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dd14c26 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 | import Mathlib
import OAI.Probability.LogConcave.Dynamics.NormSubSqThree
import OAI.Probability.LogConcave.Dynamics.SameLaw
import OAI.Probability.LogConcave.Dynamics.SharedLipschitz
namespace OAI
section
noncomputable section
namespace LogConcaveSampling.Coupling
open MeasureTheory ProbabilityTheory Function TopologicalSpace
open scoped Classical
variable {X Ω : Type*} [PseudoMetricSpace X] [SeparableSpace X] [Nonempty X]
[MeasurableSpace X] [BorelSpace X] [StandardBorelSpace X] [MeasurableSpace Ω]
{d : ℕ}
theorem SquaredAt.noisy_circuit (μ : Measure X) [IsProbabilityMeasure μ]
(ν : Measure Ω) [IsProbabilityMeasure ν]
(f : X → Ω → Point d) (g u : X → Point d)
(hf : Measurable (uncurry f)) (hg : Measurable g) (hu : Measurable u)
(ρ : X → ℝ) (hρ : ∀x,1≤ρ x) (hρlip : ∀x y,ρ y≤ρ x+dist x y)
(hρi : Integrable (fun x => (ρ x)^2) μ)
{Lf Lg E R n : ℝ} (hLf : 0≤Lf) (hLg : 0≤Lg) (hE : 0<E)
(hfl : ∀x y z,‖f x z-f y z‖≤Lf*dist x y)
(hgl : ∀x y,‖g x-g y‖≤Lg*dist x y)
(hc : ∀x,SquaredAt ν (stdGaussian (Point d)) (f x)
(fun w => g x+n • w) (E^2*(ρ x)^2))
(hi : Integrable (fun x => ‖g x-u x‖^2) μ)
(he : (∫x,‖g x-u x‖^2 ∂μ)≤R) :
SquaredAt (μ.prod ν) (μ.prod (stdGaussian (Point d))) (uncurry f)
(fun z => u z.1+n • z.2) (20*E^2*(∫x,(ρ x)^2 ∂μ)+2*R) := by
have hnmeas : Measurable (fun z : X × Point d => g z.1+n • z.2) := by fun_prop
have hc' := SquaredAt.random_anchor μ ν (stdGaussian (Point d)) f
(fun x w => g x+n • w) hf hnmeas ρ hρ hρlip hρi hLf hLg hE hfl
(fun x y w => by simpa only [add_sub_add_right_eq_sub] using hgl x y) hc
have hbase := SquaredAt.same_tape μ g u hg hu hi he
have hnoise := hbase.add_shared (stdGaussian (Point d)) hg hu (fun w => n • w) (by fun_prop)
have ht := hc'.trans hf hnmeas (by fun_prop) hnoise
convert ht using 1
ring
theorem SquaredAt.noisy_circuit_target {Γ : Type*} [MeasurableSpace Γ]
[StandardBorelSpace Γ] [Nonempty Γ]
(μ : Measure X) [IsProbabilityMeasure μ] (ν : Measure Ω) [IsProbabilityMeasure ν]
(θ : Measure Γ) [IsProbabilityMeasure θ]
(f : X → Ω → Point d) (g u : X → Point d) (v : Γ → Point d)
(hf : Measurable (uncurry f)) (hg : Measurable g) (hu : Measurable u) (hv : Measurable v)
(ρ : X → ℝ) (hρ : ∀x,1≤ρ x) (hρlip : ∀x y,ρ y≤ρ x+dist x y)
(hρi : Integrable (fun x => (ρ x)^2) μ)
{Lf Lg E R n : ℝ} (hLf : 0≤Lf) (hLg : 0≤Lg) (hE : 0<E)
(hfl : ∀x y z,‖f x z-f y z‖≤Lf*dist x y)
(hgl : ∀x y,‖g x-g y‖≤Lg*dist x y)
(hc : ∀x,SquaredAt ν (stdGaussian (Point d)) (f x)
(fun w => g x+n • w) (E^2*(ρ x)^2))
(hi : Integrable (fun x => ‖g x-u x‖^2) μ) (he : (∫x,‖g x-u x‖^2 ∂μ)≤R)
(hlaw : (μ.prod (stdGaussian (Point d))).map (fun z => u z.1+n • z.2)=θ.map v) :
SquaredAt (μ.prod ν) θ (uncurry f) v (40*E^2*(∫x,(ρ x)^2 ∂μ)+4*R) := by
have hh := (SquaredAt.noisy_circuit μ ν f g u hf hg hu ρ hρ hρlip hρi
hLf hLg hE hfl hgl hc hi he).change_ideal θ v hf (by fun_prop) hv hlaw
convert hh using 1
ring
end LogConcaveSampling.Coupling
end
end
end OAI
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