| import Mathlib |
| import OAI.Probability.LogConcave.Sampling.ProbabilityNodes |
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| namespace OAI |
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| section |
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory Quadrature |
| open scoped Classical BigOperators |
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|
| def singleCellBasisBudget (n : ℕ) : ℝ := |
| Classical.choose (exists_basis_budget (probabilityNodes n)) |
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| lemma singleCellBasisBudget_spec (n : ℕ) : |
| 1 ≤ singleCellBasisBudget n ∧ |
| ∀v∈Icc (0:ℝ) 1,∑i,|basis (probabilityNodes n) i v| ≤ singleCellBasisBudget n := |
| Classical.choose_spec (exists_basis_budget (probabilityNodes n)) |
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| variable {Ω E : Type*} [MeasurableSpace Ω] [NormedAddCommGroup E] |
| [NormedSpace ℝ E] [CompleteSpace E] [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E] {μ : Measure Ω} |
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| omit [CompleteSpace E] in |
| theorem singleCell_interpolation_rms (n : ℕ) (hn : 0<n) (C : ℝ) (hC : 1 ≤ C) |
| (hC' : ∀v∈Icc (0:ℝ) 1,∑i,|basis (probabilityNodes n) i v| ≤ C) : |
| ∀J : ℕ → ℝ → Ω → E, (∀k t,Measurable (J k t)) → |
| ∀B : ℝ,0 ≤ B → |
| (∀t∈Icc (0:ℝ) 1, |
| Integrable (fun y => ‖J 0 t y-chainTaylor (fun k v => J k v y) n 0 t‖^2) μ ∧ |
| (∫y,‖J 0 t y-chainTaylor (fun k v => J k v y) n 0 t‖^2 ∂μ) ≤ |
| (t^(n+1)/(n.factorial:ℝ))^2*B) → |
| ∀v∈Icc (0:ℝ) 1, |
| Integrable (fun y => ‖J 0 v y-interpolation (probabilityNodes n) |
| (fun i => J 0 (probabilityNodes n i) y) v‖^2) μ ∧ |
| (∫y,‖J 0 v y-interpolation (probabilityNodes n) |
| (fun i => J 0 (probabilityNodes n i) y) v‖^2 ∂μ) ≤ 4*C^2*B := by |
| intro J hJ B hB htay v hv |
| have hs (t : ℝ) (ht : t∈insert v (range (probabilityNodes n))) : t∈Icc (0:ℝ) 1 := by |
| rcases ht with rfl|⟨i,rfl⟩ |
| · exact hv |
| · exact probabilityNodes_mem hn i |
| have h := cell_chain_error_rms (probabilityNodes n) (probabilityNodes_injective hn) n |
| (by simp) J 0 1 v hB hC (hC' v hv) |
| (fun t _ => by |
| simp only [zero_add,one_mul] |
| apply Measurable.aestronglyMeasurable |
| exact (hJ 0 t).sub (chainTaylor_measurable J hJ n 0 t)) |
| (fun t ht => by simpa only [zero_add,one_mul] using (htay t (hs t ht)).1) |
| (fun t ht => by |
| simp only [zero_add,one_mul] |
| apply (htay t (hs t ht)).2.trans |
| apply mul_le_of_le_one_left hB |
| apply pow_le_one₀ (by positivity [(hs t ht).1]) |
| apply (div_le_iff₀ (by positivity : 0<(n.factorial:ℝ))).mpr |
| simp only [one_mul] |
| exact (pow_le_one₀ (hs t ht).1 (hs t ht).2).trans (by exact_mod_cast Nat.succ_le_of_lt (Nat.factorial_pos n))) |
| simpa only [zero_add,one_mul] using h |
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| theorem singleCell_quadrature_rms [SFinite μ] (n : ℕ) (f : ℝ → Ω → E) |
| (hm : Measurable (fun p : ℝ × Ω => f p.1 p.2)) |
| (hc : ∀y,ContinuousOn (fun t => f t y) (Icc (0:ℝ) 1)) |
| {B : ℝ} (hB : 0 ≤ B) |
| (hi : ∀t∈Icc (0:ℝ) 1,Integrable (fun y => ‖f t y-interpolation (probabilityNodes n) |
| (fun i => f (probabilityNodes n i) y) t‖^2) μ) |
| (hb : ∀t∈Icc (0:ℝ) 1,(∫y,‖f t y-interpolation (probabilityNodes n) |
| (fun i => f (probabilityNodes n i) y) t‖^2 ∂μ) ≤ B) |
| {θ : ℝ} (hθ : θ∈Icc (0:ℝ) 1) : |
| Integrable (fun y => ‖(∫t in 0..θ,f t y)-∑i,weight (probabilityNodes n) i 0 θ • |
| f (probabilityNodes n i) y‖^2) μ ∧ |
| (∫y,‖(∫t in 0..θ,f t y)-∑i,weight (probabilityNodes n) i 0 θ • |
| f (probabilityNodes n i) y‖^2 ∂μ) ≤ B := by |
| have hh := cellQuadrature_error_rms (probabilityNodes n) f 1 hθ.1 |
| (fun y => ((hc y).mono (by |
| intro t ht; exact ⟨ht.1,ht.2.trans hθ.2⟩)).intervalIntegrable_of_Icc hθ.1) |
| (by |
| apply Measurable.aestronglyMeasurable |
| apply hm.sub |
| unfold interpolation |
| exact Finset.measurable_sum _ (fun i _ => |
| ((basis_continuous (probabilityNodes n) i).measurable.comp measurable_fst).smul |
| (hm.comp (measurable_const.prodMk measurable_snd)))) |
| (fun t ht => hi t ⟨ht.1.le,ht.2.trans hθ.2⟩) |
| (fun t ht => hb t ⟨ht.1.le,ht.2.trans hθ.2⟩) |
| simp only [one_smul,cellQuadrature,one_mul,one_pow] at hh |
| exact ⟨hh.1,hh.2.trans (mul_le_of_le_one_left hB (pow_le_one₀ hθ.1 hθ.2))⟩ |
| end LogConcaveSampling |
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| end |
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| end |
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| end OAI |
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