| 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 |
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|
| variable {X Ω : Type*} [PseudoMetricSpace X] [SeparableSpace X] [Nonempty X] |
| [MeasurableSpace X] [BorelSpace X] [StandardBorelSpace X] [MeasurableSpace Ω] |
| {d : ℕ} |
|
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| 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 |
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| end |
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| end |
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| end OAI |
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