| import Mathlib |
| import OAI.Probability.SKGap.Localization.ConditionalRankAlgebra |
| import OAI.Probability.SKGap.Matrix.GOEBlockExpansion |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| def conditionalBlockMatrix (r j q s S a B κ ell : ℝ) (u Z : ι → ℝ) |
| (g : MatrixCoordinates ι → ℝ) : Matrix ι ι ℝ := |
| goeBlockResidual r κ ell u g+ |
| vecMulVec u ((j*sqrt (q/s)) • Z-(j*a/s) • u)+ |
| vecMulVec ((j*sqrt (q/s)) • Z-(j*a/s) • u) u+ |
| (2*j*(a+B*q)/S) • vecMulVec u u |
|
|
| def conditionalMeanCoefficient (j q s S a B BA κ : ℝ) := |
| conditionalCoefficient j |
| ((2*j*q-2*j*a/s+2*j*(a+B*q)/S+2*BA*(κ-1))/q) |
| j ((κ-1)*sqrt BA/sqrt q) |
|
|
| def conditionalScalarCoefficient (q κ ell : ℝ) := |
| conditionalCoefficient 0 ((ell-2*κ+1)/q) 0 0 |
|
|
| lemma conditionalCoefficient_add_scalar (j cM cG cN z w : ℝ) : |
| conditionalCoefficient j cM cG cN+ |
| z • conditionalCoefficient 0 w 0 0= |
| conditionalCoefficient j (cM+z*w) cG cN := by |
| ext i k |
| cases i with |
| | inl i => cases k with |
| | inl k => fin_cases i <;> fin_cases k <;> |
| simp [conditionalCoefficient] |
| | inr k => fin_cases i <;> simp [conditionalCoefficient,conditionalCoupling] |
| | inr i => cases k with |
| | inl k => fin_cases k <;> simp [conditionalCoefficient,conditionalCoupling] |
| | inr k => simp [conditionalCoefficient] |
|
|
| lemma conditionalMeanCoefficient_equality {j q s b κ : ℝ} |
| (hs : 0 < s) (hq : 0 < q) (hj : 0 ≤ j) : |
| conditionalMeanCoefficient j q s (s+j*q) (s*b) (j*b) (j*b) κ= |
| conditionalCoefficient j ((2*j*q+2*(j*b)*(κ-1))/q) j ((κ-1)*sqrt (j*b)/sqrt q) := by |
| have hS : s+j*q ≠ 0 := ne_of_gt (add_pos_of_pos_of_nonneg hs (mul_nonneg hj hq.le)) |
| unfold conditionalMeanCoefficient |
| congr 1 |
| field_simp |
| ring |
|
|
| theorem conditional_hessian_rank_form (r j a B κ ell : ℝ) |
| {q s S BA : ℝ} (hq : 0 < q) (hs : 0 < s) (hBA : 0 < BA) |
| (X u Z : ι → ℝ) (D : Matrix ι ι ℝ) (g : MatrixCoordinates ι → ℝ) : |
| let W := goeMatrix r g |
| let Q := conditionalColumns X (sqrt q • u) ((sqrt s)⁻¹ • Z) |
| (residualGramVector BA W u) |
| 1+D*(B • (1 : Matrix ι ι ℝ)-conditionalBlockMatrix r j q s S a B κ ell u Z g- |
| j • vecMulVec X X-(2*j*q) • vecMulVec u u)*D= |
| (1-D*(W-BA • (1 : Matrix ι ι ℝ))*D)- |
| D*Q*(conditionalMeanCoefficient j q s S a B BA κ+ |
| goeBilinear r u u g • conditionalScalarCoefficient q κ ell)*Qᵀ*D+ |
| (B-BA) • (D*D) := by |
| dsimp only |
| have hroot : sqrt (q/s)*sqrt s/sqrt q=1 := by |
| rw [Real.sqrt_div hq.le] |
| field_simp |
| have hz : j*sqrt (q/s)*sqrt s/sqrt q=j := by |
| calc |
| _ = j*(sqrt (q/s)*sqrt s/sqrt q) := by ring |
| _ = _ := by rw [hroot,mul_one] |
| have hrank := conditional_rank_change_basis X u Z (goeMatrix r g*ᵥu) j |
| (2*j*q-2*j*a/s+2*j*(a+B*q)/S+(ell-2*κ+1)*goeBilinear r u u g) |
| (j*sqrt (q/s)) (κ-1) hq hs hBA |
| dsimp only at hrank |
| rw [hz] at hrank |
| have hc : conditionalMeanCoefficient j q s S a B BA κ+ |
| goeBilinear r u u g • conditionalScalarCoefficient q κ ell= |
| conditionalCoefficient j |
| ((2*j*q-2*j*a/s+2*j*(a+B*q)/S+(ell-2*κ+1)*goeBilinear r u u g+2*BA*(κ-1))/q) |
| j ((κ-1)*sqrt BA/sqrt q) := by |
| unfold conditionalMeanCoefficient conditionalScalarCoefficient |
| rw [conditionalCoefficient_add_scalar] |
| congr 1 |
| ring |
| rw [hc] |
| change _ = (1-D*(goeMatrix r g-BA • (1 : Matrix ι ι ℝ))*D)-D*_*_*_*D+_ |
| have hrank' := congrArg (fun M => D*M*D) hrank |
| simp only [Matrix.mul_assoc] at hrank' |
| simp only [residualGramVector] at * |
| rw [show D*conditionalColumns X (sqrt q • u) ((sqrt s)⁻¹ • Z) |
| ((sqrt BA)⁻¹ • (goeMatrix r g*ᵥu-BA • u))* |
| conditionalCoefficient j |
| ((2*j*q-2*j*a/s+2*j*(a+B*q)/S+(ell-2*κ+1)*goeBilinear r u u g+2*BA*(κ-1))/q) |
| j ((κ-1)*sqrt BA/sqrt q)* |
| (conditionalColumns X (sqrt q • u) ((sqrt s)⁻¹ • Z) |
| ((sqrt BA)⁻¹ • (goeMatrix r g*ᵥu-BA • u)))ᵀ*D= |
| D*(j • vecMulVec X X+ |
| (2*j*q-2*j*a/s+2*j*(a+B*q)/S+(ell-2*κ+1)*goeBilinear r u u g) • vecMulVec u u+ |
| (j*sqrt (q/s)) • (vecMulVec u Z+vecMulVec Z u)+ |
| (κ-1) • (vecMulVec u (goeMatrix r g*ᵥu)+vecMulVec (goeMatrix r g*ᵥu) u))*D by |
| simpa only [Matrix.mul_assoc] using hrank'] |
| unfold conditionalBlockMatrix |
| rw [goeBlockResidual_expand] |
| have he : vecMulVec u ((j*sqrt (q/s)) • Z-(j*a/s) • u)+ |
| vecMulVec ((j*sqrt (q/s)) • Z-(j*a/s) • u) u= |
| (j*sqrt (q/s)) • (vecMulVec u Z+vecMulVec Z u)-(2*j*a/s) • vecMulVec u u := by |
| ext i k |
| simp only [Matrix.add_apply,Matrix.sub_apply,Matrix.smul_apply,vecMulVec_apply, |
| Pi.sub_apply,Pi.smul_apply,smul_eq_mul] |
| ring |
| have hcombine : goeMatrix r g+ |
| (κ-1) • (vecMulVec u (goeMatrix r g*ᵥu)+vecMulVec (goeMatrix r g*ᵥu) u)+ |
| ((ell-2*κ+1)*goeBilinear r u u g) • vecMulVec u u+ |
| vecMulVec u ((j*sqrt (q/s)) • Z-(j*a/s) • u)+ |
| vecMulVec ((j*sqrt (q/s)) • Z-(j*a/s) • u) u= |
| goeMatrix r g+(κ-1) • (vecMulVec u (goeMatrix r g*ᵥu)+vecMulVec (goeMatrix r g*ᵥu) u)+ |
| ((ell-2*κ+1)*goeBilinear r u u g) • vecMulVec u u+ |
| ((j*sqrt (q/s)) • (vecMulVec u Z+vecMulVec Z u)-(2*j*a/s) • vecMulVec u u) := by |
| rw [← he] |
| abel |
| rw [hcombine] |
| simp only [Matrix.mul_add,Matrix.mul_sub,Matrix.add_mul,Matrix.sub_mul,Matrix.mul_smul, |
| Matrix.smul_mul,Matrix.mul_one,sub_smul,add_smul] |
| abel |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Real Matrix MeasureTheory ProbabilityTheory Set Filter Topology |
| open scoped BigOperators Matrix.Norms.Frobenius |
|
|
| def conditionalKappa (j : ℝ) (p : ScalarPoint) : ℝ := sqrt ((p.2.1+p.2.2^2)/scalarS j p) |
| def conditionalEll (j : ℝ) (p : ScalarPoint) : ℝ := |
| sqrt ((p.2.1+p.2.2^2)/(scalarS j p+j*scalarQMoment p.1)) |
| def conditionalMeanData (j : ℝ) (p : ScalarPoint) (r : ℝ) : |
| Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ := |
| conditionalMeanCoefficient j (scalarQMoment p.1) (scalarS j p) |
| (scalarS j p+j*scalarQMoment p.1) (scalarA j p) |
| (j*scalarBMoment p.1) (j*scalarBMoment p.1+r) (conditionalKappa j p) |
| def conditionalScalarData (j : ℝ) (p : ScalarPoint) : |
| Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ := |
| conditionalScalarCoefficient (scalarQMoment p.1) (conditionalKappa j p) (conditionalEll j p) |
| def curveKappa (j t : ℝ) : ℝ := sqrt (t/scalarFixedPoint j t) |
| def curveCoefficient (j t : ℝ) : Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ := |
| conditionalCoefficient j |
| ((2*j*scalarQ j t+2*(j*scalarBMoment (scalarCurve j t))*(curveKappa j t-1))/scalarQ j t) |
| j ((curveKappa j t-1)*sqrt (j*scalarBMoment (scalarCurve j t))/sqrt (scalarQ j t)) |
|
|
| lemma conditionalKappa_curve {j t : ℝ} (hj : 0 < j) (hj1 : j < 1) (ht : 0 ≤ t) : |
| conditionalKappa j (scalarCurve j t,t,0)=curveKappa j t := by |
| simp only [conditionalKappa,curveKappa,scalarS_curve hj hj1 ht,zero_pow (by decide : 2 ≠ 0),add_zero] |
|
|
| lemma curveKappa_sq {j t : ℝ} (hj : 0 < j) (hj1 : j < 1) (ht : 0 < t) : |
| curveKappa j t^2=1-j*scalarQ j t/scalarFixedPoint j t := by |
| have hs := (scalar_fixed_pos hj hj1 ht).1 |
| unfold curveKappa |
| rw [Real.sq_sqrt (div_nonneg ht.le hs.le)] |
| have he := (scalarFixedPoint_spec hj hj1 ht.le).2 |
| change scalarFixedPoint j t=t+j*scalarQ j t at he |
| field_simp |
| linarith |
|
|
| lemma conditionalMeanData_curve {j t : ℝ} (hj : 0 < j) (hj1 : j < 1) (ht : 0 < t) : |
| conditionalMeanData j (scalarCurve j t,t,0) 0=curveCoefficient j t := by |
| unfold conditionalMeanData curveCoefficient |
| rw [scalarQMoment_curve hj hj1 ht.le,scalarS_curve hj hj1 ht.le, |
| scalarA_curve hj hj1 ht,conditionalKappa_curve hj hj1 ht.le,add_zero] |
| exact conditionalMeanCoefficient_equality (scalar_fixed_pos hj hj1 ht).1 |
| (scalar_fixed_pos hj hj1 ht).2 hj.le |
|
|
| lemma continuousAt_conditionalCoefficient {X : Type*} [TopologicalSpace X] |
| {j a b c : X → ℝ} {x : X} (hj : ContinuousAt j x) (ha : ContinuousAt a x) |
| (hb : ContinuousAt b x) (hc : ContinuousAt c x) : |
| ContinuousAt (fun x=>conditionalCoefficient (j x) (a x) (b x) (c x)) x := by |
| apply continuousAt_pi.mpr |
| intro i |
| apply continuousAt_pi.mpr |
| intro k |
| cases i with |
| | inl i => cases k with |
| | inl k => fin_cases i <;> fin_cases k <;> |
| simp only [conditionalCoefficient,fromBlocks_apply₁₁,Matrix.of_apply,Matrix.cons_val,Fin.reduceFinMk] <;> first | exact hj | exact ha | exact hb | exact continuousAt_const |
| | inr k => fin_cases i <;> simp [conditionalCoefficient,conditionalCoupling] <;> first | exact hc | exact continuousAt_const |
| | inr i => cases k with |
| | inl k => fin_cases k <;> simp [conditionalCoefficient,conditionalCoupling] <;> first | exact hc | exact continuousAt_const |
| | inr k => simp [conditionalCoefficient]; exact continuousAt_const |
|
|
| lemma continuousAt_conditionalMeanCoefficient {p : ℝ × (ℝ × (ℝ × (ℝ × (ℝ × (ℝ × ℝ)))))} |
| (hq : 0 < p.1) (hs : p.2.1 ≠ 0) (hS : p.2.2.1 ≠ 0) (j : ℝ) : |
| ContinuousAt (fun p : ℝ × (ℝ × (ℝ × (ℝ × (ℝ × (ℝ × ℝ))))) => |
| conditionalMeanCoefficient j p.1 p.2.1 p.2.2.1 p.2.2.2.1 p.2.2.2.2.1 |
| p.2.2.2.2.2.1 p.2.2.2.2.2.2) p := by |
| have hq0 := hq.ne' |
| have hsq := (sqrt_pos.mpr hq).ne' |
| apply continuousAt_pi.mpr |
| intro i |
| apply continuousAt_pi.mpr |
| intro k |
| cases i with |
| | inl i => cases k with |
| | inl k => fin_cases i <;> fin_cases k <;> |
| simp only [conditionalMeanCoefficient,conditionalCoefficient,fromBlocks_apply₁₁,Matrix.of_apply,Matrix.cons_val,Fin.reduceFinMk] <;> fun_prop (disch := assumption) |
| | inr k => fin_cases i <;> simp [conditionalMeanCoefficient,conditionalCoefficient,conditionalCoupling] <;> fun_prop (disch := assumption) |
| | inr i => cases k with |
| | inl k => fin_cases k <;> simp [conditionalMeanCoefficient,conditionalCoefficient,conditionalCoupling] <;> fun_prop (disch := assumption) |
| | inr k => simp [conditionalMeanCoefficient,conditionalCoefficient]; fun_prop |
|
|
| lemma continuousAt_conditionalScalarCoefficient {p : ℝ × (ℝ × ℝ)} (hq : p.1 ≠ 0) : |
| ContinuousAt (fun p : ℝ × (ℝ × ℝ) => conditionalScalarCoefficient p.1 p.2.1 p.2.2) p := by |
| unfold conditionalScalarCoefficient conditionalCoefficient conditionalCoupling |
| fun_prop (disch := assumption) |
|
|
| lemma continuousAt_conditionalData {X : Type*} [TopologicalSpace X] |
| {R j : ℝ} (hR : 0 ≤ R) (f : X → ScalarPoint) (hf : Continuous f) |
| (hdom : ∀ x,f x ∈ scalarMomentDomain R) (r : X → ℝ) (hr : Continuous r) |
| {x : X} (hq : 0 < scalarQMoment (f x).1) (hs : 0 < scalarS j (f x)) |
| (hS : 0 < scalarS j (f x)+j*scalarQMoment (f x).1) : |
| ContinuousAt (fun x=>conditionalMeanData j (f x) (r x)) x ∧ |
| ContinuousAt (fun x=>conditionalScalarData j (f x)) x := by |
| have hcq : Continuous (fun x=>scalarQMoment (f x).1) := continuous_scalarQMoment.comp (continuous_fst.comp hf) |
| have hcs : Continuous (fun x=>scalarS j (f x)) := (continuous_scalarS j).comp hf |
| have hcS : Continuous (fun x=>scalarS j (f x)+j*scalarQMoment (f x).1) := hcs.add (continuous_const.mul hcq) |
| have hca : Continuous (fun x=>scalarA j (f x)) := (continuousOn_scalarA hR j).comp_continuous hf hdom |
| have hcb : Continuous (fun x=>j*scalarBMoment (f x).1) := continuous_const.mul (continuous_scalarBMoment.comp (continuous_fst.comp hf)) |
| have hcn : Continuous (fun x=>(f x).2.1+(f x).2.2^2) := by fun_prop |
| have hk : ContinuousAt (fun x=>conditionalKappa j (f x)) x := (hcn.continuousAt.div hcs.continuousAt hs.ne').sqrt |
| have hl : ContinuousAt (fun x=>conditionalEll j (f x)) x := (hcn.continuousAt.div hcS.continuousAt hS.ne').sqrt |
| have hq' := hcq.continuousAt (x := x) |
| have hs' := hcs.continuousAt (x := x) |
| have hS' := hcS.continuousAt (x := x) |
| have ha' := hca.continuousAt (x := x) |
| have hb' := hcb.continuousAt (x := x) |
| have hba := hb'.add hr.continuousAt |
| have hM : ContinuousAt (fun x => |
| (2*j*scalarQMoment (f x).1-2*j*scalarA j (f x)/scalarS j (f x)+ |
| 2*j*(scalarA j (f x)+j*scalarBMoment (f x).1*scalarQMoment (f x).1)/(scalarS j (f x)+j*scalarQMoment (f x).1)+ |
| 2*(j*scalarBMoment (f x).1+r x)*(conditionalKappa j (f x)-1))/scalarQMoment (f x).1) x := |
| ((((continuousAt_const.mul hq').sub ((continuousAt_const.mul ha').div hs' hs.ne')).add |
| ((continuousAt_const.mul (ha'.add (hb'.mul hq'))).div hS' hS.ne')).add |
| ((continuousAt_const.mul hba).mul (hk.sub continuousAt_const))).div hq' hq.ne' |
| have hN : ContinuousAt (fun x => (conditionalKappa j (f x)-1)*sqrt (j*scalarBMoment (f x).1+r x)/sqrt (scalarQMoment (f x).1)) x := |
| ((hk.sub continuousAt_const).mul hba.sqrt).div hq'.sqrt (sqrt_pos.mpr hq).ne' |
| constructor |
| · exact continuousAt_conditionalCoefficient continuousAt_const hM continuousAt_const hN |
| · have hM' : ContinuousAt (fun x => (conditionalEll j (f x)-2*conditionalKappa j (f x)+1)/scalarQMoment (f x).1) x := |
| ((hl.sub (continuousAt_const.mul hk)).add continuousAt_const).div hq' hq.ne' |
| exact continuousAt_conditionalCoefficient continuousAt_const hM' continuousAt_const continuousAt_const |
|
|
| lemma continuous_curve_statistics {j t0 T : ℝ} (hj : 0 < j) (hj1 : j < 1) (ht0 : 0 < t0) : |
| Continuous (fun t : Icc t0 T=>scalarFixedPoint j t) ∧ |
| Continuous (fun t : Icc t0 T=>scalarQ j t) ∧ |
| Continuous (fun t : Icc t0 T=>scalarBMoment (scalarCurve j t)) ∧ |
| Continuous (fun t : Icc t0 T=>scalarEMoment (scalarCurve j t)) ∧ |
| Continuous (fun t : Icc t0 T=>curveKappa j t) ∧ |
| Continuous (fun t : Icc t0 T=>curveCoefficient j t) := by |
| let inc : Icc t0 T → Ici (0:ℝ) := fun t=>⟨t,ht0.le.trans t.2.1⟩ |
| have hi : Continuous inc := continuous_subtype_val.subtype_mk _ |
| have hs := (continuous_scalarFixedPoint_nonneg hj hj1).comp hi |
| have hcurve := (continuous_scalarCurve_nonneg hj hj1).comp hi |
| have hq : Continuous (fun t : Icc t0 T=>scalarQ j t) := by |
| convert continuous_scalarQMoment.comp hcurve using 1 |
| funext t |
| exact (scalarQMoment_curve hj hj1 (ht0.le.trans t.2.1)).symm |
| have hb := continuous_scalarBMoment.comp hcurve |
| have he := continuous_scalarEMoment.comp hcurve |
| have hk : Continuous (fun t : Icc t0 T=>curveKappa j t) := |
| (continuous_subtype_val.div hs (fun t=>(scalar_fixed_pos hj hj1 (ht0.trans_le t.2.1)).1.ne')).sqrt |
| refine ⟨hs,hq,hb,he,hk,?_⟩ |
| apply continuous_iff_continuousAt.mpr |
| intro t |
| have hqt : 0 < scalarQ j (t:ℝ) := (scalar_fixed_pos hj hj1 (ht0.trans_le t.2.1)).2 |
| have hq' := hq.continuousAt (x := t) |
| have hb' := hb.continuousAt (x := t) |
| have hk' := hk.continuousAt (x := t) |
| have hM : ContinuousAt (fun t : Icc t0 T => |
| (2*j*scalarQ j t+2*(j*scalarBMoment (scalarCurve j t))*(curveKappa j t-1))/scalarQ j t) t := |
| ((continuousAt_const.mul hq').add |
| ((continuousAt_const.mul (continuousAt_const.mul hb')).mul (hk'.sub continuousAt_const))).div hq' hqt.ne' |
| have hN : ContinuousAt (fun t : Icc t0 T => (curveKappa j t-1)*sqrt (j*scalarBMoment (scalarCurve j t))/sqrt (scalarQ j t)) t := |
| ((hk'.sub continuousAt_const).mul (continuousAt_const.mul hb').sqrt).div hq'.sqrt (sqrt_pos.mpr hqt).ne' |
| exact continuousAt_conditionalCoefficient continuousAt_const hM continuousAt_const hN |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
