real.log 2 →
Ap*probabilityMeanLipschitz lam r≤1/2 → Ah*probabilityMeanLipschitz lam r≤1/2 →
Lp*probabilityMeanLipschitz lam r≤1/2 → Lh*probabilityMeanLipschitz lam r≤1/2 →
∀N : ℕ,∀s : ProbabilityNode T h n,
let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩
∀R : ℝ,0<R → R^2≤1-T^2 → ∀m : ℕ,1 ≤ m → ∀ψ : ℝ,0<ψ → ψ*m≤1 →
let err := fun p : Point d × Point d =>
kernelActionPicard F x r T h ψ n m N s (probabilityEndpoint hT0 hT1 hh n) p-
(terminalJacobian hF x hr hl hT0.le hT1 (S,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2
Integrable (fun p => ‖err p‖^2) ((interpolationLaw F x r T).prod (stdGaussian (Point d))) ∧
(∫p,‖err p‖^2 ∂((interpolationLaw F x r T).prod (stdGaussian (Point d))))≤
kernelActionRmsBudget F x lam Ap Ah Lp r R S h ψ C J k n m N := by
obtain ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,hA⟩ := kernelActionPicard_rms n hn
refine ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,?_⟩
intro d F lam hF x r hr hlam hl hd T h hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s
dsimp only
let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩
let ep := probabilityEndpoint hT0 hT1 hh n
let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n ep,probabilityNodeTime_mem hT0 hT1 hh hn ep⟩
have hE : E=⟨T,hT0.le,le_rfl⟩ := Subtype.ext (probabilityEndpoint_time hT0 hT1 hh n hn)
intro R hR hRT m hm1 ψ hψ hψm
let fw := probabilityTransport hF x hr.le hl hT0.le hT1 S ⟨T,hT0.le,le_rfl⟩
let inp := fun y : Point (d+d) => (fw (productPointEquiv d d y).1,(productPointEquiv d d y).2)
let err := fun p : Point d × Point d => kernelActionPicard F x r T h ψ n m N s ep p-
(terminalJacobian hF x hr hl hT0.le hT1 (S,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2
have ha := hA hF x hr hlam hl hd hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s ep R hR hRT m hm1 ψ hψ hψm
let aerr := fun f : Point d → Point d => fun y : Point (d+d) =>
kernelActionPicard F x r T h ψ n m N s ep
(f (productPointEquiv d d y).1,(productPointEquiv d d y).2)-kernelTransportAction f y
change Integrable (fun y => ‖aerr (probabilityTransport hF x hr.le hl hT0.le hT1 S E) y‖^2)
(gibbs (centeringPotential F x r S)) ∧
(∫y,‖aerr (probabilityTransport hF x hr.le hl hT0.le hT1 S E) y‖^2
∂gibbs (centeringPotential F x r S))≤kernelActionRmsBudget F x lam Ap Ah Lp r R S h ψ C J k n m N at ha
rw [hE] at ha
have heq (y : Point (d+d)) : err (inp y)=kernelActionPicard F x r T h ψ n m N s ep (inp y)-kernelTransportAction fw y := by
dsimp only [err,inp]
rw [kernelTransportAction_terminal,terminalForward_eq]
have hfw : Measurable fw := (probabilityTransport_continuous hF x hr.le hl hT0.le hT1 S ⟨T,hT0.le,le_rfl⟩).measurable
have hinp : Measurable inp := (hfw.comp ((productPointEquiv d d).continuous.measurable.fst)).prodMk
((productPointEquiv d d).continuous.measurable.snd)
have hlaw := centering_transport_joint_law hF x hr.le hl hT0.le hT1 S ⟨T,hT0.le,le_rfl⟩
have hK : Continuous (fun p : Point d × Point d =>
(terminalJacobian hF x hr hl hT0.le hT1 (S,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2) :=
(((terminalJacobian_smooth hF x hr hl hT0.le hT1).continuous.comp
(continuous_const.prodMk continuous_fst)).sub continuous_const).clm_apply continuous_snd
have he : Measurable err := ((kernelActionPicard_continuous hF x hr.le hl hT0 hT1 hh ψ n hn m N s ep).sub hK).measurable
have hi : Integrable (fun y => ‖err (inp y)‖^2) (gibbs (centeringPotential F x r S)) := by
simpa only [heq] using ha.1
have hb : (∫y,‖err (inp y)‖^2 ∂gibbs (centeringPotential F x r S))≤
kernelActionRmsBudget F x lam Ap Ah Lp r R S h ψ C J k n m N := by
simpa only [heq] using ha.2
exact rms_bound_pushforward hinp hlaw he hi hb
end LogConcaveSampling
end
end
section
noncomputable section
namespace LogConcaveSampling
open Set MeasureTheory ProbabilityTheory RMSIntegral Quadrature
open scoped Classical BigOperators NNReal
def kernelQuadraturePicard {d : ℕ} (F : Point d → ℝ) (x : Point d)
(r T h ψ : ℝ) (n m N : ℕ) (e : ProbabilityNode T h n)
(w : ProbabilityNode T h n → ℝ) (p : Point d × Point d) : Point d :=
∑s,w s • kernelActionPicard F x r T h ψ n m N s e p
theorem kernelQuadraturePicard_numerical_rms (n : ℕ) (hn : 0<n) :
∃Ap Ah Lp Lh : ℝ≥0,∃C J : ℝ,0≤C ∧ 1≤J ∧ ∃k : ℕ,
∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam,
∀(x : Point d) {r : ℝ},∀hr : 0<r,0<lam → ∀hl : (lam:ℝ)*r^2≤1/2,1≤d →
∀{T h : ℝ},∀hT0 : 0<T,∀hT1 : T<1,∀hh : 0<h,h≤Real.log 2 →
Ap*probabilityMeanLipschitz lam r≤1/2 → Ah*probabilityMeanLipschitz lam r≤1/2 →
Lp*probabilityMeanLipschitz lam r≤1/2 → Lh*probabilityMeanLipschitz lam r≤1/2 →
∀N : ℕ,∀R : ℝ,0<R → R^2≤1-T^2 → ∀m : ℕ,1 ≤ m → ∀ψ : ℝ,0<ψ → ψ*m≤1 →
∀w : ProbabilityNode T h n → ℝ,
let err := fun p : Point d × Point d =>
kernelQuadraturePicard F x r T h ψ n m N (probabilityEndpoint hT0 hT1 hh n) w p-
∑s,w s • (terminalJacobian hF x hr hl hT0.le hT1
(probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2
Integrable (fun p => ‖err p‖^2) ((interpolationLaw F x r T).prod (stdGaussian (Point d))) ∧
(∫p,‖err p‖^2 ∂((interpolationLaw F x r T).prod (stdGaussian (Point d))))≤
(∑s,|w s|)*(∑s,|w s| *
kernelActionRmsBudget F x lam Ap Ah Lp r R (probabilityNodeTime T h n s) h ψ C J k n m N) := by
obtain ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,hA⟩ := kernelActionPicard_terminal_rms n hn
refine ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,?_⟩
intro d F lam hF x r hr hlam hl hd T h hT0 hT1 hh hsmall hqp hqh hqlp hqlh N R hR hRT m hm1 ψ hψ hψm w
dsimp only
let ep := probabilityEndpoint hT0 hT1 hh n
let f := fun (s : ProbabilityNode T h n) (p : Point d × Point d) =>
kernelActionPicard F x r T h ψ n m N s ep p-
(terminalJacobian hF x hr hl hT0.le hT1
(probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2
have hs (s : ProbabilityNode T h n) :=
hA hF x hr hlam hl hd hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s R hR hRT m hm1 ψ hψ hψm
have hfc (s : ProbabilityNode T h n) : Continuous (f s) := by
have hK : Continuous (fun p : Point d × Point d =>
(terminalJacobian hF x hr hl hT0.le hT1
(probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2) :=
(((terminalJacobian_smooth hF x hr hl hT0.le hT1).continuous.comp
(continuous_const.prodMk continuous_fst)).sub continuous_const).clm_apply continuous_snd
exact (kernelActionPicard_continuous hF x hr.le hl hT0 hT1 hh ψ n hn m N s ep).sub hK
have he (p : Point d × Point d) :
kernelQuadraturePicard F x r T h ψ n m N ep w p-
∑s,w s • (terminalJacobian hF x hr hl hT0.le hT1
(probabilityNodeTime T h n s,p.1)-ContinuousLinearMap.id ℝ (Point d)) p.2=
∑s,w s • f s p := by
simp only [kernelQuadraturePicard,f,smul_sub,Finset.sum_sub_distrib]
simp_rw [show probabilityEndpoint hT0 hT1 hh n=ep from rfl,he]
exact weighted_sum_sq_varying w f
(fun s => kernelActionRmsBudget F x lam Ap Ah Lp r R (probabilityNodeTime T h n s) h ψ C J k n m N)
(fun s => (hfc s).aestronglyMeasurable) (fun s => (hs s).1) (fun s => (hs s).2)
end LogConcaveSampling
end
end
section
noncomputable section
namespace LogConcaveSampling
open Quadrature
open scoped Classical BigOperators NNReal
theorem kernelQuadraturePicard_lipschitz (n : ℕ) (hn : 0<n) :
∃A B : ℝ≥0,∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0},
Primitive F lam → ∀(x : Point d) {r T h ψ : ℝ},0≤r →
(lam:ℝ)*r^2≤1/2 → 0<T → T<1 → 0<h → 0<ψ →
∀m : ℕ,ψ*m≤1 →
A*probabilityMeanLipschitz lam r≤1/2 →
B*probabilityMeanLipschitz lam r≤1/2 →
∀N : ℕ,∀e : ProbabilityNode T h n,∀w : ProbabilityNode T h n → ℝ,
LipschitzWith ((∑s,‖w s‖₊)*
((∑i : Fin (m+1),‖derivativeWeight (angleNodes m) i/ψ‖₊)*
(16*(A*probabilityMeanLipschitz lam r))))
(kernelQuadraturePicard F x r T h ψ n m N e w) := by
obtain ⟨A,B,hAB⟩ := kernelActionPicard_lipschitz n hn
refine ⟨A,B,?_⟩
intro d F lam hF x r T h ψ hr hl hT0 hT1 hh hψ m hψm hq hq' N e w
exact finite_weighted_lipschitz w
(fun s => kernelActionPicard F x r T h ψ n m N s e)
(fun s => hAB hF x hr hl hT0 hT1 hh hψ m hψm hq hq' N s e)
end LogConcaveSampling
end
end
section
noncomputable section
namespace LogConcaveSampling.FinitePicard
open MeasureTheory
open scoped BigOperators NNReal RealInnerProductSpace
variable {Ω E I : Type*} [MeasurableSpace Ω] {ν : Measure Ω} [IsFiniteMeasure ν]
[NormedAddCommGroup E] [InnerProductSpace ℝ E] [Fintype I]
lemma iterateLp_ae (w : I → I → ℝ) (φ : I → E → E) {K : ℝ≥0}
(hφ : ∀i,LipschitzWith K (φ i)) (a : I → Lp E 2 ν) (n : ℕ) :
∀ᵐω ∂ν,∀i, ((stepLp w φ hφ a)^[n] a i) ω=
nodes w φ (fun ω j => a j ω) n ω i := by
induction n with
| zero => exact Filter.Eventually.of_forall (fun _ _ => rfl)
| succ n ih =>
have hs : ∀ᵐω ∂ν,∀i,
(stepLp w φ hφ a ((stepLp w φ hφ a)^[n] a) i) ω=
step w φ (fun j => a j ω) (fun j => (((stepLp w φ hφ a)^[n] a) j) ω) i := by
rw [ae_all_iff]
exact stepLp_ae w φ hφ a _
filter_upwards [hs,ih] with ω hs hi
intro i
rw [Function.iterate_succ_apply',hs i]
simp only [nodes,hi]
theorem actual_nodes_rms (w : I → I → ℝ) (φ : I → E → E) {A K : ℝ≥0}
(hφ : ∀i,LipschitzWith K (φ i)) (a e : I → Lp E 2 ν)
(hw : ∀i,∑j,|w i j|≤A) (hq : (A*K:ℝ≥0)≤1/2)
{D : ℝ} (hD : 0≤D)
(he : ∀i,(∫ω,‖(stepLp w φ hφ a e i) ω-e i ω‖^2 ∂ν)≤D^2)
(n : ℕ) (i : I) :
(∫ω,‖nodes w φ (fun ω j => a j ω) n ω i-e i ω‖^2 ∂ν)≤
(2*D+(A*K:ℝ≥0)^n*dist a e)^2 := by
have hh := rms_picard_defect w φ hφ a e hw hq hD
(fun i => nodewise_rms_le hD (he i)) n
have hj := (dist_le_pi_dist ((stepLp w φ hφ a)^[n] a) e i).trans hh
have hn : 0≤2*D+(A*K:ℝ≥0)^n*dist a e := by positivity
have heq : (∫ω,‖nodes w φ (fun ω j => a j ω) n ω i-e i ω‖^2 ∂ν)=
dist (((stepLp w φ hφ a)^[n] a) i) (e i)^2 := by
rw [dist_eq_norm,lp_norm_sq_integral]
apply integral_congr_ae
filter_upwards [iterateLp_ae w φ hφ a n,
Lp.coeFn_sub (((stepLp w φ hφ a)^[n] a) i) (e i)] with ω h1 h2
simp only [h2,Pi.sub_apply,h1 i]
rw [heq]
exact pow_le_pow_left₀ dist_nonneg hj 2
end LogConcaveSampling.FinitePicard
end
end
end
end OAI
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