| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.LiftedVectors |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open MeasureTheory ProbabilityTheory |
| open scoped BigOperators |
| namespace MinUncut.GaussianBudget |
| variable {X C : Type*} [Fintype X] [Fintype C] |
|
|
| abbrev productLaw (X C : Type*) [Fintype X] [Fintype C] := |
| (Measure.pi (fun _ : X => gaussianReal 0 1)).prod |
| ((Measure.pi (fun _ : X => gaussianReal 0 1)).prod (Measure.pi (fun _ : C => gaussianReal 0 1))) |
|
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| def packEquiv : ((X → ℝ) × ((X → ℝ) × (C → ℝ))) ≃ᵐ (Coordinates X C → ℝ) := |
| ((MeasurableEquiv.refl (X → ℝ)).prodCongr |
| (MeasurableEquiv.sumPiEquivProdPi (fun _ : X ⊕ C => ℝ)).symm).trans |
| (MeasurableEquiv.sumPiEquivProdPi (fun _ : X ⊕ (X ⊕ C) => ℝ)).symm |
|
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| lemma pack_preserving : MeasurePreserving (packEquiv (X := X) (C := C)) |
| (productLaw X C) (Measure.pi (fun _ : Coordinates X C => gaussianReal 0 1)) := by |
| exact (measurePreserving_sumPiEquivProdPi_symm (fun _ : X ⊕ (X ⊕ C) => gaussianReal 0 1)).comp |
| ((MeasurePreserving.id _).prod (measurePreserving_sumPiEquivProdPi_symm (fun _ : X ⊕ C => gaussianReal 0 1))) |
|
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| lemma score_packed (v : X → ℝ) (σ η : ℝ) (z : C) |
| (p : (X → ℝ) × ((X → ℝ) × (C → ℝ))) : |
| score (liftVector v σ η z) (packEquiv p)=score v p.1+σ*score v p.2.1+η*p.2.2 z := by |
| classical |
| simp only [score,Fintype.sum_sum_type,liftVector,packEquiv,MeasurableEquiv.trans_apply] |
| change (∑ x, v x*p.1 x)+((∑ x, (σ*v x)*p.2.1 x)+∑ c, (if c=z then η else 0)*p.2.2 c)=_ |
| simp only [ite_mul,zero_mul,Finset.sum_ite_eq',Finset.mem_univ,ite_true,mul_assoc,← Finset.mul_sum] |
| ring |
|
|
| lemma integral_pack_sign (v w : Coordinates X C → ℝ) : |
| (∫ c, ∫ g, ∫ l, signFailure (score v (packEquiv (c,g,l))) |
| (score w (packEquiv (c,g,l))) ∂Measure.pi (fun _ : C => gaussianReal 0 1) |
| ∂Measure.pi (fun _ : X => gaussianReal 0 1) ∂Measure.pi (fun _ : X => gaussianReal 0 1)) = |
| ∫ q, signFailure (score v q) (score w q) ∂Measure.pi (fun _ : Coordinates X C => gaussianReal 0 1) := by |
| have hv : Measurable (score v) := by unfold score; fun_prop |
| have hw : Measurable (score w) := by unfold score; fun_prop |
| have hp := pack_preserving (X := X) (C := C) |
| have hi : Integrable (fun p : (X → ℝ) × ((X → ℝ) × (C → ℝ)) => |
| signFailure (score v (packEquiv p)) (score w (packEquiv p))) (productLaw X C) := |
| signFailure_integrable (hv.comp hp.measurable) (hw.comp hp.measurable) |
| rw [← hp.integral_comp packEquiv.measurableEmbedding (fun q => signFailure (score v q) (score w q))] |
| rw [integral_prod _ hi] |
| apply integral_congr_ae |
| filter_upwards [hi.prod_right_ae] with c hc |
| exact (integral_prod _ hc).symm |
|
|
| lemma first_triple_bound {v : X → ℝ} (hv : ‖vector v‖=1) {σ η : ℝ} |
| (hσ : 0≤σ) (hη : 0≤η) (z : C) : |
| (∫ c, ∫ g, ∫ l, signFailure (score v c) (score v c+σ*score v g+η*l z) |
| ∂Measure.pi (fun _ : C => gaussianReal 0 1) ∂Measure.pi (fun _ : X => gaussianReal 0 1) |
| ∂Measure.pi (fun _ : X => gaussianReal 0 1)) ≤ 4*(σ+η) := by |
| have h := lift_sign_first hv hσ hη z |
| rw [← integral_pack_sign] at h |
| simpa only [score_packed,zero_mul,add_zero] using h |
|
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| lemma second_triple_bound {v w : X → ℝ} (hv : ‖vector v‖=1) (hw : ‖vector w‖=1) |
| {σ η : ℝ} (hσ : 0≤σ) (hσ1 : σ≤1) (hη : 0≤η) (z z' : C) : |
| (∫ c, ∫ g, ∫ l, signFailure (score v c+σ*score v g+η*l z) |
| (score w c+σ*score w g+η*l z') |
| ∂Measure.pi (fun _ : C => gaussianReal 0 1) ∂Measure.pi (fun _ : X => gaussianReal 0 1) |
| ∂Measure.pi (fun _ : X => gaussianReal 0 1)) ≤ 4*‖vector v-vector w‖+4*η := by |
| have h := lift_sign_second hv hw hσ hσ1 hη z z' |
| rw [← integral_pack_sign] at h |
| simpa only [score_packed] using h |
| end MinUncut.GaussianBudget |
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| end |
| end |
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| end OAI |
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