| import Mathlib | |
| import OAI.Probability.LogConcave.Sampling.SkewCenteringFieldSmooth | |
| namespace OAI | |
| section | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling.LinearGrowthODE | |
| open Set MeasureTheory | |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] | |
| omit [MeasurableSpace E] [BorelSpace E] in | |
| lemma flow_restrict {f : E → E} (hf : ContDiff ℝ 1 f) {K B a b c : ℝ} | |
| (hK : 0≤K) (hB : 0≤B) (hg : ∀x,‖f x‖≤K*‖x‖+B) (hab : a≤b) | |
| (hc : c∈Icc a b) (x : E) : | |
| EqOn (flow hf hK hB hg hab x) (flow hf hK hB hg hc.1 x) (Icc a c) := by | |
| apply unique hf (flow_continuous hf hK hB hg hab x).continuousOn | |
| (flow_continuous hf hK hB hg hc.1 x).continuousOn | |
| · intro t ht | |
| exact (flow_deriv hf hK hB hg hab x t ⟨ht.1,ht.2.trans hc.2⟩).mono | |
| (Icc_subset_Icc le_rfl hc.2) | |
| · exact flow_deriv hf hK hB hg hc.1 x | |
| · rw [flow_initial,flow_initial] | |
| theorem stationary_at {f : E → E} (hf : ContDiff ℝ 1 f) {K B a b : ℝ} | |
| (hK : 0≤K) (hB : 0≤B) (hg : ∀x,‖f x‖≤K*‖x‖+B) (hab : a≤b) | |
| {μ : Measure E} [IsFiniteMeasure μ] (hfi : Integrable (fun x => ‖f x‖) μ) | |
| (hflux : ∀ψ : E → ℝ,ContDiff ℝ 1 ψ → HasCompactSupport ψ → | |
| (∫x,(fderiv ℝ ψ x) (f x) ∂μ)=0) {t : ℝ} (ht : t∈Icc a b) : | |
| μ.map (fun y => flow hf hK hB hg hab y t)=μ := by | |
| obtain he|hlt := ht.1.eq_or_lt | |
| · subst t | |
| simp only [flow_initial] | |
| exact Measure.map_id | |
| · have hEq : (fun y => flow hf hK hB hg hab y t)= | |
| (fun y => flow hf hK hB hg hlt.le y t) := by | |
| funext y | |
| exact flow_restrict hf hK hB hg hab ht y ⟨ht.1,le_rfl⟩ | |
| rw [hEq] | |
| exact stationary_law hf hK hB hg hlt hfi hflux | |
| end LogConcaveSampling.LinearGrowthODE | |
| end | |
| end | |
| section | |
| noncomputable section | |
| namespace LogConcaveSampling | |
| open Set Function MeasureTheory ProbabilityTheory | |
| open scoped NNReal RealInnerProductSpace | |
| theorem exists_skew_centering_flow {d : ℕ} {μ : Measure (Point d)} [IsFiniteMeasure μ] | |
| {K : Point d → Point d →L[ℝ] Point d} {m : Point d → Point d} | |
| (hK : ContDiff ℝ 1 K) (hm : ContDiff ℝ 1 m) {C : ℝ} {L : ℝ≥0} | |
| (hC : 0≤C) (hKb : ∀y,‖K y‖≤C) (hmL : LipschitzWith L m) | |
| (hμ : Integrable (fun y => ‖y‖) μ) | |
| (hstein : ∀u : Point d,∀φ : Point d → ℝ,ContDiff ℝ 1 φ → HasCompactSupport φ → | |
| (∫y,inner ℝ u (m y)*φ y ∂μ)=∫y,fderiv ℝ φ y ((K y).adjoint u) ∂μ) | |
| {s a b : ℝ} (hs : 0<s) (hab : a≤b) : | |
| ∃Ψ : (Point d × Point d) → ℝ → (Point d × Point d), | |
| (∀p,Continuous (Ψ p) ∧ Ψ p a=p ∧ ∀t∈Icc a b, | |
| HasDerivWithinAt (Ψ p) (skewCenteringField K m s (Ψ p t)) (Icc a b) t) ∧ | |
| (∀t∈Icc a b,Continuous (fun p => Ψ p t)) ∧ | |
| (∀t∈Icc a b,(μ.prod (stdGaussian (Point d))).map (fun p => Ψ p t)=μ.prod (stdGaussian (Point d))) := by | |
| let f := skewCenteringField K m s | |
| have hf : ContDiff ℝ 1 f := skewCenteringField_smooth hK hm s | |
| have hk : 0≤ s⁻¹*(C+L) := by positivity | |
| have hb : 0≤ s⁻¹*‖m 0‖ := by positivity | |
| have hg : ∀p,‖f p‖≤ s⁻¹*(C+L)*‖p‖+s⁻¹*‖m 0‖ := skewCenteringField_growth hC hKb hmL hs | |
| let Ψ := LinearGrowthODE.flow hf hk hb hg hab | |
| refine ⟨Ψ,?_,?_,?_⟩ | |
| · intro p | |
| exact ⟨LinearGrowthODE.flow_continuous hf hk hb hg hab p, | |
| LinearGrowthODE.flow_initial hf hk hb hg hab p, | |
| LinearGrowthODE.flow_deriv hf hk hb hg hab p⟩ | |
| · exact fun t ht => LinearGrowthODE.flow_continuous_initial hf hk hb hg hab t ht | |
| · have hi : Integrable (fun g : Point d => ‖g‖) (stdGaussian (Point d)) := IsGaussian.integrable_id.norm | |
| have hprod := integrable_product_norm hμ hi | |
| have hfi : Integrable (fun p => ‖f p‖) (μ.prod (stdGaussian (Point d))) := by | |
| apply ((hprod.const_mul (s⁻¹*(C+L))).add (integrable_const (s⁻¹*‖m 0‖))).mono' hf.continuous.norm.aestronglyMeasurable | |
| filter_upwards [] with p | |
| rw [norm_norm] | |
| exact hg p | |
| exact fun t ht => LinearGrowthODE.stationary_at hf hk hb hg hab hfi | |
| (fun φ hφ hφc => product_stein_flux hK.continuous hm.continuous hstein s hφ hφc) ht | |
| theorem centering_momentum_identity {d : ℕ} {K : Point d → Point d →L[ℝ] Point d} | |
| {M : Point d → Point d} (hM : Continuous M) (u : Point d) {s : ℝ} (hs : 0<s) | |
| {α : ℝ → Point d × Point d} (hα : Continuous α) | |
| (hd : ∀t∈Icc (0:ℝ) 1,HasDerivWithinAt α | |
| (skewCenteringField K (fun y => M y-u) s (α t)) (Icc (0:ℝ) 1) t) : | |
| s • (α 0).2+(∫t in (0:ℝ)..1,M (α t).1)=u+s • (α 1).2 := by | |
| let β := fun t : ℝ => s • (α t).2+t • u | |
| have hβ : Continuous β := (hα.snd.const_smul s).add (continuous_id.smul (continuous_const (y:=u))) | |
| have hdβ (t : ℝ) (ht : t∈Ioo (0:ℝ) 1) : HasDerivAt β (M (α t).1) t := by | |
| have hh := (hd t ⟨ht.1.le,ht.2.le⟩).hasDerivAt (Icc_mem_nhds ht.1 ht.2) | |
| have hG : HasDerivAt (fun t => (α t).2) (s⁻¹ • (M (α t).1-u)) t := hh.snd | |
| have he := (hG.const_smul s).add ((hasDerivAt_id t).smul_const u) | |
| convert he using 1 <;> first | rfl | simp [smul_smul, hs.ne'] | |
| have hFTC := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le (by norm_num : (0:ℝ)≤1) | |
| hβ.continuousOn hdβ ((hM.comp hα.fst).intervalIntegrable 0 1) | |
| change (∫t in (0:ℝ)..1,M (α t).1)=β 1-β 0 at hFTC | |
| rw [hFTC] | |
| simp only [β,one_smul,zero_smul,add_zero] | |
| abel | |
| end LogConcaveSampling | |
| end | |
| end | |
| end | |
| end OAI | |