| import Mathlib |
| import OAI.Probability.LogConcave.Complexity.ProbabilityRmsBudget |
|
|
| namespace OAI |
|
|
| section |
| section |
| noncomputable section |
| namespace LogConcaveSampling.RMSIntegral |
| open MeasureTheory |
| open scoped NNReal |
|
|
| variable {Ω E H : Type*} [MeasurableSpace Ω] {μ : Measure Ω} |
| [NormedAddCommGroup E] [SecondCountableTopology E] [MeasurableSpace E] [BorelSpace E] |
| [NormedAddCommGroup H] [SecondCountableTopology H] [MeasurableSpace H] [BorelSpace H] |
|
|
| omit [SecondCountableTopology E] in |
| lemma lipschitz_error_sq {f g : Ω → E} (hf : Measurable f) (hg : Measurable g) |
| {Q : E → H} {K : ℝ≥0} (hQ : LipschitzWith K Q) |
| (hi : Integrable (fun y => ‖f y-g y‖^2) μ) |
| {B : ℝ} (hB : (∫y,‖f y-g y‖^2 ∂μ)≤B) : |
| Integrable (fun y => ‖Q (f y)-Q (g y)‖^2) μ ∧ |
| (∫y,‖Q (f y)-Q (g y)‖^2 ∂μ)≤(K:ℝ)^2*B := by |
| have hpt (y : Ω) : ‖Q (f y)-Q (g y)‖^2≤(K:ℝ)^2*‖f y-g y‖^2 := by |
| simpa only [dist_eq_norm,mul_pow] using pow_le_pow_left₀ (dist_nonneg) |
| (hQ.dist_le_mul (f y) (g y)) 2 |
| have hs : Integrable (fun y => ‖Q (f y)-Q (g y)‖^2) μ := |
| (hi.const_mul ((K:ℝ)^2)).mono' |
| (((hQ.continuous.measurable.comp hf).sub (hQ.continuous.measurable.comp hg)).norm.pow_const 2).aestronglyMeasurable |
| (Filter.Eventually.of_forall (fun y => by simpa only [Real.norm_eq_abs,abs_sq] using hpt y)) |
| refine ⟨hs,?_⟩ |
| have hb := integral_mono hs (hi.const_mul ((K:ℝ)^2)) hpt |
| rw [integral_const_mul] at hb |
| exact hb.trans (mul_le_mul_of_nonneg_left hB (sq_nonneg _)) |
|
|
| lemma add_error_sq {f g k : Ω → E} (hf : Measurable f) (hg : Measurable g) (hk : Measurable k) |
| (hfg : Integrable (fun y => ‖f y-g y‖^2) μ) |
| (hgk : Integrable (fun y => ‖g y-k y‖^2) μ) |
| {A B : ℝ} (hA : (∫y,‖f y-g y‖^2 ∂μ)≤A) (hB : (∫y,‖g y-k y‖^2 ∂μ)≤B) : |
| Integrable (fun y => ‖f y-k y‖^2) μ ∧ (∫y,‖f y-k y‖^2 ∂μ)≤2*A+2*B := by |
| have he (y : Ω) : f y-k y=(f y-g y)-(k y-g y) := by abel |
| have hi : Integrable (fun y => ‖k y-g y‖^2) μ := by simpa only [norm_sub_rev] using hgk |
| have hb : (∫y,‖k y-g y‖^2 ∂μ)≤B := by simpa only [norm_sub_rev] using hB |
| simp_rw [he] |
| exact sub_sq_bound (hf.sub hg).aestronglyMeasurable (hk.sub hg).aestronglyMeasurable hfg hi hA hb |
|
|
| omit [SecondCountableTopology E] in |
| lemma compose_error_sq {f g : Ω → E} {k : Ω → H} |
| (hf : Measurable f) (hg : Measurable g) (hk : Measurable k) |
| {Q : E → H} {K : ℝ≥0} (hQ : LipschitzWith K Q) |
| (hfg : Integrable (fun y => ‖f y-g y‖^2) μ) |
| (hgk : Integrable (fun y => ‖Q (g y)-k y‖^2) μ) |
| {A B : ℝ} (hA : (∫y,‖f y-g y‖^2 ∂μ)≤A) (hB : (∫y,‖Q (g y)-k y‖^2 ∂μ)≤B) : |
| Integrable (fun y => ‖Q (f y)-k y‖^2) μ ∧ |
| (∫y,‖Q (f y)-k y‖^2 ∂μ)≤2*(K:ℝ)^2*A+2*B := by |
| have hh := lipschitz_error_sq hf hg hQ hfg hA |
| simpa only [Function.comp_def,mul_assoc] using add_error_sq (hQ.continuous.measurable.comp hf) (hQ.continuous.measurable.comp hg) hk |
| hh.1 hgk hh.2 hB |
| end LogConcaveSampling.RMSIntegral |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open Set MeasureTheory ProbabilityTheory RMSIntegral |
| open scoped Classical NNReal |
|
|
| def kernelCorrectionPicard {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (r T h z : ℝ) (n N : ℕ) (s e : ProbabilityNode T h n) |
| (p : Point d × Point d) : Point d := |
| let b := probabilityAnchoredPicard F x r T h n N e p.1 s |
| let w := harmonicPicard F x r (probabilityNodeTime T h n s) z n N (b,p.2) (Fin.last n) |
| probabilityAnchoredPicard F x r T h n (N+1) s w e-w |
|
|
| def harmonicRmsBudget {d : ℕ} (F : Point d → ℝ) (x : Point d) |
| (lam A : ℝ≥0) (r R t z C : ℝ) (n N : ℕ) : ℝ := |
| 2*(r*t)*Real.sqrt (4*C^2*(z^(n+1)/(n.factorial:ℝ))^2* |
| ((d*((lam:ℝ)*r)^2)*(R⁻¹)^(4*(n+1))*harmonicMeanBudget (n+1)))+ |
| (A*probabilityMeanLipschitz lam r:ℝ≥0)^N*(r*t)*Real.sqrt (harmonicStationaryMoment F x r t) |
|
|
| theorem kernelCorrectionPicard_rms (n : ℕ) (hn : 0<n) : |
| ∃Ap Ah Lp Lh : ℝ≥0,∃C J : ℝ,0≤C ∧ 1≤J ∧ ∃k : ℕ, |
| ∀{d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0},∀hF : Primitive F lam, |
| ∀(x : Point d) {r : ℝ},∀hr : 0<r,0<lam → ∀hl : (lam:ℝ)*r^2≤1/2,1≤d → |
| ∀{T h : ℝ},∀hT0 : 0<T,∀hT1 : T<1,∀hh : 0<h,h≤Real.log 2 → |
| Ap*probabilityMeanLipschitz lam r≤1/2 → Ah*probabilityMeanLipschitz lam r≤1/2 → |
| Lp*probabilityMeanLipschitz lam r≤1/2 → Lh*probabilityMeanLipschitz lam r≤1/2 → |
| ∀N : ℕ,∀s e : ProbabilityNode T h n, |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n e,probabilityNodeTime_mem hT0 hT1 hh hn e⟩ |
| let μ := gibbs (centeringPotential F x r S) |
| let fw := probabilityTransport hF x hr.le hl hT0.le hT1 S E |
| ∀R : ℝ,0<R → R^2≤1-(S:ℝ)^2 → |
| ∀Ξ : Point (d+d) → ℝ → Point (d+d), |
| (∀y,Continuous (Ξ y)) → (∀y,Ξ y 0=y) → |
| (∀y v,v∈Icc (0:ℝ) 1 → HasDerivWithinAt (Ξ y) |
| (skewLieField (centeringPotential F x r S) (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,μ.map (fun y => Ξ y v)=μ) → |
| ∀z : ℝ,0≤z → z≤1 → |
| let w := fun y => (productPointEquiv d d (Ξ y z)).1 |
| let err := fun y => kernelCorrectionPicard F x r T h z n N s e |
| (fw (productPointEquiv d d y).1,(productPointEquiv d d y).2)-(fw (w y)-w y) |
| Integrable (fun y => ‖err y‖^2) μ ∧ |
| (∫y,‖err y‖^2 ∂μ)≤ |
| 2*(2*(Lp*probabilityMeanLipschitz lam r):ℝ≥0)^2* |
| (32*(probabilityRmsBudget F x lam Ap r h C k n N)^2+ |
| 2*(harmonicRmsBudget F x lam Ah r R S z J n N)^2)+ |
| 2*(probabilityRmsBudget F x lam Ap r h C k n (N+1))^2 := by |
| obtain ⟨Ap,C,hC,k,hP⟩ := kernel_probability_errors n hn |
| obtain ⟨Ah,J,hJ,hH⟩ := harmonicPicard_stationary_rms n hn |
| obtain ⟨Lp,hLp⟩ := probabilityAnchored_lipschitz n hn |
| obtain ⟨Lh,hLh⟩ := harmonicPicard_lipschitz n hn |
| refine ⟨Ap,Ah,Lp,Lh,C,J,hC,hJ,k,?_⟩ |
| intro d F lam hF x r hr hlam hl hd T h hT0 hT1 hh hsmall hqp hqh hqlp hqlh N s e |
| dsimp only |
| let S : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n s,probabilityNodeTime_mem hT0 hT1 hh hn s⟩ |
| let E : Icc (0:ℝ) T := ⟨probabilityNodeTime T h n e,probabilityNodeTime_mem hT0 hT1 hh hn e⟩ |
| let μ := gibbs (centeringPotential F x r S) |
| let fw := probabilityTransport hF x hr.le hl hT0.le hT1 S E |
| intro R hR hRT Ξ hc hinit hder hm hlaw z hz hz1 |
| let pos := fun y : Point (d+d) => (productPointEquiv d d y).1 |
| let mom := fun y : Point (d+d) => (productPointEquiv d d y).2 |
| let b := fun y => probabilityAnchoredPicard F x r T h n N e (fw (pos y)) s |
| let H := fun p => harmonicPicard F x r S z n N p (Fin.last n) |
| let Q := fun v => probabilityAnchoredPicard F x r T h n (N+1) s v e-v |
| let w := fun y => pos (Ξ y z) |
| let u := fun y => H (b y,mom y) |
| have hpos : Measurable pos := (productPointEquiv d d).continuous.measurable.fst |
| have hmom : Measurable mom := (productPointEquiv d d).continuous.measurable.snd |
| have hfw : Continuous fw := probabilityTransport_continuous hF x hr.le hl hT0.le hT1 S E |
| have hb : Measurable b := (probabilityAnchored_continuous hF x hr.le hl hT0 hT1 hh n hn N e s).measurable.comp |
| (hfw.measurable.comp hpos) |
| have hW : Measurable w := hpos.comp (hm.comp (measurable_const.prodMk measurable_id)) |
| have hHL : LipschitzWith 4 H := by |
| have he := (LipschitzWith.eval (Fin.last n)).comp |
| (hLh hF x hr.le hl S.2.1 (S.2.2.trans_lt hT1) hz hz1 hqlh N).1 |
| convert he using 1 <;> first | rfl | simp only [one_mul] |
| have hQL : LipschitzWith (2*(Lp*probabilityMeanLipschitz lam r)) Q := by |
| have he := (LipschitzWith.eval e).comp (hLp hF x hr.le hl hT0 hT1 hh hqlp N s).2 |
| convert he using 1 <;> first | rfl | simp only [one_mul] |
| have hu : Measurable u := hHL.continuous.measurable.comp (hb.prodMk hmom) |
| have hprob := hP hF x hr hlam hl hd hT0 hT1 hh hsmall hqp N s e |
| have hnorm (y : Point (d+d)) : ‖(b y,mom y)-(pos y,mom y)‖^2=‖b y-pos y‖^2 := by |
| simp only [Prod.mk_sub_mk,sub_self,Prod.norm_def,norm_zero,max_eq_left (norm_nonneg _)] |
| have hi : Integrable (fun y => ‖(b y,mom y)-(pos y,mom y)‖^2) μ := by |
| simpa only [hnorm] using hprob.1.1 |
| have hbnd : (∫y,‖(b y,mom y)-(pos y,mom y)‖^2 ∂μ)≤ |
| (probabilityRmsBudget F x lam Ap r h C k n N)^2 := by |
| simpa only [hnorm] using hprob.1.2 |
| have hnum := lipschitz_error_sq (hb.prodMk hmom) (hpos.prodMk hmom) hHL hi hbnd |
| have hlast : probabilityNodes n (Fin.last n)=1 := by |
| change (n:ℝ)/(n:ℝ)=1 |
| exact div_self (by exact_mod_cast hn.ne') |
| have hharm := hH hF x hr hlam hl hR hRT S.2.1 (S.2.2.trans_lt hT1) |
| Ξ hc hinit hder hm hlaw z hz hz1 hqh N (Fin.last n) |
| rw [hlast,mul_one] at hharm |
| have hharm' : Integrable (fun y => ‖H (pos y,mom y)-w y‖^2) μ ∧ |
| (∫y,‖H (pos y,mom y)-w y‖^2 ∂μ)≤(harmonicRmsBudget F x lam Ah r R S z J n N)^2 := hharm |
| have hstate := add_error_sq hu (hHL.continuous.measurable.comp (hpos.prodMk hmom)) hW |
| hnum.1 hharm'.1 hnum.2 hharm'.2 |
| have hstate' : Integrable (fun y => ‖u y-w y‖^2) μ ∧ |
| (∫y,‖u y-w y‖^2 ∂μ)≤32*(probabilityRmsBudget F x lam Ap r h C k n N)^2+ |
| 2*(harmonicRmsBudget F x lam Ah r R S z J n N)^2 := by |
| simpa only [NNReal.coe_ofNat,show (4:ℝ)^2=16 by norm_num,show (2:ℝ)*16=32 by norm_num,←mul_assoc] using hstate |
| have hforward := hprob.2 (fun y => Ξ y z) (hm.comp (measurable_const.prodMk measurable_id)) (hlaw z ⟨hz,hz1⟩) |
| have hcancel (y : Point (d+d)) : Q (w y)-(fw (w y)-w y)= |
| probabilityAnchoredPicard F x r T h n (N+1) s (w y) e-fw (w y) := by dsimp [Q]; abel |
| have hforw' : Integrable (fun y => ‖Q (w y)-(fw (w y)-w y)‖^2) μ ∧ |
| (∫y,‖Q (w y)-(fw (w y)-w y)‖^2 ∂μ)≤(probabilityRmsBudget F x lam Ap r h C k n (N+1))^2 := by |
| simpa only [hcancel] using hforward |
| exact compose_error_sq hu hW ((hfw.measurable.comp hW).sub hW) hQL |
| hstate'.1 hforw'.1 hstate'.2 hforw'.2 |
|
|
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| section |
|
|
| noncomputable section |
| namespace LogConcaveSampling |
| open Set |
| open scoped NNReal |
|
|
| lemma harmonicPicard_continuous {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r ρ : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hρ0 : 0≤ρ) (hρ1 : ρ<1) |
| (z : ℝ) (n N : ℕ) (i : Fin (n+1)) : |
| Continuous (fun p => harmonicPicard F x r ρ z n N p i) := by |
| apply (continuous_apply i).comp |
| apply FinitePicard.nodes_continuous |
| · exact fun _ => (harmonicMeanVelocity_lipschitz hF x hr hl hρ0 hρ1).continuous |
| · exact (harmonicFree_lipschitz n z).continuous |
|
|
| lemma kernelCorrectionPicard_continuous {d : ℕ} {F : Point d → ℝ} {lam : ℝ≥0} |
| (hF : Primitive F lam) (x : Point d) {r T h : ℝ} (hr : 0≤r) |
| (hl : (lam:ℝ)*r^2≤1/2) (hT0 : 0<T) (hT1 : T<1) (hh : 0<h) |
| (z : ℝ) (n : ℕ) (hn : 0<n) (N : ℕ) (s e : ProbabilityNode T h n) : |
| Continuous (kernelCorrectionPicard F x r T h z n N s e) := by |
| have hb : Continuous (fun p : Point d × Point d => probabilityAnchoredPicard F x r T h n N e p.1 s) := |
| (probabilityAnchored_continuous hF x hr hl hT0 hT1 hh n hn N e s).comp continuous_fst |
| have ht := probabilityNodeTime_mem hT0 hT1 hh hn s |
| have hw := (harmonicPicard_continuous hF x hr hl ht.1 (ht.2.trans_lt hT1) z n N (Fin.last n)).comp |
| (hb.prodMk continuous_snd) |
| exact ((probabilityAnchored_continuous hF x hr hl hT0 hT1 hh n hn (N+1) s e).comp hw).sub hw |
| end LogConcaveSampling |
|
|
| end |
|
|
| end |
|
|
| end |
|
|
| end OAI |
|
|
