| import Mathlib |
| import OAI.Probability.LogConcave.Analysis.InterpolationPotential |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| open MeasureTheory Filter |
| open scoped ENNReal NNReal Topology |
|
|
| section UpperProof |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped ENNReal NNReal RealInnerProductSpace Topology |
|
|
| namespace LogConcaveSampling |
| open MeasureTheory |
| open scoped RealInnerProductSpace |
|
|
| lemma interpolation_conditional_exponent {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (r ρ : ℝ) (y z : Point d) (ha : 1-ρ^2≠0) : |
| -primitivePotential F x r z-‖y-ρ • z‖^2/(2*(1-ρ^2))= |
| -‖y‖^2/2-conditionalPotential F x r ρ y z := by |
| simp only [primitivePotential,conditionalPotential,norm_sub_sq_real,norm_smul, |
| Real.norm_eq_abs,mul_pow,sq_abs,inner_smul_right] |
| rw [real_inner_comm z y] |
| field_simp |
| ring |
|
|
| lemma weighted_integral_zero_of_gibbs {d : ℕ} {E : Type*} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Point d → ℝ} |
| (hc : Continuous H) (hi : Integrable (fun z => Real.exp (-H z))) |
| {u : Point d → E} (hu : (∫ z,u z ∂gibbs H)=0) : |
| (∫ z,Real.exp (-H z) • u z)=0 := by |
| rw [integral_gibbs_vector hc hi] at hu |
| exact (smul_eq_zero.mp hu).resolve_left (inv_ne_zero (integral_exp_pos hi).ne') |
|
|
| def jointInterpolationLaw {d : ℕ} (F : Point d → ℝ) (x : Point d) (r ρ : ℝ) := |
| (((gibbs (primitivePotential F x r)).prod (stdGaussian (Point d))).map |
| (fun p => (p.1,ρ • p.1+Real.sqrt (1-ρ^2) • p.2))) |
|
|
| theorem interpolation_joint_integral_zero {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1) |
| {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| {u : Point d × Point d → E} |
| (hi : Integrable u (jointInterpolationLaw F x r ρ)) |
| (hu : ∀ y,(∫ z,u (z,y) ∂gibbs (conditionalPotential F x r ρ y))=0) : |
| (∫ p,u p ∂jointInterpolationLaw F x r ρ)=0 := by |
| unfold jointInterpolationLaw at hi ⊢ |
| have ha := (probability_time hρ0 hρ1).1 |
| have hσ := (Real.sqrt_pos.mpr ha).ne' |
| have hp0 := (partition_pos_of_continuous (hF.continuous_potential x r)).ne' |
| have hpt := partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr (by linarith)) |
| let c := (partition (primitivePotential F x r))⁻¹* |
| EulerDensity.noiseCoefficient d (Real.sqrt (1-ρ^2)) |
| have hc0 : c≠0 := mul_ne_zero (ENNReal.inv_ne_zero.mpr hpt) |
| (EulerDensity.noiseCoefficient_positive d hσ).ne' |
| have hct : c≠⊤ := ENNReal.mul_ne_top (ENNReal.inv_ne_top.mpr hp0) |
| (EulerDensity.noiseCoefficient_finite _ _) |
| let w := fun p : Point d × Point d => Real.exp (-primitivePotential F x r p.1- |
| ‖p.2-ρ • p.1‖^2/(2*(1-ρ^2))) |
| have hw0 : ∀ p,0≤w p := fun _ => (Real.exp_pos _).le |
| have hw : Measurable w := by |
| have := (hF.continuous_potential x r).measurable |
| dsimp [w]; fun_prop |
| have hden : Measurable (fun p => ENNReal.ofReal (w p)) := ENNReal.measurable_ofReal.comp hw |
| have hd := EulerDensity.joint_density (hF.continuous_potential x r).measurable |
| (fun z => ρ • z) (by fun_prop) hσ |
| simp only [Real.sq_sqrt ha.le] at hd |
| change _=c • ((volume : Measure (Point d)).prod volume).withDensity |
| (fun p => ENNReal.ofReal (w p)) at hd |
| rw [hd] at hi ⊢ |
| have hif : Integrable (fun p => w p • u p) ((volume : Measure (Point d)).prod volume) := by |
| have hi' := (integrable_smul_measure hc0 hct).mp hi |
| rw [integrable_withDensity_iff_integrable_smul' (hden) |
| (Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] at hi' |
| simpa only [ENNReal.toReal_ofReal (hw0 _)] using hi' |
| rw [integral_smul_measure,integral_withDensity_eq_integral_toReal_smul |
| hden (Filter.Eventually.of_forall fun _ => ENNReal.ofReal_lt_top)] |
| simp only [ENNReal.toReal_ofReal (hw0 _)] |
| rw [integral_prod_symm _ hif] |
| have hz : ∀ y,(∫ z,w (z,y) • u (z,y))=0 := by |
| intro y |
| have hci := (conditional_hasGaussianLowerTail hF x hr hl hρ0 hρ1 y).integrable_exp |
| (conditionalPotential_smooth hF x r ρ y).continuous |
| have hh := weighted_integral_zero_of_gibbs |
| (conditionalPotential_smooth hF x r ρ y).continuous hci (hu y) |
| have he : ∀ z,w (z,y)=Real.exp (-‖y‖^2/2)*Real.exp (-conditionalPotential F x r ρ y z) := by |
| intro z; dsimp [w] |
| rw [interpolation_conditional_exponent F x r ρ y z ha.ne',sub_eq_add_neg,Real.exp_add] |
| simp_rw [he,mul_smul] |
| rw [integral_smul,hh,smul_zero] |
| simp_rw [hz] |
| simp |
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
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| end OAI |
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