| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.NodePolynomial |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Set MeasureTheory |
| open scoped BigOperators |
|
|
| variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| omit [CompleteSpace E] in |
| lemma chainTaylor_hasDerivAt (J : ℕ → ℝ → E) (n : ℕ) (a x : ℝ) : |
| HasDerivAt (chainTaylor J (n+1) a) |
| (chainTaylor (fun k => J (k+1)) n a x) x := by |
| have he (t : ℝ) : chainTaylor J (n+1) a t= |
| (∑k∈Finset.range (n+1), (((k+1).factorial:ℝ)⁻¹*(t-a)^(k+1)) • J (k+1) a)+J 0 a := by |
| simp only [chainTaylor,Finset.sum_range_succ',Nat.factorial_zero,Nat.cast_one, |
| inv_one,pow_zero,mul_one,one_smul] |
| rw [funext he] |
| have h (k : ℕ) : HasDerivAt |
| (fun t : ℝ => (((k+1).factorial:ℝ)⁻¹*(t-a)^(k+1)) • J (k+1) a) |
| (((k.factorial:ℝ)⁻¹*(x-a)^k) • J (k+1) a) x := by |
| have hh := ((((hasDerivAt_id x).sub_const a).pow (k+1)).const_mul |
| (((k+1).factorial:ℝ)⁻¹)).smul_const (J (k+1) a) |
| convert hh using 1 |
| congr 1 |
| rw [Nat.factorial_succ,Nat.cast_mul,Nat.cast_add,Nat.cast_one] |
| have hk : (k:ℝ)+1≠0 := by positivity |
| have hkf : (k.factorial:ℝ)≠0 := by positivity |
| simp only [Nat.add_sub_cancel,mul_one] |
| field_simp |
| rfl |
| exact (HasDerivAt.fun_sum (fun k _ => h k)).add_const (J 0 a) |
|
|
| omit [CompleteSpace E] in |
| lemma chainTaylor_deriv (J : ℕ → ℝ → E) (n : ℕ) (a x : ℝ) : |
| deriv (chainTaylor J (n+1) a) x=chainTaylor (fun k => J (k+1)) n a x := |
| (chainTaylor_hasDerivAt J n a x).deriv |
|
|
| omit [CompleteSpace E] in |
| lemma chainTaylor_second_deriv (J : ℕ → ℝ → E) (n : ℕ) (a x : ℝ) : |
| deriv (deriv (chainTaylor J (n+2) a)) x=chainTaylor (fun k => J (k+2)) n a x := by |
| rw [show deriv (chainTaylor J (n+2) a)=chainTaylor (fun k => J (k+1)) (n+1) a from |
| funext (chainTaylor_deriv J (n+1) a)] |
| exact chainTaylor_deriv (fun k => J (k+1)) n a x |
|
|
| omit [CompleteSpace E] in |
| lemma chainTaylor_joint_continuous {X : Type*} [TopologicalSpace X] |
| (J : ℕ → ℝ → X → E) (a : ℝ) (hJ : ∀k,Continuous (J k a)) (n : ℕ) : |
| Continuous (fun p : ℝ × X => chainTaylor (fun k t => J k t p.2) n a p.1) := by |
| unfold chainTaylor |
| exact continuous_finsetSum _ (fun k _ => |
| (continuous_const.mul ((continuous_fst.sub continuous_const).pow k)).smul ((hJ k).comp continuous_snd)) |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Set MeasureTheory Polynomial |
| open scoped BigOperators |
|
|
| variable {I E : Type*} [Fintype I] [DecidableEq I] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| def basisDerivative (u : I → ℝ) (i : I) (t : ℝ) : ℝ := |
| (Lagrange.basis Finset.univ u i).derivative.eval t |
|
|
| def derivativeInterpolation (u : I → ℝ) (v : I → E) (t : ℝ) : E := |
| ∑i,basisDerivative u i t • v i |
|
|
| lemma basisDerivative_continuous (u : I → ℝ) (i : I) : Continuous (basisDerivative u i) := |
| Polynomial.continuous _ |
|
|
| omit [CompleteSpace E] in |
| lemma interpolation_hasDerivAt (u : I → ℝ) (v : I → E) (t : ℝ) : |
| HasDerivAt (interpolation u v) (derivativeInterpolation u v t) t := by |
| exact HasDerivAt.fun_sum (fun i _ => |
| ((Lagrange.basis Finset.univ u i).hasDerivAt t).smul_const (v i)) |
|
|
| omit [CompleteSpace E] in |
| lemma derivativeInterpolation_chainTaylor (u : I → ℝ) (hu : Function.Injective u) |
| (n : ℕ) (hn : n+2≤Fintype.card I) (J : ℕ → ℝ → E) (a ℓ t : ℝ) : |
| derivativeInterpolation u (fun i => chainTaylor J (n+1) a (a+ℓ*u i)) t= |
| ℓ • chainTaylor (fun k => J (k+1)) n a (a+ℓ*t) := by |
| have he : interpolation u (fun i => chainTaylor J (n+1) a (a+ℓ*u i))= |
| fun t => chainTaylor J (n+1) a (a+ℓ*t) := |
| funext (reproduces_cell_chainTaylor u hu (n+1) hn J a ℓ) |
| have h := interpolation_hasDerivAt u (fun i => chainTaylor J (n+1) a (a+ℓ*u i)) t |
| rw [he] at h |
| have ht := ((chainTaylor_hasDerivAt J n a (a+ℓ*t)).scomp t |
| (((hasDerivAt_id t).const_mul ℓ).const_add a)) |
| have ht' : HasDerivAt (fun t => chainTaylor J (n+1) a (a+ℓ*t)) |
| (ℓ • chainTaylor (fun k => J (k+1)) n a (a+ℓ*t)) t := by |
| simpa only [Function.comp_def,mul_one,id_eq] using ht |
| exact h.unique ht' |
|
|
| omit [CompleteSpace E] in |
| lemma derivativeInterpolation_sub (u : I → ℝ) (v w : I → E) (t : ℝ) : |
| derivativeInterpolation u (fun i => v i-w i) t= |
| derivativeInterpolation u v t-derivativeInterpolation u w t := by |
| simp only [derivativeInterpolation,smul_sub,Finset.sum_sub_distrib] |
|
|
| omit [CompleteSpace E] in |
| lemma derivative_cell_chain_error_eq (u : I → ℝ) (hu : Function.Injective u) |
| (n : ℕ) (hn : n+2≤Fintype.card I) (J : ℕ → ℝ → E) |
| (a t : ℝ) {ℓ : ℝ} (hℓ : ℓ≠0) : |
| J 1 (a+ℓ*t)-ℓ⁻¹ • derivativeInterpolation u (fun i => J 0 (a+ℓ*u i)) t= |
| (J 1 (a+ℓ*t)-chainTaylor (fun k => J (k+1)) n a (a+ℓ*t))- |
| ℓ⁻¹ • derivativeInterpolation u |
| (fun i => J 0 (a+ℓ*u i)-chainTaylor J (n+1) a (a+ℓ*u i)) t := by |
| rw [derivativeInterpolation_sub,derivativeInterpolation_chainTaylor u hu n hn, |
| smul_sub,smul_smul,inv_mul_cancel₀ hℓ,one_smul] |
| abel |
|
|
| lemma exists_basisDerivative_budget (u : I → ℝ) : |
| ∃C : ℝ,1≤C ∧ ∀t∈Icc (0:ℝ) 1,∑i,|basisDerivative u i t|≤C := by |
| have hc : Continuous (fun t => ∑i,|basisDerivative u i t|) := |
| continuous_finsetSum _ (fun i _ => (basisDerivative_continuous u i).abs) |
| obtain ⟨C,hC⟩ := isCompact_Icc.exists_bound_of_continuousOn hc.continuousOn |
| refine ⟨max C 1,le_max_right _ _,fun t ht => ?_⟩ |
| have hn : 0≤∑i,|basisDerivative u i t| := Finset.sum_nonneg (fun _ _ => abs_nonneg _) |
| have hh : (∑i,|basisDerivative u i t|)≤C := by |
| simpa only [Real.norm_eq_abs,abs_of_nonneg hn] using hC t ht |
| exact hh.trans (le_max_left _ _) |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory Filter |
| open scoped BigOperators |
|
|
| namespace RMSIntegral |
| variable {Ω E : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| [NormedAddCommGroup E] [NormedSpace ℝ E] |
| lemma smul_sq_bound (c : ℝ) {f : Ω → E} |
| (hi : Integrable (fun z => ‖f z‖^2) μ) {B : ℝ} |
| (hB : (∫z,‖f z‖^2 ∂μ)≤B) : |
| Integrable (fun z => ‖c • f z‖^2) μ ∧ |
| (∫z,‖c • f z‖^2 ∂μ)≤c^2*B := by |
| simp only [norm_smul,Real.norm_eq_abs,mul_pow,sq_abs] |
| exact ⟨hi.const_mul _,by rw [integral_const_mul]; exact mul_le_mul_of_nonneg_left hB (sq_nonneg c)⟩ |
| end RMSIntegral |
|
|
| namespace Quadrature |
| variable {Ω E : Type*} [TopologicalSpace Ω] [MeasurableSpace Ω] [BorelSpace Ω] [SecondCountableTopology Ω] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| {μ : Measure Ω} [SFinite μ] |
|
|
| lemma chainTaylor_cell_rms (J : ℕ → ℝ × Ω → E) |
| (hJc : ∀k,Continuous (J k)) |
| (hJ : ∀k t z,HasDerivAt (fun t => J k (t,z)) (J (k+1) (t,z)) t) |
| (n : ℕ) {a x b B : ℝ} (hax : a≤x) (hxb : x≤b) (hB0 : 0≤B) |
| (hi : ∀t∈Icc a b,Integrable (fun z => ‖J (n+1) (t,z)‖^2) μ) |
| (hB : ∀t∈Icc a b,(∫z,‖J (n+1) (t,z)‖^2 ∂μ)≤B) : |
| Integrable (fun z => ‖J 0 (x,z)-chainTaylor (fun k t => J k (t,z)) n a x‖^2) μ ∧ |
| (∫z,‖J 0 (x,z)-chainTaylor (fun k t => J k (t,z)) n a x‖^2 ∂μ)≤ |
| ((x-a)^(n+1))^2*B := by |
| have hh := chainTaylor_remainder_rms (J:=fun k t z => J k (t,z)) (s:=univ) (n:=n) |
| hax hB0 (subset_univ _) (fun z k t _ => (hJ k t z).hasDerivWithinAt) |
| (hJc (n+1)).aestronglyMeasurable |
| (fun t ht => hi t ⟨ht.1.le,ht.2.trans hxb⟩) |
| (fun t ht => hB t ⟨ht.1.le,ht.2.trans hxb⟩) |
| refine ⟨hh.1,hh.2.trans ?_⟩ |
| apply mul_le_mul_of_nonneg_right _ hB0 |
| apply pow_le_pow_left₀ (div_nonneg (pow_nonneg (sub_nonneg.mpr hax) _) (Nat.cast_nonneg _)) |
| apply div_le_self (pow_nonneg (sub_nonneg.mpr hax) _) |
| exact_mod_cast Nat.factorial_pos n |
|
|
| lemma chainTaylor_regularized_derivative_rms (J : ℕ → ℝ × Ω → E) |
| (hJc : ∀k,Continuous (J k)) |
| (hJ : ∀k t z,HasDerivAt (fun t => J k (t,z)) (J (k+1) (t,z)) t) |
| (n : ℕ) {ℓ z B : ℝ} (hℓ : 0<ℓ) (hz : z∈Icc (0:ℝ) 1) (hB0 : 0≤B) |
| (hi : ∀t∈Icc (0:ℝ) ℓ,Integrable (fun y => ‖J (n+2) (t,y)‖^2) μ) |
| (hB : ∀t∈Icc (0:ℝ) ℓ,(∫y,‖J (n+2) (t,y)‖^2 ∂μ)≤B) : |
| let f := fun y => z⁻¹ • (J 1 (ℓ*z,y)- |
| chainTaylor (fun k t => J (k+1) (t,y)) n 0 (ℓ*z)) |
| Integrable (fun y => ‖f y‖^2) μ ∧ |
| (∫y,‖f y‖^2 ∂μ)≤(ℓ^(n+1))^2*B := by |
| dsimp only |
| have h0 : 0≤ℓ*z := mul_nonneg hℓ.le hz.1 |
| have h1 : ℓ*z≤ℓ := by nlinarith [hz.2] |
| have hh := chainTaylor_cell_rms (fun k => J (k+1)) (fun k => hJc (k+1)) |
| (fun k t y => hJ (k+1) t y) n h0 h1 hB0 hi hB |
| simp only [Nat.zero_add,sub_zero] at hh |
| have hv := RMSIntegral.smul_sq_bound z⁻¹ hh.1 hh.2 |
| refine ⟨hv.1,hv.2.trans ?_⟩ |
| rw [←mul_assoc] |
| apply mul_le_mul_of_nonneg_right _ hB0 |
| by_cases hzne : z=0 |
| · simp only [hzne,inv_zero,zero_pow (by norm_num : (2:ℕ) ≠ 0),zero_mul] |
| exact sq_nonneg _ |
| · have he : z⁻¹^2*((ℓ*z)^(n+1))^2=(ℓ^(n+1))^2*(z^n)^2 := by |
| rw [mul_pow,pow_succ z] |
| field_simp |
| rw [he] |
| apply mul_le_of_le_one_right (sq_nonneg _) |
| exact pow_le_one₀ (pow_nonneg hz.1 _) (pow_le_one₀ hz.1 hz.2) |
| end Quadrature |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Set MeasureTheory Filter |
| open scoped BigOperators |
|
|
| variable {I Ω E : Type*} [Fintype I] [DecidableEq I] |
| [TopologicalSpace Ω] [MeasurableSpace Ω] [BorelSpace Ω] [SecondCountableTopology Ω] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| {μ : Measure Ω} [SFinite μ] |
|
|
| theorem derivative_cell_error_rms (u : I → ℝ) (hu : Function.Injective u) |
| (hun : ∀i,u i∈Icc (0:ℝ) 1) (n : ℕ) (hcard : n+2≤Fintype.card I) |
| (J : ℕ → ℝ × Ω → E) (hJc : ∀k,Continuous (J k)) |
| (hJ : ∀k t z,HasDerivAt (fun t => J k (t,z)) (J (k+1) (t,z)) t) |
| {a ℓ t B : ℝ} (hℓ : 0<ℓ) (ht : t∈Icc (0:ℝ) 1) (hB0 : 0≤B) |
| (hi : ∀s∈Icc a (a+ℓ),Integrable (fun z => ‖J (n+2) (s,z)‖^2) μ) |
| (hB : ∀s∈Icc a (a+ℓ),(∫z,‖J (n+2) (s,z)‖^2 ∂μ)≤B) : |
| let err := fun z => J 1 (a+ℓ*t,z)-ℓ⁻¹ • |
| derivativeInterpolation u (fun i => J 0 (a+ℓ*u i,z)) t |
| Integrable (fun z => ‖err z‖^2) μ ∧ |
| (∫z,‖err z‖^2 ∂μ)≤ |
| 2*(1+(∑i,|basisDerivative u i t|)^2)*(ℓ^(n+1))^2*B := by |
| dsimp only |
| let P := fun (t : ℝ) (z : Ω) => J 0 (t,z)-chainTaylor (fun k v => J k (v,z)) (n+1) a t |
| let R := fun z => J 1 (a+ℓ*t,z)-chainTaylor (fun k v => J (k+1) (v,z)) n a (a+ℓ*t) |
| have hpoint (v : ℝ) (hv : v∈Icc (0:ℝ) 1) : |
| a≤a+ℓ*v ∧ a+ℓ*v≤a+ℓ := by |
| constructor <;> nlinarith [hv.1,hv.2] |
| have hR := chainTaylor_cell_rms (fun k => J (k+1)) (fun k => hJc (k+1)) |
| (fun k s z => hJ (k+1) s z) n (hpoint t ht).1 (hpoint t ht).2 hB0 hi hB |
| have hRb : (∫z,‖R z‖^2 ∂μ)≤(ℓ^(n+1))^2*B := by |
| refine hR.2.trans ?_ |
| apply mul_le_mul_of_nonneg_right _ hB0 |
| apply pow_le_pow_left₀ (pow_nonneg (sub_nonneg.mpr (hpoint t ht).1) _) |
| exact pow_le_pow_left₀ (sub_nonneg.mpr (hpoint t ht).1) (by linarith [(hpoint t ht).2]) _ |
| have hnode (i : I) : Integrable (fun z => ‖P (a+ℓ*u i) z‖^2) μ ∧ |
| (∫z,‖P (a+ℓ*u i) z‖^2 ∂μ)≤(ℓ^(n+2))^2*B := by |
| have hh := chainTaylor_cell_rms J hJc hJ (n+1) |
| (hpoint (u i) (hun i)).1 (hpoint (u i) (hun i)).2 hB0 hi hB |
| refine ⟨hh.1,hh.2.trans ?_⟩ |
| apply mul_le_mul_of_nonneg_right _ hB0 |
| apply pow_le_pow_left₀ (pow_nonneg (sub_nonneg.mpr (hpoint (u i) (hun i)).1) _) |
| exact pow_le_pow_left₀ (sub_nonneg.mpr (hpoint (u i) (hun i)).1) |
| (by linarith [(hpoint (u i) (hun i)).2]) _ |
| have hPc : Continuous (fun p : ℝ × Ω => P p.1 p.2) := |
| (hJc 0).sub (chainTaylor_joint_continuous (fun k v z => J k (v,z)) a |
| (fun k => (hJc k).comp (continuous_const.prodMk continuous_id)) (n+1)) |
| have hnm (i : I) : AEStronglyMeasurable (P (a+ℓ*u i)) μ := |
| (hPc.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| have hsum := RMSIntegral.weighted_sum_sq (fun i => basisDerivative u i t) |
| (fun i => P (a+ℓ*u i)) hnm (fun i => (hnode i).1) (fun i => (hnode i).2) |
| have hscaled := RMSIntegral.smul_sq_bound ℓ⁻¹ hsum.1 hsum.2 |
| have hRc : Continuous R := |
| ((hJc 1).comp (continuous_const.prodMk continuous_id)).sub |
| ((chainTaylor_joint_continuous (fun k v z => J (k+1) (v,z)) a |
| (fun k => (hJc (k+1)).comp (continuous_const.prodMk continuous_id)) n).comp |
| (continuous_const.prodMk continuous_id)) |
| have hsm : AEStronglyMeasurable |
| (fun z => ℓ⁻¹ • ∑i,basisDerivative u i t • P (a+ℓ*u i) z) μ := by |
| exact (Finset.aestronglyMeasurable_fun_sum _ (fun i _ => (hnm i).const_smul _)).const_smul _ |
| have hdiff := RMSIntegral.sub_sq_bound hRc.aestronglyMeasurable hsm |
| hR.1 hscaled.1 hRb hscaled.2 |
| have he (z : Ω) := derivative_cell_chain_error_eq u hu n hcard |
| (fun k s => J k (s,z)) a t hℓ.ne' |
| simp_rw [he] |
| refine ⟨hdiff.1,hdiff.2.trans_eq ?_⟩ |
| rw [pow_succ ℓ (n+1)] |
| field_simp |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Set MeasureTheory Filter |
| open scoped BigOperators |
|
|
| variable {I E : Type*} [Fintype I] [DecidableEq I] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| omit [CompleteSpace E] in |
| lemma regularized_intervalIntegrable {f : ℝ → E} (hf : Continuous f) |
| (hd : DifferentiableAt ℝ f 0) (h0 : f 0=0) : |
| IntervalIntegrable (fun t => t⁻¹ • f t) volume 0 1 := by |
| have hc : Continuous (dslope f 0) := continuousOn_univ.mp |
| ((continuousOn_dslope (s:=univ) (by simp)).mpr ⟨hf.continuousOn,hd⟩) |
| apply (hc.intervalIntegrable 0 1).congr_uIoo |
| intro t ht |
| rw [uIoo_of_le (by norm_num : (0:ℝ)≤1)] at ht |
| simp only [dslope_of_ne f ht.1.ne',slope_def_module,h0,sub_zero] |
|
|
| lemma zeroHermite_chainTaylor_integral (u : I → ℝ) (hu : Function.Injective u) |
| (i₀ : I) (hu0 : u i₀=0) (n : ℕ) (hcard : n+1≤Fintype.card I) |
| (J : ℕ → ℝ → E) (hJ : J 1 0=0) {ℓ : ℝ} (hℓ : 0<ℓ) : |
| (∫t in (0:ℝ)..1,t⁻¹ • chainTaylor (fun k => J (k+1)) n 0 (ℓ*t))= |
| ∑i,(zeroHermiteWeight u i₀ i/ℓ) • chainTaylor J (n+1) 0 (ℓ*u i) := by |
| let v := fun i => chainTaylor J (n+1) 0 (ℓ*u i) |
| have he := funext (zeroHermite_reproduces_chainTaylor u hu i₀ hu0 (n+1) |
| (by omega) hcard J hJ ℓ) |
| have heD (t : ℝ) : deriv (zeroHermiteInterpolation u i₀ v) t= |
| ℓ • chainTaylor (fun k => J (k+1)) n 0 (ℓ*t) := by |
| dsimp only [v] |
| rw [he] |
| simpa only [Function.comp_def,mul_one] using |
| ((chainTaylor_hasDerivAt J n 0 (ℓ*t)).scomp t |
| ((hasDerivAt_id t).const_mul ℓ)).deriv |
| calc |
| _ = ∫t in (0:ℝ)..1,ℓ⁻¹ • zeroHermiteAction u i₀ v t := by |
| apply intervalIntegral.integral_congr_Ioo_of_le (by norm_num) |
| intro t ht |
| dsimp only |
| rw [zeroHermiteAction_eq u hu i₀ hu0 v ht.1.ne',heD] |
| simp only [smul_smul] |
| congr 1 |
| field_simp |
| _ = ℓ⁻¹ • ∑i,zeroHermiteWeight u i₀ i • v i := by |
| rw [intervalIntegral.integral_smul,zeroHermiteAction_integral] |
| _ = _ := by |
| simp only [Finset.smul_sum,smul_smul,div_eq_mul_inv,mul_comm,v] |
|
|
| lemma zeroHermite_derivative_error_eq (u : I → ℝ) (hu : Function.Injective u) |
| (i₀ : I) (hu0 : u i₀=0) (n : ℕ) (hcard : n+1≤Fintype.card I) |
| (J : ℕ → ℝ → E) (hJc : ∀k,Continuous (J k)) |
| (hJ : ∀k t,HasDerivAt (J k) (J (k+1) t) t) (hz : J 1 0=0) |
| {ℓ : ℝ} (hℓ : 0<ℓ) : |
| (∫t in (0:ℝ)..1,t⁻¹ • J 1 (ℓ*t))- |
| (∑i,(zeroHermiteWeight u i₀ i/ℓ) • J 0 (ℓ*u i))= |
| (∫t in (0:ℝ)..1,t⁻¹ • (J 1 (ℓ*t)-chainTaylor (fun k => J (k+1)) n 0 (ℓ*t)))- |
| (∑i,(zeroHermiteWeight u i₀ i/ℓ) • (J 0 (ℓ*u i)-chainTaylor J (n+1) 0 (ℓ*u i))) := by |
| have h1 := regularized_intervalIntegrable ((hJc 1).comp (continuous_const.mul continuous_id)) |
| (((hJ 1 (ℓ*0)).scomp 0 ((hasDerivAt_id 0).const_mul ℓ)).differentiableAt) |
| (by simpa using hz) |
| change IntervalIntegrable (fun t => t⁻¹ • J 1 (ℓ*t)) volume 0 1 at h1 |
| have hp : Continuous (fun t => chainTaylor (fun k => J (k+1)) n 0 (ℓ*t)) := by |
| unfold chainTaylor |
| fun_prop |
| have hpd : DifferentiableAt ℝ (fun t => chainTaylor (fun k => J (k+1)) n 0 (ℓ*t)) 0 := by |
| unfold chainTaylor |
| fun_prop |
| have hp0 : chainTaylor (fun k => J (k+1)) n 0 (ℓ*0)=0 := by |
| simp [chainTaylor,Finset.sum_range_succ',hz] |
| have h2 := regularized_intervalIntegrable hp hpd hp0 |
| simp only [smul_sub,Finset.sum_sub_distrib] |
| rw [intervalIntegral.integral_sub h1 h2,zeroHermite_chainTaylor_integral u hu i₀ hu0 n hcard J hz hℓ] |
| abel |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
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| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Set MeasureTheory Filter |
| open scoped BigOperators |
|
|
| variable {I Ω E : Type*} [Fintype I] [DecidableEq I] |
| [TopologicalSpace Ω] [MeasurableSpace Ω] [BorelSpace Ω] [SecondCountableTopology Ω] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| {μ : Measure Ω} [SFinite μ] |
|
|
| theorem zeroHermite_derivative_error_rms (u : I → ℝ) (hu : Function.Injective u) |
| (i₀ : I) (hu0 : u i₀=0) (hun : ∀i,u i∈Icc (0:ℝ) 1) |
| (n : ℕ) (hcard : n+1≤Fintype.card I) |
| (J : ℕ → ℝ × Ω → E) (hJc : ∀k,Continuous (J k)) |
| (hJ : ∀k t z,HasDerivAt (fun t => J k (t,z)) (J (k+1) (t,z)) t) |
| (hz : ∀z,J 1 (0,z)=0) {ℓ B : ℝ} (hℓ : 0<ℓ) (hB0 : 0≤B) |
| (hi : ∀t∈Icc (0:ℝ) ℓ,Integrable (fun z => ‖J (n+2) (t,z)‖^2) μ) |
| (hB : ∀t∈Icc (0:ℝ) ℓ,(∫z,‖J (n+2) (t,z)‖^2 ∂μ)≤B) : |
| let err := fun z => (∫t in (0:ℝ)..1,t⁻¹ • J 1 (ℓ*t,z))- |
| ∑i,(zeroHermiteWeight u i₀ i/ℓ) • J 0 (ℓ*u i,z) |
| Integrable (fun z => ‖err z‖^2) μ ∧ |
| (∫z,‖err z‖^2 ∂μ)≤ |
| 2*(1+(∑i,|zeroHermiteWeight u i₀ i|)^2)*(ℓ^(n+1))^2*B := by |
| dsimp only |
| let R := fun (t : ℝ) (z : Ω) => J 1 (ℓ*t,z)- |
| chainTaylor (fun k v => J (k+1) (v,z)) n 0 (ℓ*t) |
| let f := fun p : ℝ × Ω => p.1⁻¹ • R p.1 p.2 |
| let η : Measure ℝ := volume.restrict (Ioc (0:ℝ) 1) |
| have hRc : Continuous (fun p : ℝ × Ω => R p.1 p.2) := by |
| apply Continuous.sub |
| · exact (hJc 1).comp ((continuous_const.mul continuous_fst).prodMk continuous_snd) |
| · exact (chainTaylor_joint_continuous (fun k v z => J (k+1) (v,z)) 0 |
| (fun k => (hJc (k+1)).comp (continuous_const.prodMk continuous_id)) n).comp |
| ((continuous_const.mul continuous_fst).prodMk continuous_snd) |
| have hfm : AEStronglyMeasurable f (η.prod μ) := |
| measurable_fst.inv.aestronglyMeasurable.smul hRc.aestronglyMeasurable |
| have hrms (t : ℝ) (ht : t∈Icc (0:ℝ) 1) := |
| chainTaylor_regularized_derivative_rms J hJc hJ n hℓ ht hB0 hi hB |
| have hint := RMSIntegral.time_seed_uniform hfm |
| (ae_restrict_of_forall_mem measurableSet_Ioc (fun t ht => (hrms t ⟨ht.1.le,ht.2⟩).1)) |
| (ae_restrict_of_forall_mem measurableSet_Ioc (fun t ht => (hrms t ⟨ht.1.le,ht.2⟩).2)) |
| have heta : η.real univ=1 := by |
| simp only [η,Measure.real,Measure.restrict_apply_univ,Real.volume_Ioc,sub_zero, |
| ENNReal.ofReal_one,ENNReal.toReal_one] |
| rw [heta,one_pow,one_mul] at hint |
| let P := fun (t : ℝ) (z : Ω) => J 0 (t,z)-chainTaylor (fun k v => J k (v,z)) (n+1) 0 t |
| have hPc : Continuous (fun p : ℝ × Ω => P p.1 p.2) := |
| (hJc 0).sub (chainTaylor_joint_continuous (fun k v z => J k (v,z)) 0 |
| (fun k => (hJc k).comp (continuous_const.prodMk continuous_id)) (n+1)) |
| have hnode (i : I) : Integrable (fun z => ‖P (ℓ*u i) z‖^2) μ ∧ |
| (∫z,‖P (ℓ*u i) z‖^2 ∂μ)≤(ℓ^(n+2))^2*B := by |
| have h0 : 0≤ℓ*u i := mul_nonneg hℓ.le (hun i).1 |
| have h1 : ℓ*u i≤ℓ := mul_le_of_le_one_right hℓ.le (hun i).2 |
| obtain ⟨hi',hb'⟩ := chainTaylor_cell_rms J hJc hJ (n+1) h0 h1 hB0 hi hB |
| refine ⟨hi',hb'.trans ?_⟩ |
| simp only [sub_zero] |
| exact mul_le_mul_of_nonneg_right |
| (pow_le_pow_left₀ (pow_nonneg h0 _) (pow_le_pow_left₀ h0 h1 _) 2) hB0 |
| have hnm (i : I) : AEStronglyMeasurable (P (ℓ*u i)) μ := |
| (hPc.comp (continuous_const.prodMk continuous_id)).aestronglyMeasurable |
| have hsum := RMSIntegral.weighted_sum_sq (fun i => zeroHermiteWeight u i₀ i/ℓ) |
| (fun i => P (ℓ*u i)) hnm (fun i => (hnode i).1) (fun i => (hnode i).2) |
| rw [zeroHermite_weights_abs u i₀ hℓ] at hsum |
| have hsm : AEStronglyMeasurable |
| (fun z => ∑i,(zeroHermiteWeight u i₀ i/ℓ) • P (ℓ*u i) z) μ := by |
| exact Finset.aestronglyMeasurable_fun_sum _ (fun i _ => (hnm i).const_smul _) |
| have hdiff := RMSIntegral.sub_sq_bound hfm.prod_swap.integral_prod_right' hsm hint.1 hsum.1 hint.2 hsum.2 |
| have he (z : Ω) := zeroHermite_derivative_error_eq u hu i₀ hu0 n hcard |
| (fun k t => J k (t,z)) (fun k => (hJc k).comp (continuous_id.prodMk continuous_const)) |
| (fun k t => hJ k t z) (hz z) hℓ |
| simp_rw [he,intervalIntegral.integral_of_le (by norm_num : (0:ℝ)≤1)] |
| refine ⟨hdiff.1,hdiff.2.trans_eq ?_⟩ |
| rw [pow_succ ℓ (n+1)] |
| field_simp |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
