| import Mathlib |
| import OAI.Probability.LogConcave.Numerics.ChainTaylor |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Set MeasureTheory |
| open scoped BigOperators |
|
|
| variable {I : Type*} [Fintype I] [DecidableEq I] |
|
|
| def compositeAngle {N : ℕ} (u : I → ℝ) (i : Fin N × I) (c : Fin N) : ℝ := |
| if c < i.1 then 1 else if c = i.1 then u i.2 else 0 |
|
|
| omit [Fintype I] [DecidableEq I] in |
| lemma compositeAngle_mem {N : ℕ} {u : I → ℝ} (hu : ∀i,u i∈Icc (0:ℝ) 1) |
| (i : Fin N × I) (c : Fin N) : compositeAngle u i c∈Icc (0:ℝ) 1 := by |
| unfold compositeAngle |
| split_ifs <;> simp_all |
|
|
| def compositeWeight {N : ℕ} (u : I → ℝ) (ℓ : Fin N → ℝ) |
| (i j : Fin N × I) : ℝ := ℓ j.1*weight u j.2 0 (compositeAngle u i j.1) |
|
|
| lemma compositeWeight_budget {N : ℕ} {u : I → ℝ} (hu : ∀i,u i∈Icc (0:ℝ) 1) |
| {ℓ : Fin N → ℝ} (hℓ : ∀c,0≤ℓ c) (hsum : (∑c,ℓ c)≤1) |
| {C : ℝ} (hC : 0≤C) (hbasis : ∀t∈Icc (0:ℝ) 1,∑j,|basis u j t|≤C) |
| (i : Fin N × I) : ∑j,|compositeWeight u ℓ i j|≤(Fintype.card I:ℝ)*C := by |
| rw [Fintype.sum_prod_type] |
| have hc (c : Fin N) : ∑j,|compositeWeight u ℓ i (c,j)|≤ℓ c*((Fintype.card I:ℝ)*C) := by |
| simp only [compositeWeight,abs_mul,abs_of_nonneg (hℓ c),←Finset.mul_sum] |
| apply mul_le_mul_of_nonneg_left _ (hℓ c) |
| have ht := compositeAngle_mem hu i c |
| exact (weights_abs_bound u ht.1 ht.2 hbasis).trans |
| (mul_le_of_le_one_right (mul_nonneg (Nat.cast_nonneg _) hC) ht.2) |
| calc |
| _ ≤ ∑c,ℓ c*((Fintype.card I:ℝ)*C) := Finset.sum_le_sum (fun c _ => hc c) |
| _ = (∑c,ℓ c)*((Fintype.card I:ℝ)*C) := by rw [Finset.sum_mul] |
| _ ≤ _ := mul_le_of_le_one_left (mul_nonneg (Nat.cast_nonneg _) hC) hsum |
|
|
| lemma compositeWeight_sum {N : ℕ} {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] |
| [CompleteSpace E] (u : I → ℝ) (ℓ : Fin N → ℝ) (v : Fin N × I → E) (i : Fin N × I) : |
| (∑j,compositeWeight u ℓ i j • v j)= |
| ∑c,cellQuadrature u (fun j => v (c,j)) (ℓ c) (compositeAngle u i c) := by |
| rw [Fintype.sum_prod_type] |
| rfl |
|
|
| lemma composite_error_rms {N : ℕ} {Ω E : Type*} [MeasurableSpace Ω] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ν : Measure Ω} |
| (ℓ : Fin N → ℝ) (R : Fin N → Ω → E) {B : ℝ} |
| (hℓ : ∀c,0≤ℓ c) (hsum : (∑c,ℓ c)≤1) (hB : 0≤B) |
| (hm : ∀c,AEStronglyMeasurable (R c) ν) |
| (hi : ∀c,Integrable (fun z => ‖R c z‖^2) ν) |
| (hb : ∀c,(∫z,‖R c z‖^2 ∂ν)≤B) : |
| Integrable (fun z => ‖∑c,ℓ c • R c z‖^2) ν ∧ |
| (∫z,‖∑c,ℓ c • R c z‖^2 ∂ν)≤B := by |
| have he := RMSIntegral.weighted_sum_sq ℓ R hm hi hb |
| simp only [abs_of_nonneg (hℓ _)] at he |
| refine ⟨he.1,he.2.trans ?_⟩ |
| have hn : 0≤∑c,ℓ c := Finset.sum_nonneg (fun c _ => hℓ c) |
| exact mul_le_of_le_one_left hB (by nlinarith) |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open Set MeasureTheory |
| open scoped BigOperators |
|
|
| variable {I E : Type*} [Fintype I] [DecidableEq I] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
|
|
| omit [Fintype I] [DecidableEq I] [CompleteSpace E] in |
| lemma compositeIntegral_sum {N : ℕ} (u : I → ℝ) (t : ℕ → ℝ) |
| (f : ℝ → E) (hf : Continuous f) (i : Fin N × I) : |
| (∑c : Fin N,(t (c.val+1)-t c.val) • |
| ∫s in 0..compositeAngle u i c,f (t c.val+(t (c.val+1)-t c.val)*s))= |
| ∫s in t 0..(t i.1.val+(t (i.1.val+1)-t i.1.val)*u i.2),f s := by |
| let v := t i.1.val+(t (i.1.val+1)-t i.1.val)*u i.2 |
| let τ : ℕ → ℝ := fun k => if k≤ i.1.val then t k else v |
| have hc (c : Fin N) : (t (c.val+1)-t c.val) • |
| (∫s in 0..compositeAngle u i c,f (t c.val+(t (c.val+1)-t c.val)*s))= |
| ∫s in τ c.val..τ (c.val+1),f s := by |
| rw [intervalIntegral.smul_integral_comp_add_mul] |
| simp only [mul_zero,add_zero] |
| rcases lt_trichotomy c i.1 with hlt | heq | hgt |
| · have hnat : c.val< i.1.val := hlt |
| have hnext : c.val+1≤ i.1.val := by omega |
| simp only [compositeAngle,ite_eq_left hlt,τ,ite_eq_left hnat.le, |
| ite_eq_left hnext,mul_one,add_sub_cancel] |
| · subst c |
| have hnot : ¬i.1.val+1≤ i.1.val := by omega |
| simp only [compositeAngle,lt_self_iff_false,ite_false,τ,le_refl,ite_true,ite_eq_right hnot,v] |
| · have hnot : ¬c< i.1 := not_lt_of_gt hgt |
| have hne : c≠i.1 := ne_of_gt hgt |
| have hnat : ¬c.val≤ i.1.val := not_le_of_gt hgt |
| have hnext : ¬c.val+1≤ i.1.val := by omega |
| simp only [compositeAngle,ite_eq_right hnot,ite_eq_right hne,τ, |
| ite_eq_right hnat,ite_eq_right hnext,mul_zero,add_zero, |
| intervalIntegral.integral_same] |
| simp_rw [hc] |
| rw [Fin.sum_univ_eq_sum_range (fun k => ∫s in τ k..τ (k+1),f s), |
| intervalIntegral.sum_integral_adjacent_intervals (fun _ _ => hf.intervalIntegrable _ _)] |
| have hnot : ¬N≤ i.1.val := not_le_of_gt i.1.2 |
| simp only [τ,Nat.zero_le,ite_true,ite_eq_right hnot,v] |
|
|
| lemma compositeQuadrature_defect {N : ℕ} (u : I → ℝ) (t : ℕ → ℝ) |
| (f : ℝ → E) (hf : Continuous f) (i : Fin N × I) : |
| (∫s in t 0..(t i.1.val+(t (i.1.val+1)-t i.1.val)*u i.2),f s)- |
| (∑j : Fin N × I,compositeWeight u (fun c => t (c.val+1)-t c.val) i j • |
| f (t j.1.val+(t (j.1.val+1)-t j.1.val)*u j.2))= |
| ∑c : Fin N,(t (c.val+1)-t c.val) • |
| (∫s in 0..compositeAngle u i c,f (t c.val+(t (c.val+1)-t c.val)*s)- |
| interpolation u (fun j => f (t c.val+(t (c.val+1)-t c.val)*u j)) s) := by |
| rw [←compositeIntegral_sum u t f hf i,compositeWeight_sum,←Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro _ _ |
| rw [cellQuadrature_eq_integral,←smul_sub,intervalIntegral.integral_sub] |
| · exact (hf.comp (continuous_const.add (continuous_const.mul continuous_id))).intervalIntegrable _ _ |
| · exact (interpolation_continuous _ _).intervalIntegrable _ _ |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open MeasureTheory Set Polynomial |
| open scoped BigOperators |
|
|
| variable {I Ω E : Type*} [Fintype I] [DecidableEq I] [MeasurableSpace Ω] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] {ν : Measure Ω} |
|
|
| def derivativeWeight (u : I → ℝ) (i : I) : ℝ := |
| (Lagrange.basis Finset.univ u i).derivative.eval 0 |
|
|
| def derivativeStencil (u : I → ℝ) (ψ : ℝ) (v : I → E) : E := |
| ∑i,(derivativeWeight u i/ψ) • v i |
|
|
| lemma derivative_reproduces_scalar (u : I → ℝ) (hu : Function.Injective u) |
| (p : ℝ[X]) (hp : p.degree<Fintype.card I) : |
| (∑i,derivativeWeight u i*p.eval (u i))=p.derivative.eval 0 := by |
| have he := congrArg (fun q : ℝ[X] => q.derivative.eval 0) |
| (Lagrange.eq_interpolate (s:=Finset.univ) hu.injOn (by simpa using hp)) |
| simpa only [Lagrange.interpolate_apply,derivative_sum,derivative_mul, |
| derivative_C,zero_mul,mul_zero,zero_add,add_zero,eval_finsetSum,eval_mul,eval_C,derivativeWeight,mul_comm] using he.symm |
|
|
| lemma derivative_reproduces_power (u : I → ℝ) (hu : Function.Injective u) (k : ℕ) |
| (hk : k<Fintype.card I) : |
| (∑i,derivativeWeight u i*(u i)^k)=if k=1 then 1 else 0 := by |
| have hh := derivative_reproduces_scalar u hu (X^k) (by simpa using hk) |
| simp only [eval_pow,eval_X,derivative_X_pow,eval_mul,eval_C] at hh |
| rw [hh] |
| by_cases h0 : k=0 |
| · simp [h0] |
| by_cases h1 : k=1 |
| · simp [h1] |
| have hk1 : k-1≠0 := by omega |
| simp [h1,hk1] |
|
|
| lemma derivative_weights_norm (u : I → ℝ) {ψ : ℝ} (hψ : 0<ψ) : |
| (∑i,|derivativeWeight u i/ψ|)=(∑i,|derivativeWeight u i|)/ψ := by |
| simp only [abs_div,abs_of_pos hψ,Finset.sum_div] |
|
|
| omit [CompleteSpace E] in |
| lemma derivative_reproduces_vector (u : I → ℝ) (hu : Function.Injective u) |
| (n : ℕ) (hn : 1≤n) (hm : n+1≤Fintype.card I) {ψ : ℝ} (hψ : ψ≠0) |
| (v : Fin (n+1) → E) : |
| derivativeStencil u ψ (fun i => ∑k : Fin (n+1),(ψ*u i)^(k:ℕ) • v k)=v ⟨1,by omega⟩ := by |
| unfold derivativeStencil |
| simp_rw [Finset.smul_sum,smul_smul,mul_pow] |
| rw [Finset.sum_comm] |
| have he (k : Fin (n+1)) : |
| (∑i,((derivativeWeight u i/ψ)*(ψ^(k:ℕ)*(u i)^(k:ℕ))) • v k)= |
| (if (k:ℕ)=1 then 1 else 0) • v k := by |
| rw [←Finset.sum_smul] |
| have hc : (∑i,(derivativeWeight u i/ψ)*(ψ^(k:ℕ)*(u i)^(k:ℕ)))= |
| (ψ^(k:ℕ)/ψ)*∑i,derivativeWeight u i*(u i)^(k:ℕ) := by |
| rw [Finset.mul_sum] |
| apply Finset.sum_congr rfl |
| intro _ _ |
| ring |
| rw [hc,derivative_reproduces_power u hu k (lt_of_lt_of_le k.2 hm)] |
| by_cases hk : (k:ℕ)=1 |
| · simp [hk,hψ] |
| · simp [hk] |
| simp_rw [he] |
| simp only [ite_smul,one_smul,zero_smul] |
| rw [Finset.sum_eq_single ⟨1,by omega⟩] |
| · simp |
| · intro b _ hne |
| have h : (b:ℕ)≠1 := fun h => hne (Fin.ext h) |
| simp [h] |
| · simp |
|
|
| omit [CompleteSpace E] in |
| theorem derivative_stencil_error_rms (u : I → ℝ) (hu : Function.Injective u) |
| (n : ℕ) (hn : 1≤n) (hcard : n+1≤Fintype.card I) {ψ B : ℝ} |
| (hψ : 0<ψ) (J : ℕ → ℝ → Ω → E) |
| (hm : ∀i,AEStronglyMeasurable (fun z => J 0 (ψ*u i) z- |
| chainTaylor (fun k t => J k t z) n 0 (ψ*u i)) ν) |
| (hi : ∀i,Integrable (fun z => ‖J 0 (ψ*u i) z- |
| chainTaylor (fun k t => J k t z) n 0 (ψ*u i)‖^2) ν) |
| (hB : ∀i,(∫z,‖J 0 (ψ*u i) z- |
| chainTaylor (fun k t => J k t z) n 0 (ψ*u i)‖^2 ∂ν)≤B) : |
| Integrable (fun z => ‖derivativeStencil u ψ (fun i => J 0 (ψ*u i) z)-J 1 0 z‖^2) ν ∧ |
| (∫z,‖derivativeStencil u ψ (fun i => J 0 (ψ*u i) z)-J 1 0 z‖^2 ∂ν)≤ |
| ((∑i,|derivativeWeight u i|)/ψ)^2*B := by |
| have he (z : Ω) : derivativeStencil u ψ (fun i => |
| chainTaylor (fun k t => J k t z) n 0 (ψ*u i))=J 1 0 z := by |
| have hh := derivative_reproduces_vector u hu n hn hcard hψ.ne' |
| (fun k => ((k:ℕ).factorial:ℝ)⁻¹ • J k 0 z) |
| have ht (i : I) : chainTaylor (fun k t => J k t z) n 0 (ψ*u i)= |
| ∑k : Fin (n+1),(ψ*u i)^(k:ℕ) • (((k:ℕ).factorial:ℝ)⁻¹ • J k 0 z) := by |
| unfold chainTaylor |
| rw [←Fin.sum_univ_eq_sum_range] |
| apply Finset.sum_congr rfl |
| intro _ _ |
| simp only [sub_zero,mul_smul] |
| exact smul_comm _ _ _ |
| simp_rw [ht] |
| simpa only [Nat.factorial_one,Nat.cast_one,inv_one,one_smul] using hh |
| have hc (z : Ω) : derivativeStencil u ψ (fun i => J 0 (ψ*u i) z)-J 1 0 z= |
| ∑i,(derivativeWeight u i/ψ) • (J 0 (ψ*u i) z- |
| chainTaylor (fun k t => J k t z) n 0 (ψ*u i)) := by |
| rw [←he z] |
| simp only [derivativeStencil,smul_sub,Finset.sum_sub_distrib] |
| simp_rw [hc] |
| have hh := RMSIntegral.weighted_sum_sq (fun i => derivativeWeight u i/ψ) _ hm hi hB |
| rw [derivative_weights_norm u hψ] at hh |
| exact hh |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling.Quadrature |
| open MeasureTheory Set |
|
|
| variable {β E : Type*} [MeasurableSpace β] |
| [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E] |
| {ν : Measure β} [SFinite ν] |
|
|
| theorem endpoint_quotient_rms {J : ℕ → ℝ → β → E} {s : Set ℝ} {x B : ℝ} |
| (hx : 0<x) (hB0 : 0≤B) (hsub : uIcc (0:ℝ) x⊆s) |
| (hf : ∀z n t,t∈s → HasDerivWithinAt (fun t => J n t z) (J (n+1) t z) s t) |
| (hzero : ∀z,J 1 0 z=0) |
| (hm : AEStronglyMeasurable (fun p : ℝ × β => J 2 p.1 p.2) |
| ((volume.restrict (Ioc 0 x)).prod ν)) |
| (hi : ∀t∈Ioc 0 x,Integrable (fun z => ‖J 2 t z‖^2) ν) |
| (hB : ∀t∈Ioc 0 x,(∫z,‖J 2 t z‖^2 ∂ν)≤B) : |
| Integrable (fun z => ‖x⁻¹ • J 1 x z‖^2) ν ∧ |
| (∫z,‖x⁻¹ • J 1 x z‖^2 ∂ν)≤B := by |
| have hchain : ∀z n t,t∈s → HasDerivWithinAt (fun t => J (n+1) t z) (J (n+1+1) t z) s t := |
| fun z n t ht => hf z (n+1) t ht |
| have he := chainTaylor_remainder_rms (J:=fun n => J (n+1)) (n:=0) |
| hx.le hB0 hsub hchain hm hi hB |
| simp only [chainTaylor,Finset.sum_range_one,Nat.factorial_zero,Nat.cast_one, |
| inv_one,pow_zero,mul_one,one_smul,hzero,sub_zero,zero_add,pow_one,div_one] at he |
| have hnorm (z : β) : ‖x⁻¹ • J 1 x z‖^2=(x⁻¹)^2*‖J 1 x z‖^2 := by |
| rw [norm_smul,Real.norm_eq_abs,mul_pow,sq_abs] |
| simp_rw [hnorm] |
| refine ⟨he.1.const_mul _,?_⟩ |
| rw [integral_const_mul] |
| calc |
| _ ≤ (x⁻¹)^2*(x^2*B) := mul_le_mul_of_nonneg_left he.2 (sq_nonneg _) |
| _ = B := by field_simp |
|
|
| theorem endpoint_integral_rms {J : ℕ → ℝ → β → E} {s : Set ℝ} {b B : ℝ} |
| (hb : 0≤b) (hB0 : 0≤B) (hsub : Icc (0:ℝ) b⊆s) |
| (hf : ∀z n t,t∈s → HasDerivWithinAt (fun t => J n t z) (J (n+1) t z) s t) |
| (hzero : ∀z,J 1 0 z=0) |
| (hm1 : AEStronglyMeasurable (fun p : ℝ × β => J 1 p.1 p.2) |
| ((volume.restrict (Ioc 0 b)).prod ν)) |
| (hm2 : AEStronglyMeasurable (fun p : ℝ × β => J 2 p.1 p.2) |
| ((volume.restrict (Ioc 0 b)).prod ν)) |
| (hi : ∀t∈Ioc 0 b,Integrable (fun z => ‖J 2 t z‖^2) ν) |
| (hB : ∀t∈Ioc 0 b,(∫z,‖J 2 t z‖^2 ∂ν)≤B) : |
| Integrable (fun z => ‖∫t in (0:ℝ)..b,t⁻¹ • J 1 t z‖^2) ν ∧ |
| (∫z,‖∫t in (0:ℝ)..b,t⁻¹ • J 1 t z‖^2 ∂ν)≤b^2*B := by |
| let μ : Measure ℝ := volume.restrict (Ioc 0 b) |
| have hout (t : ℝ) (ht : t∈Ioc 0 b) : |
| Integrable (fun z => ‖t⁻¹ • J 1 t z‖^2) ν ∧ |
| (∫z,‖t⁻¹ • J 1 t z‖^2 ∂ν)≤B := by |
| have htb : Ioc (0:ℝ) t⊆Ioc 0 b := Ioc_subset_Ioc_right ht.2 |
| apply endpoint_quotient_rms ht.1 hB0 |
| (by rw [uIcc_of_le ht.1.le]; exact (Icc_subset_Icc_right ht.2).trans hsub) |
| hf hzero |
| · exact hm2.mono_measure (Measure.prod_mono (Measure.restrict_mono htb le_rfl) le_rfl) |
| · exact fun u hu => hi u (htb hu) |
| · exact fun u hu => hB u (htb hu) |
| have hm : AEStronglyMeasurable (fun p : ℝ × β => p.1⁻¹ • J 1 p.1 p.2) (μ.prod ν) := |
| measurable_fst.inv.aestronglyMeasurable.smul hm1 |
| have hs : ∀ᵐ t ∂μ,Integrable (fun z => ‖t⁻¹ • J 1 t z‖^2) ν := by |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht |
| exact (hout t ht).1 |
| have hbound : ∀ᵐ t ∂μ,(∫z,‖t⁻¹ • J 1 t z‖^2 ∂ν)≤B := by |
| filter_upwards [ae_restrict_mem measurableSet_Ioc] with t ht |
| exact (hout t ht).2 |
| have he := RMSIntegral.time_seed_uniform hm hs hbound |
| have hμ : μ.real univ=b := by |
| dsimp only [μ,Measure.real] |
| rw [Measure.restrict_apply_univ] |
| exact (Real.volume_real_Ioc_of_le hb).trans (sub_zero b) |
| simpa only [intervalIntegral.integral_of_le hb,hμ] using he |
| end LogConcaveSampling.Quadrature |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
