| import Mathlib |
| import OAI.Probability.LogConcave.Dynamics.ProbabilityFlowIntegral |
| import OAI.Probability.LogConcave.Sampling.MeasureSmoothLipschitz |
| import OAI.Probability.LogConcave.Sampling.ClosedFieldMean |
|
|
| 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 |
| open Function MeasureTheory Set Filter |
| open scoped Topology NNReal |
|
|
| namespace LogConcaveSampling |
| open Set Function MeasureTheory Filter ProbabilityTheory |
| open scoped Topology |
|
|
| def fullProbabilityFlow {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≤1/2) (t : ℝ) (z : Point d) : Point d := |
| GlobalODE.flow (a:=0) (b:=1) (fun t _ => fullProbabilityVelocity_lipschitz hF x hr hl t) |
| (fullProbabilityVelocity_continuous hF x hr hl) ⟨0,le_rfl,by norm_num⟩ z t |
|
|
| lemma fullProbabilityFlow_time_continuous {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≤1/2) (z : Point d) : Continuous (fun t => fullProbabilityFlow hF x hr hl t z) := |
| GlobalODE.flow_continuous _ _ _ _ |
|
|
| lemma fullProbabilityFlow_lipschitz {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r t : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (ht : t∈Icc 0 1) : |
| LipschitzWith ⟨Real.exp (((Real.pi^2/2)*(lam:ℝ)*r^2)*|t|),(Real.exp_pos _).le⟩ |
| (fullProbabilityFlow hF x hr hl t) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro u v |
| have hh := GlobalODE.trajectory_dist (a:=0) (b:=1) |
| (fun t _ => fullProbabilityVelocity_lipschitz hF x hr hl t) |
| (fullProbabilityFlow_time_continuous hF x hr hl u).continuousOn |
| (GlobalODE.flow_deriv (a:=0) (b:=1) |
| (fun t _ => fullProbabilityVelocity_lipschitz hF x hr hl t) |
| (fullProbabilityVelocity_continuous hF x hr hl) ⟨0,le_rfl,by norm_num⟩ u) |
| (fullProbabilityFlow_time_continuous hF x hr hl v).continuousOn |
| (GlobalODE.flow_deriv (a:=0) (b:=1) |
| (fun t _ => fullProbabilityVelocity_lipschitz hF x hr hl t) |
| (fullProbabilityVelocity_continuous hF x hr hl) ⟨0,le_rfl,by norm_num⟩ v) |
| ⟨0,le_rfl,by norm_num⟩ ⟨t,ht⟩ |
| have hu0 : fullProbabilityFlow hF x hr hl 0 u=u := GlobalODE.flow_initial _ _ _ _ |
| have hv0 : fullProbabilityFlow hF x hr hl 0 v=v := GlobalODE.flow_initial _ _ _ _ |
| change dist (fullProbabilityFlow hF x hr hl t u) (fullProbabilityFlow hF x hr hl t v) ≤ |
| dist (fullProbabilityFlow hF x hr hl 0 u) (fullProbabilityFlow hF x hr hl 0 v)*_ at hh |
| rw [hu0,hv0,sub_zero,mul_comm] at hh |
| exact hh |
|
|
| lemma fullProbabilityFlow_eq {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) : |
| fullProbabilityFlow hF x hr hl T=probabilityFlow hF x hr hl hT0 hT1 := by |
| funext z |
| let α := fun t => fullProbabilityFlow hF x hr hl t z |
| let hL := fun t (_ : t∈Icc 0 T) => clampedProbabilityVelocity_lipschitz hF x hr hl hT0 hT1 t |
| let hc := clampedProbabilityVelocity_continuous hF x hr hl hT0 hT1 |
| let β := GlobalODE.flow hL hc ⟨0,le_rfl,hT0⟩ z |
| have hsub : Icc (0:ℝ) T ⊆ Icc 0 1 := Icc_subset_Icc_right hT1.le |
| have hdα : ∀ t∈Icc (0:ℝ) T, HasDerivWithinAt α |
| (clampedProbabilityVelocity F x r T hT0 t (α t)) (Icc 0 T) t := by |
| intro t ht |
| have hh := GlobalODE.flow_deriv (a:=0) (b:=1) |
| (fun t _ => fullProbabilityVelocity_lipschitz hF x hr hl t) |
| (fullProbabilityVelocity_continuous hF x hr hl) ⟨0,le_rfl,by norm_num⟩ z t (hsub ht) |
| have he : fullProbabilityVelocity F x r t=clampedProbabilityVelocity F x r T hT0 t := by |
| funext y |
| rw [fullProbabilityVelocity_eq F x r ht.1 (ht.2.trans_lt hT1),clampedProbabilityVelocity_eq F x r T hT0 ht] |
| rw [he] at hh |
| exact hh.mono hsub |
| have hh := GlobalODE.trajectory_dist hL |
| (fullProbabilityFlow_time_continuous hF x hr hl z).continuousOn hdα |
| (GlobalODE.flow_continuous hL hc ⟨0,le_rfl,hT0⟩ z).continuousOn |
| (GlobalODE.flow_deriv hL hc ⟨0,le_rfl,hT0⟩ z) |
| ⟨0,le_rfl,hT0⟩ ⟨T,hT0,le_rfl⟩ |
| have ha0 : α 0=z := GlobalODE.flow_initial _ _ _ _ |
| have hb0 : β 0=z := GlobalODE.flow_initial _ _ _ _ |
| change dist (α T) (β T)≤dist (α 0) (β 0)*_ at hh |
| rw [ha0,hb0,dist_self,zero_mul,dist_le_zero] at hh |
| exact hh |
|
|
| lemma fullProbabilityFlow_integral {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) |
| {φ : Point d → ℝ} (hφ : Differentiable ℝ φ) {L : ℝ≥0} (hφL : LipschitzWith L φ) |
| {B : ℝ} (hB : ∀ z,|φ z|≤B) : |
| (∫ z,φ (fullProbabilityFlow hF x hr hl T z) ∂stdGaussian (Point d))= |
| ∫ p,φ (interpolationPath T p) ∂(gibbs (primitivePotential F x r)).prod (stdGaussian (Point d)) := by |
| rw [fullProbabilityFlow_eq hF x hr hl hT0.le hT1] |
| have hh := probabilityFlow_integral hF x hr hl hT0 hT1 hφ hφL hB |
| change (∫ y, φ y ∂Measure.map (interpolationPath T) |
| ((gibbs (primitivePotential F x r)).prod (stdGaussian (Point d)))) = _ at hh |
| rw [integral_map (interpolationPath_measurable T).aemeasurable |
| hφ.continuous.aestronglyMeasurable] at hh |
| exact hh.symm |
|
|
| lemma fullProbabilityFlow_endpoint_integral {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≤1/2) |
| {φ : Point d → ℝ} (hφ : Differentiable ℝ φ) {L : ℝ≥0} (hφL : LipschitzWith L φ) |
| {B : ℝ} (hB : ∀ z,|φ z|≤B) : |
| (∫ z,φ (fullProbabilityFlow hF x hr hl 1 z) ∂stdGaussian (Point d))= |
| ∫ z,φ z ∂gibbs (primitivePotential F x r) := 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))) |
| let t : ℕ → ℝ := fun n => 1-(1/2:ℝ)^(n+1) |
| have ht0 (n : ℕ) : 0<t n := by |
| have hh : (1/2:ℝ)^(n+1)<1 := pow_lt_one₀ (by norm_num) (by norm_num) (by omega) |
| dsimp [t]; linarith |
| have ht1 (n : ℕ) : t n<1 := by |
| have hh : 0 < (1/2:ℝ)^(n+1) := by positivity |
| dsimp [t]; linarith |
| have ht : Tendsto t atTop (𝓝 1) := by |
| have hh := (tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num : 0≤(1/2:ℝ)) (by norm_num)).mul_const (1/2:ℝ) |
| simpa only [zero_mul,sub_zero,←pow_succ] using (tendsto_const_nhds (x:=(1:ℝ))).sub hh |
| have hleft : Tendsto (fun n => ∫ z,φ (fullProbabilityFlow hF x hr hl (t n) z) ∂ν) |
| atTop (𝓝 (∫ z,φ (fullProbabilityFlow hF x hr hl 1 z) ∂ν)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ => B) |
| (fun n => (hφ.continuous.comp (fullProbabilityFlow_lipschitz hF x hr hl ⟨(ht0 n).le,(ht1 n).le⟩).continuous).aestronglyMeasurable) |
| (integrable_const B) |
| · intro n |
| exact Filter.Eventually.of_forall (fun z => by simpa only [Function.comp_apply,Real.norm_eq_abs] using hB (fullProbabilityFlow hF x hr hl (t n) z)) |
| · exact Filter.Eventually.of_forall (fun z => (hφ.continuous.comp |
| (fullProbabilityFlow_time_continuous hF x hr hl z)).continuousAt.tendsto.comp ht) |
| have hright : Tendsto (fun n => ∫ p,φ (interpolationPath (t n) p) ∂μ.prod ν) |
| atTop (𝓝 (∫ p,φ p.1 ∂μ.prod ν)) := by |
| apply tendsto_integral_of_dominated_convergence (fun _ => B) |
| (fun n => (hφ.continuous.comp (by unfold interpolationPath; fun_prop)).aestronglyMeasurable) |
| (integrable_const B) |
| · intro n |
| exact Filter.Eventually.of_forall (fun p => by simpa only [Function.comp_apply,Real.norm_eq_abs] using hB (interpolationPath (t n) p)) |
| · filter_upwards [] with p |
| have hc : Continuous (fun t : ℝ => φ (interpolationPath t p)) := by |
| apply hφ.continuous.comp |
| unfold interpolationPath |
| fun_prop |
| simpa [Function.comp_def,interpolationPath] using hc.continuousAt.tendsto.comp ht |
| have he : (fun n => ∫ p,φ (interpolationPath (t n) p) ∂μ.prod ν)= |
| (fun n => ∫ z,φ (fullProbabilityFlow hF x hr hl (t n) z) ∂ν) := by |
| funext n |
| exact (fullProbabilityFlow_integral hF x hr hl (ht0 n) (ht1 n) hφ hφL hB).symm |
| rw [he] at hright |
| have hh := tendsto_nhds_unique hleft hright |
| simpa only [integral_fun_fst,probReal_univ,one_smul] using hh |
| |
| theorem fullProbabilityFlow_pushforward {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≤1/2) : |
| (stdGaussian (Point d)).map (fullProbabilityFlow hF x hr hl 1)=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 (by linarith))) |
| have hc := (fullProbabilityFlow_lipschitz hF x hr hl (t:=1) ⟨by norm_num,le_rfl⟩).continuous |
| apply measure_eq_of_smooth_lipschitz |
| intro φ hφ ⟨L,hL⟩ ⟨B,hB⟩ |
| rw [integral_map hc.measurable.aemeasurable hφ.continuous.aestronglyMeasurable] |
| exact fullProbabilityFlow_endpoint_integral hF x hr hl (hφ.differentiable (by norm_num)) hL hB |
| end LogConcaveSampling |
|
|
| end UpperProof |
| end |
| end |
| end |
|
|
| end OAI |
|
|
