| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.JointInterpolationLaw |
|
|
| 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 ProbabilityTheory |
| open scoped RealInnerProductSpace |
|
|
| lemma integrable_lipschitz_comp_of_integrable {Ω E G : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsFiniteMeasure μ] [NormedAddCommGroup E] [NormedAddCommGroup G] |
| {X : Ω → E} (hX : Integrable X μ) {g : E → G} {L : ℝ≥0} (hg : LipschitzWith L g) : |
| Integrable (fun ω => g (X ω)) μ := by |
| apply ((integrable_const ‖g 0‖).add (hX.norm.const_mul (L:ℝ))).mono' |
| (hg.continuous.comp_aestronglyMeasurable hX.aestronglyMeasurable) |
| filter_upwards [] with ω |
| have hh := hg.dist_le_mul (X ω) 0 |
| rw [dist_eq_norm,dist_zero_right] at hh |
| calc |
| ‖g (X ω)‖ ≤ ‖g 0‖+‖g (X ω)-g 0‖ := norm_le_norm_add_norm_sub' _ _ |
| _ ≤ ‖g 0‖+(L:ℝ)*‖X ω‖ := by linarith |
|
|
| lemma primitive_identity_integrable {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r : ℝ} (hr : 0≤r) (hl : (lam:ℝ)*r^2<1) : |
| Integrable (fun z => z) (gibbs (primitivePotential F x r)) := by |
| let := probability_gibbs_of_partition |
| (partition_pos_of_continuous (hF.continuous_potential x r)).ne' |
| (partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr hl)) |
| exact (Appell.HasGrowth.lipschitz (LipschitzWith.id (α:=Point d))).integrable |
| continuous_id.aestronglyMeasurable (hF.hasExpMoments x hr hl).hasMoments |
|
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| def interpolationResidual {d : ℕ} (F : Point d → ℝ) (x : Point d) (r ρ : ℝ) |
| (p : Point d × Point d) : Point d := |
| (1/(1-ρ^2)) • (p.1-ρ • p.2)+r • conditionalFieldMean F x r ρ p.2 |
|
|
| lemma interpolationResidual_integrable {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) : |
| Integrable (interpolationResidual F x r ρ) (jointInterpolationLaw F x r ρ) := by |
| let μ := gibbs (primitivePotential F x r) |
| let ν := stdGaussian (Point d) |
| let := probability_gibbs_of_partition |
| (partition_pos_of_continuous (hF.continuous_potential x r)).ne' |
| (partition_ne_top_of_integrable (hF.integrable_exp_neg_potential x hr (by linarith))) |
| have hi := primitive_identity_integrable hF x hr (by linarith : (lam:ℝ)*r^2<1) |
| have hz : Integrable (fun p : Point d × Point d => p.1) (μ.prod ν) := hi.comp_fst ν |
| have hg : Integrable (fun p : Point d × Point d => p.2) (μ.prod ν) := |
| (IsGaussian.integrable_id (μ:=ν)).comp_snd μ |
| have hy := (hz.smul ρ).add (hg.smul (Real.sqrt (1-ρ^2))) |
| have hM := conditionalFieldMean_lipschitz hF x hr hl hρ0 hρ1 |
| have hm := integrable_lipschitz_comp_of_integrable hy hM |
| have hu := ((hz.sub (hy.smul ρ)).smul (1/(1-ρ^2))).add (hm.smul r) |
| have hc : Continuous (interpolationResidual F x r ρ) := by |
| unfold interpolationResidual |
| exact ((continuous_fst.sub (continuous_snd.const_smul ρ)).const_smul (1/(1-ρ^2))).add |
| ((hM.continuous.comp continuous_snd).const_smul r) |
| apply (integrable_map_measure hc.aestronglyMeasurable (by fun_prop)).2 |
| exact hu |
|
|
| lemma conditional_residual_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) (y : Point d) : |
| (∫ z,interpolationResidual F x r ρ (z,y) ∂gibbs (conditionalPotential F x r ρ y))=0 := by |
| let := conditionalLaw_probability hF x hr hl hρ0 hρ1 y |
| have hi := conditional_identity_integrable hF x hr hl hρ0 hρ1 y |
| have hh := integral_scaled_affine_add hi (integrable_const |
| (conditionalFieldMean F x r ρ y)) (1/(1-ρ^2)) r (ρ • y) |
| rw [show (∫ z,interpolationResidual F x r ρ (z,y) ∂gibbs (conditionalPotential F x r ρ y))= |
| (1/(1-ρ^2)) • ((∫ z,z ∂gibbs (conditionalPotential F x r ρ y))-ρ • y)+ |
| r • (∫ _,conditionalFieldMean F x r ρ y ∂gibbs (conditionalPotential F x r ρ y)) from hh, |
| integral_const,probReal_univ,one_smul,conditional_mean_identity hF x hr hl hρ0 hρ1] |
| have ha := (probability_time hρ0 hρ1).1 |
| rw [sub_sub_cancel_left,smul_neg,smul_smul] |
| have he : 1/(1-ρ^2)*((1-ρ^2)*r)=r := by field_simp |
| rw [he,neg_add_cancel] |
|
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| lemma integrable_scaled_affine_const {d : ℕ} {μ : Measure (Point d)} [IsFiniteMeasure μ] |
| (hi : Integrable (fun z => z) μ) (c : ℝ) (a b : Point d) : |
| Integrable (fun z => c • (z-a)+b) μ := |
| ((hi.sub (integrable_const a)).smul c).add (integrable_const b) |
|
|
| lemma conditional_residual_integrable {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) (y : Point d) : |
| Integrable (fun z => interpolationResidual F x r ρ (z,y)) |
| (gibbs (conditionalPotential F x r ρ y)) := by |
| let := conditionalLaw_probability hF x hr hl hρ0 hρ1 y |
| exact integrable_scaled_affine_const (conditional_identity_integrable hF x hr hl hρ0 hρ1 y) |
| (1/(1-ρ^2)) (ρ • y) (r • conditionalFieldMean F x r ρ y) |
|
|
| theorem interpolation_flux_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) |
| {T : Point d → Point d →L[ℝ] ℝ} (hT : Measurable T) |
| {C : ℝ} (hC : ∀ y,‖T y‖≤C) : |
| (∫ p,T p.2 (interpolationResidual F x r ρ p) ∂jointInterpolationLaw F x r ρ)=0 := by |
| have hi := interpolationResidual_integrable hF x hr hl hρ0 hρ1 |
| have hM := conditionalFieldMean_lipschitz hF x hr hl hρ0 hρ1 |
| have hc : Continuous (interpolationResidual F x r ρ) := by |
| unfold interpolationResidual |
| exact ((continuous_fst.sub (continuous_snd.const_smul ρ)).const_smul (1/(1-ρ^2))).add |
| ((hM.continuous.comp continuous_snd).const_smul r) |
| have hm : Measurable (fun p => T p.2 (interpolationResidual F x r ρ p)) := |
| (continuous_fst.clm_apply continuous_snd).measurable.comp |
| ((hT.comp measurable_snd).prodMk hc.measurable) |
| apply interpolation_joint_integral_zero hF x hr hl hρ0 hρ1 |
| · refine (hi.norm.const_mul C).mono' hm.aestronglyMeasurable (Filter.Eventually.of_forall fun p => ?_) |
| exact ((T p.2).le_opNorm _).trans (mul_le_mul_of_nonneg_right (hC p.2) (norm_nonneg _)) |
| · intro y |
| have hrz := conditional_residual_integrable hF x hr hl hρ0 hρ1 y |
| rw [(T y).integral_comp_comm hrz,conditional_residual_zero hF x hr hl hρ0 hρ1,map_zero] |
| end LogConcaveSampling |
|
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| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| end UpperProof |
| end |
| end |
| end |
|
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| end OAI |
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