| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.ProbabilityNodes |
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| namespace OAI |
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| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory |
| open scoped Classical BigOperators RealInnerProductSpace NNReal |
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| theorem harmonic_integral_equation {E : Type*} [NormedAddCommGroup E] |
| [NormedSpace ℝ E] [CompleteSpace E] {Y P : ℝ → E} {f : E → E} |
| (hY : Continuous Y) (hP : Continuous P) (hf : Continuous f) |
| (hderY : ∀v∈Icc (0:ℝ) 1,HasDerivWithinAt Y (P v) (Icc (0:ℝ) 1) v) |
| (hderP : ∀v∈Icc (0:ℝ) 1,HasDerivWithinAt P (-Y v-f (Y v)) (Icc (0:ℝ) 1) v) |
| {t : ℝ} (ht : t∈Icc (0:ℝ) 1) : |
| Y t=Real.cos t • Y 0+Real.sin t • P 0- |
| ∫v in 0..t,Real.sin (t-v) • f (Y v) := by |
| let g : ℝ → E := fun v => Real.cos (t-v) • Y v+Real.sin (t-v) • P v |
| have hg : Continuous g := by unfold g; fun_prop |
| have hfg : Continuous (fun v => -Real.sin (t-v) • f (Y v)) := by fun_prop |
| have he := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le ht.1 hg.continuousOn |
| (f':=fun v => -Real.sin (t-v) • f (Y v)) |
| (fun v hv => by |
| have hm : v∈Icc (0:ℝ) 1 := ⟨hv.1.le,hv.2.le.trans ht.2⟩ |
| have hn := Icc_mem_nhds hv.1 (hv.2.trans_le ht.2) |
| have hDY := (hderY v hm).hasDerivAt hn |
| have hDP := (hderP v hm).hasDerivAt hn |
| have hcos : HasDerivAt (fun u => Real.cos (t-u)) (Real.sin (t-v)) v := by |
| convert ((hasDerivAt_const v t).sub (hasDerivAt_id v)).cos using 1 <;> simp |
| have hsin : HasDerivAt (fun u => Real.sin (t-u)) (-Real.cos (t-v)) v := by |
| convert ((hasDerivAt_const v t).sub (hasDerivAt_id v)).sin using 1 <;> simp |
| convert (hcos.smul hDY).add (hsin.smul hDP) using 1 |
| module) |
| (hfg.intervalIntegrable 0 t) |
| change (∫v in 0..t,-Real.sin (t-v) • f (Y v))=g t-g 0 at he |
| simp only [g,sub_self,Real.cos_zero,Real.sin_zero,one_smul,zero_smul,add_zero,sub_zero] at he |
| simp_rw [neg_smul] at he |
| rw [intervalIntegral.integral_neg] at he |
| exact (sub_eq_iff_eq_add.mp he.symm).trans (by abel) |
|
|
| theorem conditional_harmonic_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) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) |
| (hc : ∀y,Continuous (Ξ y)) |
| (hder : ∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (productPotential (interpolationPotential F x r T) (fun z : Point d => ‖z‖^2/2)) (harmonicSkew d) (Ξ y v)) (Icc (0:ℝ) 1) v) |
| (y : Point (d+d)) {t : ℝ} (ht : t∈Icc (0:ℝ) 1) : |
| let W := fun v => (productPointEquiv d d (Ξ y v)).1 |
| let P := fun v => (productPointEquiv d d (Ξ y v)).2 |
| W t=Real.cos t • W 0+Real.sin t • P 0- |
| (r*T) • (∫v in 0..t,Real.sin (t-v) • conditionalFieldMean F x r T (W v)) := by |
| intro W P |
| have hH : Differentiable ℝ (interpolationPotential F x r T) := |
| (interpolationPotential_smooth hF x hr hl hT0 hT1).differentiable (by norm_cast) |
| have hD (v : ℝ) (hv : v∈Icc (0:ℝ) 1) : |
| HasDerivWithinAt (fun s => productPointEquiv d d (Ξ y s)) |
| (P v,-W v-(r*T) • conditionalFieldMean F x r T (W v)) (Icc (0:ℝ) 1) v := by |
| have h := (productPointEquiv d d).toContinuousLinearMap.hasFDerivAt.comp_hasDerivWithinAt v (hder y v hv) |
| change HasDerivWithinAt (fun s => productPointEquiv d d (Ξ y s)) |
| (productPointEquiv d d (skewLieField |
| (productPotential (interpolationPotential F x r T) (fun z : Point d => ‖z‖^2/2)) |
| (harmonicSkew d) (Ξ y v))) (Icc (0:ℝ) 1) v at h |
| rw [harmonicSkew_field hH] at h |
| rw [interpolationPotential_gradient hF x hr hl hT0 hT1] at h |
| simpa only [W,P,neg_add_rev,sub_eq_add_neg,add_comm] using h |
| have he := harmonic_integral_equation ((productPointEquiv d d).continuous.comp (hc y)).fst |
| ((productPointEquiv d d).continuous.comp (hc y)).snd |
| ((conditionalFieldMean_lipschitz hF x hr hl hT0 hT1).continuous.const_smul (r*T)) |
| (fun v hv => (hD v hv).fst) (fun v hv => (hD v hv).snd) ht |
| change W t=Real.cos t • W 0+Real.sin t • P 0- |
| ∫v in 0..t,Real.sin (t-v) • ((r*T) • conditionalFieldMean F x r T (W v)) at he |
| simp_rw [smul_comm (Real.sin (t-_)) (r*T)] at he |
| rw [intervalIntegral.integral_smul] at he |
| exact he |
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| end LogConcaveSampling |
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| end |
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| end |
|
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| section |
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| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory TensorEnergy Quadrature |
| open scoped Classical BigOperators NNReal RealInnerProductSpace |
|
|
| local instance harmonicMeanDefectDecidableEqUnit : DecidableEq Unit := Classical.decEq _ |
|
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| def harmonicMeanDefect {d : ℕ} (F : Point d → ℝ) (x : Point d) (r T z : ℝ) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) (n : ℕ) (i : Fin (n+1)) (y : Point (d+d)) : Point d := |
| let M := fun t => conditionalFieldMean F x r T (productPointEquiv d d (Ξ y (z*t))).1 |
| z • (∫t in 0..probabilityNodes n i,Real.sin (z*(probabilityNodes n i-t)) • M t)- |
| ∑j,harmonicWeight n z i j • M (probabilityNodes n j) |
|
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| lemma interpolation_error_measurable {I E Ω : Type*} [Fintype I] [DecidableEq I] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace E] [BorelSpace E] |
| [MeasurableSpace Ω] [SecondCountableTopology E] (u : I → ℝ) (f : ℝ × Ω → E) |
| (hf : Measurable f) : Measurable (fun p : ℝ × Ω => f p- |
| interpolation u (fun j => f (u j,p.2)) p.1) := by |
| apply hf.sub |
| apply Finset.measurable_sum |
| intro j hj |
| exact ((basis_continuous u j).measurable.comp measurable_fst).smul |
| (hf.comp (measurable_const.prodMk measurable_snd)) |
|
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| lemma harmonicMean_measurable {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T z : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) |
| (hm : Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1)) : |
| Measurable (fun p : ℝ × Point (d+d) => |
| conditionalFieldMean F x r T (productPointEquiv d d (Ξ p.2 (z*p.1))).1) := by |
| have h1 : Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 (z*p.1)) := |
| hm.comp ((measurable_const.mul measurable_fst).prodMk measurable_snd) |
| have h2 : Measurable (fun p : ℝ × Point (d+d) => (productPointEquiv d d (Ξ p.2 (z*p.1))).1) := |
| ((productPointEquiv d d).continuous.measurable.comp h1).fst |
| exact (conditionalFieldMean_lipschitz hF x hr hl hT0 hT1).continuous.measurable.comp h2 |
|
|
| theorem harmonic_mean_quadrature_rms (n : ℕ) (hn : 0<n) : |
| ∃C : ℝ,1≤C ∧ ∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0}, |
| ∀_ : Primitive F lam,∀(x : Point d) {r R T : ℝ}, |
| 0<r → 0<lam → (lam:ℝ)*r^2≤1/2 → 0<R → R^2≤1-T^2 → 0≤T → T<1 → |
| ∀Ξ : Point (d+d) → ℝ → Point (d+d), |
| (∀y,Continuous (Ξ y)) → |
| (∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r T) (harmonicSkew d) (Ξ y v)) (Icc (0:ℝ) 1) v) → |
| Measurable (fun p : ℝ × Point (d+d) => Ξ p.2 p.1) → |
| (∀v∈Icc (0:ℝ) 1,(gibbs (centeringPotential F x r T)).map (fun y => Ξ y v)= |
| gibbs (centeringPotential F x r T)) → |
| ∀z : ℝ,0≤z → z≤1 → ∀i : Fin (n+1), |
| Integrable (fun y => ‖harmonicMeanDefect F x r T z Ξ n i y‖^2) |
| (gibbs (centeringPotential F x r T)) ∧ |
| (∫y,‖harmonicMeanDefect F x r T z Ξ n i y‖^2 ∂gibbs (centeringPotential F x r T))≤ |
| 4*C^2*(z^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)*(R⁻¹)^(4*(n+1))*harmonicMeanBudget (n+1)) := by |
| obtain ⟨C,hC,hC'⟩ := harmonic_point_interpolation_rms n hn |
| refine ⟨C,hC,?_⟩ |
| intro d F lam hF x r R T hr hlam hl hR hRT hT0 hT1 Ξ hc hder hm hlaw z hz hz1 i |
| have hH := productPotential_polySmooth |
| (interpolationPotential_polySmooth hF x hr hlam hl hT0 hT1) (gaussianPotential_polySmooth d) |
| have ht := productPotential_lowerTail |
| (interpolationPotential_lowerTail hF x hr.le (by linarith) hT0 hT1) (gaussianPotential_lowerTail d) |
| let : IsProbabilityMeasure (gibbs (centeringPotential F x r T)) := |
| probability_gibbs_of_gaussianTail hH.smooth.continuous ht |
| let M := fun t y => conditionalFieldMean F x r T (productPointEquiv d d (Ξ y (z*t))).1 |
| let B := 4*C^2*(z^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)*(R⁻¹)^(4*(n+1))*harmonicMeanBudget (n+1)) |
| have hB0 : 0≤B := by dsimp [B]; positivity [harmonicMeanBudget_nonneg (n+1)] |
| have hMcont := (conditionalFieldMean_lipschitz hF x hr.le hl hT0 hT1).continuous |
| have hMc (y : Point (d+d)) : Continuous (fun t => M t y) := |
| hMcont.comp (((productPointEquiv d d).continuous.comp |
| ((hc y).comp (continuous_const.mul continuous_id))).fst) |
| have hMm := harmonicMean_measurable (z:=z) hF x hr.le hl hT0 hT1 Ξ hm |
| have hErr := interpolation_error_measurable (probabilityNodes n) _ hMm |
| have hi := probabilityNodes_mem hn i |
| have hmem (t : ℝ) (ht : t∈Ioc 0 (probabilityNodes n i)) : t∈Icc (0:ℝ) 1 := |
| ⟨ht.1.le,ht.2.trans hi.2⟩ |
| have he := harmonic_quadrature_error_rms n z i M hi.1 hB0 hMc hErr.aestronglyMeasurable |
| (fun t ht => (hC' hF x hr hlam hl hR hRT hT0 hT1 Ξ hder hm hlaw z hz hz1 t (hmem t ht)).1) |
| (fun t ht => (hC' hF x hr hlam hl hR hRT hT0 hT1 Ξ hder hm hlaw z hz hz1 t (hmem t ht)).2) |
| refine ⟨he.1,he.2.trans ?_⟩ |
| have hz2 : z^2≤1 := pow_le_one₀ hz hz1 |
| have hi2 : (probabilityNodes n i)^2≤1 := pow_le_one₀ hi.1 hi.2 |
| exact (mul_le_mul_of_nonneg_right ((mul_le_of_le_one_left (sq_nonneg _) hz2).trans hi2) hB0).trans_eq (one_mul _) |
|
|
| lemma harmonicPicard_exact_defect {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T z : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0≤T) (hT1 : T<1) |
| (Ξ : Point (d+d) → ℝ → Point (d+d)) (hc : ∀y,Continuous (Ξ y)) |
| (hinit : ∀y,Ξ y 0=y) |
| (hder : ∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r T) (harmonicSkew d) (Ξ y v)) (Icc (0:ℝ) 1) v) |
| (hz : 0≤z) (hz1 : z≤1) (n : ℕ) (hn : 0<n) (i : Fin (n+1)) (y : Point (d+d)) : |
| FinitePicard.step (harmonicWeight n z) (fun _ => harmonicMeanVelocity F x r T) |
| (harmonicFree n z (productPointEquiv d d y)) |
| (fun j => (productPointEquiv d d (Ξ y (z*probabilityNodes n j))).1) i- |
| (productPointEquiv d d (Ξ y (z*probabilityNodes n i))).1= |
| (r*T) • harmonicMeanDefect F x r T z Ξ n i y := by |
| have hi := probabilityNodes_mem hn i |
| have hzi : z*probabilityNodes n i∈Icc (0:ℝ) 1 := |
| ⟨mul_nonneg hz hi.1,(mul_le_of_le_one_right hz hi.2).trans hz1⟩ |
| have he := conditional_harmonic_integral hF x hr hl hT0 hT1 Ξ hc hder y hzi |
| dsimp only at he |
| rw [hinit] at he |
| have hchange := intervalIntegral.smul_integral_comp_mul_left |
| (fun v => Real.sin (z*probabilityNodes n i-v) • |
| conditionalFieldMean F x r T (productPointEquiv d d (Ξ y v)).1) |
| (a:=0) (b:=probabilityNodes n i) z |
| simp only [mul_zero] at hchange |
| simp_rw [←mul_sub] at hchange |
| rw [←hchange] at he |
| change (harmonicFree n z (productPointEquiv d d y) i+ |
| ∑j,harmonicWeight n z i j • harmonicMeanVelocity F x r T |
| (productPointEquiv d d (Ξ y (z*probabilityNodes n j))).1)-_= _ |
| have hvel (u : Point d) : harmonicMeanVelocity F x r T u=-(r*T) • conditionalFieldMean F x r T u := by |
| simp only [harmonicMeanVelocity,probabilityMeanVelocity,smul_smul,mul_neg,mul_comm] |
| simp_rw [hvel,smul_comm (harmonicWeight n z i _) (-(r*T))] |
| rw [←Finset.smul_sum,neg_smul,he] |
| dsimp only [harmonicFree,harmonicMeanDefect] |
| rw [smul_sub] |
| abel |
| end LogConcaveSampling |
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| end |
|
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| end |
|
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| end |
|
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| end OAI |
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|
