| import Mathlib |
| import OAI.Probability.LogConcave.Complexity.SingleCellBasisBudget |
| import OAI.Probability.LogConcave.Sampling.CenteringVelocityJoint |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory Quadrature TensorEnergy |
| open scoped Classical BigOperators NNReal |
|
|
| structure StationaryPath {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] (μ : Measure E) (V : E → E) where |
| path : E → ℝ → E |
| continuous : ∀y,Continuous (path y) |
| initial : ∀y,path y 0=y |
| derivative : ∀y t,t∈Icc (0:ℝ) 1 → HasDerivWithinAt (path y) (V (path y t)) (Icc (0:ℝ) 1) t |
| measurable : Measurable (fun p : ℝ × E => path p.2 p.1) |
| law : ∀t∈Icc (0:ℝ) 1,μ.map (fun y => path y t)=μ |
|
|
| namespace StationaryPath |
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [MeasurableSpace E] |
| {μ : Measure E} {V : E → E} (X : StationaryPath μ V) |
|
|
| lemma measurable_slice (t : ℝ) : Measurable (fun y => X.path y t) := |
| X.measurable.comp (measurable_const.prodMk measurable_id) |
|
|
| lemma measurePreserving {t : ℝ} (ht : t∈Icc (0:ℝ) 1) : |
| MeasurePreserving (fun y => X.path y t) μ μ := ⟨X.measurable_slice t,X.law t ht⟩ |
|
|
| lemma memLp_slice {t : ℝ} (ht : t∈Icc (0:ℝ) 1) (hi : MemLp (id : E → E) 2 μ) : |
| MemLp (fun y => X.path y t) 2 μ := hi.comp_measurePreserving (X.measurePreserving ht) |
|
|
| lemma integral_eq [CompleteSpace E] (hV : Continuous V) (y : E) {t : ℝ} |
| (ht : t∈Icc (0:ℝ) 1) : (∫u in 0..t,V (X.path y u))=X.path y t-y := by |
| have he := intervalIntegral.integral_eq_sub_of_hasDerivAt_of_le ht.1 |
| (X.continuous y).continuousOn |
| (fun u hu => (X.derivative y u ⟨hu.1.le,hu.2.le.trans ht.2⟩).hasDerivAt |
| (Icc_mem_nhds hu.1 (hu.2.trans_le ht.2))) |
| ((hV.comp (X.continuous y)).intervalIntegrable 0 t) |
| simpa only [X.initial] using he |
| end StationaryPath |
|
|
| lemma centeringTrueVelocity_continuous {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T : ℝ} (hr : 0<r) (hlam : 0<lam) |
| (hl : (lam:ℝ)*r^2 ≤ 1/2) (hT0 : 0 ≤ T) (hT1 : T<1) (s : ℝ) : |
| Continuous (skewLieField (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s)) := by |
| let E := productPointEquiv d d |
| have hK := (centeringKernel_smooth hF x hr hlam hl hT0 hT1).continuous |
| have hM : Continuous (fun z => conditionalFieldMean F x r T z-primitiveExpectedField F x r) := |
| (conditionalFieldMean_lipschitz hF x hr.le hl hT0 hT1).continuous.sub continuous_const |
| have hc : Continuous (skewCenteringField (centeringKernel hF x hr hl hT0 hT1) |
| (fun z => conditionalFieldMean F x r T z-primitiveExpectedField F x r) s) := by |
| exact ((((ContinuousLinearMap.adjoint.continuous.comp hK).comp continuous_fst).clm_apply |
| continuous_snd).const_smul (-s⁻¹)).prodMk ((hM.comp continuous_fst).const_smul s⁻¹) |
| convert! E.symm.continuous.comp (hc.comp E.continuous) using 1 |
| funext y |
| change _=E.symm (skewCenteringField (centeringKernel hF x hr hl hT0 hT1) |
| (fun z => conditionalFieldMean F x r T z-primitiveExpectedField F x r) s (E y)) |
| dsimp only [E] |
| rw [←actual_centering_joint_field hF x hr hlam hl hT0 hT1 s y] |
| exact (E.symm_apply_apply _).symm |
|
|
| theorem exists_centering_stationaryPath {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T s : ℝ} (hr : 0<r) (hlam : 0<lam) |
| (hl : (lam:ℝ)*r^2 ≤ 1/2) (hT0 : 0 ≤ T) (hT1 : T<1) (hs : 0<s) : |
| Nonempty (StationaryPath (gibbs (centeringPotential F x r T)) |
| (skewLieField (centeringPotential F x r T) |
| (jointSkew (centeringKernel hF x hr hl hT0 hT1) s))) := by |
| obtain ⟨Ξ,hc,hd,hm,hlaw⟩ := exists_actual_joint_centering_flow hF x hr hlam hl hT0 hT1 hs |
| exact ⟨⟨Ξ,fun y => (hc y).1,fun y => (hc y).2,hd,hm,hlaw⟩⟩ |
|
|
| def centeringWeight (n : ℕ) (i j : Fin (n+1)) : ℝ := |
| weight (probabilityNodes n) j 0 (probabilityNodes n i) |
|
|
| def centeringRowBudget (n : ℕ) : ℝ≥0 := |
| ⟨(n+1)*singleCellBasisBudget n,by positivity [(singleCellBasisBudget_spec n).1]⟩ |
|
|
| lemma centeringWeight_bound (n : ℕ) (hn : 0<n) (i : Fin (n+1)) : |
| ∑j,|centeringWeight n i j| ≤ centeringRowBudget n := by |
| change ∑j,|centeringWeight n i j| ≤ ((n:ℝ)+1)*singleCellBasisBudget n |
| have hi := probabilityNodes_mem hn i |
| have hb := weights_abs_bound (probabilityNodes n) hi.1 hi.2 (singleCellBasisBudget_spec n).2 |
| simpa only [centeringWeight,centeringRowBudget,NNReal.coe_mk,Fintype.card_fin,Nat.cast_add,Nat.cast_one] using |
| hb.trans (mul_le_of_le_one_right (by positivity [(singleCellBasisBudget_spec n).1]) hi.2) |
|
|
| lemma centeringFullWeight_bound (n : ℕ) : |
| ∑j,|weight (probabilityNodes n) j 0 1| ≤ centeringRowBudget n := by |
| change ∑j,|weight (probabilityNodes n) j 0 1| ≤ ((n:ℝ)+1)*singleCellBasisBudget n |
| simpa only [centeringRowBudget,NNReal.coe_mk,Fintype.card_fin,Nat.cast_add,Nat.cast_one,mul_one] using |
| weights_abs_bound (probabilityNodes n) (by norm_num : (0:ℝ) ≤ 1) le_rfl (singleCellBasisBudget_spec n).2 |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end OAI |
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|
