| import Mathlib |
| import OAI.Probability.SKBarriers.Scalar.PositiveTimeChain |
| import OAI.Probability.SKBarriers.Replicas.TripleScaleStats |
| import OAI.Probability.SKBarriers.Hierarchy.TimeIncrementBridge |
| import OAI.Probability.SKBarriers.Parisi.CDFCramer |
| import OAI.Probability.SKBarriers.Parisi.CDFSquareSusceptibility |
| import OAI.Probability.SKBarriers.Scalar.ScalarConditionalStability |
| |
| namespace OAI |
|
|
| section |
|
|
| noncomputable section |
| open scoped BigOperators NNReal Topology |
| open MeasureTheory ProbabilityTheory Filter Set |
| namespace SK.Analytic |
| attribute [local instance 2000] parameterNormedGroup parameterNormedSpace |
|
|
| theorem scalarTimeChainAverage_hessian_square_cramer (β : ℝ) (l j : List (ℝ × ℝ≥0)) |
| {f : ℝ → ℝ} (hf : BoundedDerivs f) (hc : ScalarSpinConvex f) |
| (hj : ∀ p∈j,p.1∈Icc (0:ℝ) 1) : |
| let H := scalarTimeChain β j f |
| β^2*(chainDuration j:ℝ)*scalarTimeChainAverage β l H (fun x => (rootHessian 0 H x)^2) 0 ≤ |
| (1+2*(β^2*(chainDuration j:ℝ)))*(scalarTimeChainAverage β (l++j) f (fun x => (rootGradient 0 f x)^2) 0- |
| scalarTimeChainAverage β l H (fun x => (rootGradient 0 H x)^2) 0) := by |
| have G := scalarIncrementAverage_hessian_square_cramer |
| (scaleIncrementChain β (rawTimeChain l)) (scaleIncrementChain β (rawTimeChain j)) hf hc |
| (scaleIncrementChain_mass β _ (rawTimeChain_mass _ hj)) |
| dsimp only at G ⊢ |
| rw [← scaleIncrementChain_append,← rawTimeChain_append] at G |
| simpa only [rawVariance_scale,rawVariance_rawTimeChain,scalarIncrementChain_rawTimeChain, |
| scalarIncrementAverage_rawTimeChain] using G |
|
|
| theorem scalarCDF_conditional_stability_quantile {k : ℕ} (β : ℝ) (Q : Fin (k+1) → ℝ) |
| (hQ : Q∈admissibleQuantiles k) {r q : ℝ} (hr : 0≤r) (hrq : r≤q) (hq : q≤1) : |
| β^2*(q-r)*scalarCDFSusceptibilitySquareAverage β (quantileCDF k Q) r ≤ |
| (1+2*(β^2*(q-r)))*(scalarCDFOverlap β (quantileCDF k Q) q- |
| scalarCDFOverlap β (quantileCDF k Q) r) := by |
| let α := quantileCDF k Q |
| have ha := quantileCDF_bounds k Q |
| have hm := quantileCDF_monotone k Q |
| let sr := Real.toNNReal (1-r) |
| let sq := Real.toNNReal (1-q) |
| let tq := Real.toNNReal q |
| let tr := Real.toNNReal r |
| let tj := Real.toNNReal (q-r) |
| have hsr : (sr:ℝ)=1-r := Real.coe_toNNReal _ (by linarith) |
| have hsq : (sq:ℝ)=1-q := Real.coe_toNNReal _ (by linarith) |
| have htq : (tq:ℝ)=q := Real.coe_toNNReal _ (by linarith) |
| have htr : (tr:ℝ)=r := Real.coe_toNNReal _ hr |
| have htj : (tj:ℝ)=q-r := Real.coe_toNNReal _ (sub_nonneg.mpr hrq) |
| have hsr1 : sr≤1 := by rw [← NNReal.coe_le_coe,hsr,NNReal.coe_one] |
| have hsq1 : sq≤1 := by rw [← NNReal.coe_le_coe,hsq,NNReal.coe_one] |
| have htr1 : tr≤1 := by rw [← NNReal.coe_le_coe,htr,NNReal.coe_one] |
| have htq1 : tq≤1 := by rw [← NNReal.coe_le_coe,htq,NNReal.coe_one] |
| have htj1 : tj≤1 := by rw [← NNReal.coe_le_coe,htj,NNReal.coe_one] |
| obtain ⟨l,hdl,hml,_,hmodelL⟩ := quantile_interval_chain β Q hQ (s:=0) (t:=r) le_rfl hr (hrq.trans hq) |
| obtain ⟨j,hdj,hmj,_,hmodelJ⟩ := quantile_interval_chain β Q hQ hr hrq hq |
| have hdl' : chainDuration l=tr := NNReal.coe_injective (by rw [htr,hdl,sub_zero]) |
| have hdj' : chainDuration j=tj := NNReal.coe_injective (by rw [htj,hdj]) |
| have hdlj : chainDuration (l++j)=tq := by |
| apply NNReal.coe_injective |
| rw [chainDuration_append,NNReal.coe_add,hdl,hdj,htq] |
| have hmlj : ∀ p∈l++j,p.1∈Icc (0:ℝ) 1 := by |
| intro p hp |
| have hmodelLJ : TimeChainModels α 0 (l++j) := hmodelL.append (by simpa only [hdl,zero_add,sub_zero] using hmodelJ) |
| let F := scalarCDFValue β α q sq |
| let H := scalarCDFValue β α r sr |
| have hF : BoundedDerivs F := scalarCDFValue_regular β ha hm q sq hsq1 |
| have hH : BoundedDerivs H := scalarCDFValue_regular β ha hm r sr hsr1 |
| have hFc : ScalarSpinConvex F := scalarCDFValue_spinConvex β ha hm q sq hsq1 |
| have he : scalarTimeChain β j F=H := by |
| funext x |
| rw [← scalarCDFOperator_eq_chain hF (scalarCDFValue_lipschitz β ha hm q sq hsq1) |
| β ha hm j hmj r tj htj1 hdj' hmodelJ x] |
| exact (congrFun (scalarCDFValue_suffix_semigroup_quantile β Q hQ hr hrq hq) x).symm |
| have havr : scalarTimeChainAverage β l H (fun x => (rootGradient 0 H x)^2) 0=scalarCDFOverlap β α r := by |
| have ht := scalarCDFGradient_sq_regular β ha hm r sr hsr1 |
| change scalarTimeChainAverage β l (scalarCDFValue β α r sr) _ 0=_ |
| rw [scalarCDFValue_rootGradient β ha hm r sr hsr1] |
| exact (scalarCDFAverage_eq_chainAverage hH ht.1 ht.2.1 |
| (scalarCDFValue_lipschitz β ha hm r sr hsr1) (B:=1) ht.2.2.1 (C:=2) ht.2.2.2 |
| β ha hm l hml 0 tr htr1 hdl' hmodelL 0).symm |
| have havq : scalarTimeChainAverage β (l++j) F (fun x => (rootGradient 0 F x)^2) 0=scalarCDFOverlap β α q := by |
| have ht := scalarCDFGradient_sq_regular β ha hm q sq hsq1 |
| change scalarTimeChainAverage β (l++j) (scalarCDFValue β α q sq) _ 0=_ |
| rw [scalarCDFValue_rootGradient β ha hm q sq hsq1] |
| exact (scalarCDFAverage_eq_chainAverage hF ht.1 ht.2.1 |
| (scalarCDFValue_lipschitz β ha hm q sq hsq1) (B:=1) ht.2.2.1 (C:=2) ht.2.2.2 |
| β ha hm (l++j) hmlj 0 tq htq1 hdlj hmodelLJ 0).symm |
| have havχ : scalarTimeChainAverage β l H (fun x => (rootHessian 0 H x)^2) 0= |
| scalarCDFSusceptibilitySquareAverage β α r := by |
| change scalarTimeChainAverage β l (scalarCDFValue β α r sr) _ 0=_ |
| rw [scalarCDFValue_rootHessian β ha hm r sr hsr1] |
| exact (scalarCDFAverage_eq_chainAverage_lipschitz hH |
| (scalarCDFValue_lipschitz β ha hm r sr hsr1) |
| (scalarCDFHessian_square_lipschitz β ha hm r sr hsr1) |
| (B:=1) (scalarCDFHessian_square_bound β ha hm r sr hsr1) |
| β ha hm l hml 0 tr htr1 hdl' hmodelL 0).symm |
| have G := scalarTimeChainAverage_hessian_square_cramer β l j hF hFc hmj |
| dsimp only at G |
| rw [he,hdj,havr,havq,havχ] at G |
| exact G |
| |
| theorem scalarCDF_conditional_stability (β : ℝ) (α : StieltjesFunction ℝ) |
| (ha : ∀ z,α z∈Icc (0:ℝ) 1) (h1 : α 1=1) {r q : ℝ} |
| (hr : 0≤r) (hrq : r≤q) (hq : q≤1) : |
| β^2*(q-r)*scalarCDFSusceptibilitySquareAverage β α r ≤ |
| (1+2*(β^2*(q-r)))*(scalarCDFOverlap β α q-scalarCDFOverlap β α r) := by |
| let a (n : ℕ) := quantileCDF n (uniformCDFQuantiles n α) |
| have hn (n : ℕ) := quantileCDF_bounds n (uniformCDFQuantiles n α) |
| have hnm (n : ℕ) := quantileCDF_monotone n (uniformCDFQuantiles n α) |
| have hD := uniformCDFQuantiles_L1_tendsto α ha h1 |
| have Hχ := scalarCDFSusceptibilitySquareAverage_tendsto β ha α.mono hn hnm hD ⟨hr,hrq.trans hq⟩ |
| have HΓ := scalarCDFOverlap_tendstoUniformlyOn β ha α.mono hn hnm hD |
| have HL := Hχ.const_mul (β^2*(q-r)) |
| have HR := ((HΓ.tendsto_at ⟨hr.trans hrq,hq⟩).sub (HΓ.tendsto_at ⟨hr,hrq.trans hq⟩)).const_mul (1+2*(β^2*(q-r))) |
| exact le_of_tendsto_of_tendsto HL HR (Eventually.of_forall (fun n => |
| scalarCDF_conditional_stability_quantile β (uniformCDFQuantiles n α) |
| (uniformCDFQuantiles_admissible n α h1) hr hrq hq)) |
| |
| end SK.Analytic |
| |
| end |
| end |
| |
| end OAI |
| |
