| import Mathlib |
| import OAI.Probability.SKGap.Gaussian.GOEScalarTail |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix MeasureTheory ProbabilityTheory Real Set |
| open scoped BigOperators MatrixOrder Matrix.Norms.Frobenius |
|
|
| theorem compact_conditional_uniform_tail {α : Type*} [TopologicalSpace α] [CompactSpace α] |
| {j A : ℝ} (hj : 0 < j) (hj1 : j < 1) (hA : 0 < A) |
| (hsub : sqrt j*A < 1) |
| (b e s q κ : α → ℝ) (hb : Continuous b) (he : Continuous e) |
| (hs : Continuous s) (hq : Continuous q) (hk : Continuous κ) |
| (hb0 : ∀ x,0 < b x) (hb1 : ∀ x,b x ≤ 1) (he0 : ∀ x,0 ≤ e x) |
| (hs0 : ∀ x,0 < s x) (hq0 : ∀ x,0 < q x) |
| (hkq : ∀ x,κ x^2=1-j*q x/s x) : |
| ∃ δ ρ : ℝ, 0 < δ ∧ δ < 1 ∧ 0 < ρ ∧ |
| ∀ U : ℝ, 0 ≤ U → ∀ ε τ : ℝ, 0 < ε → 0 < τ → ∃ c : ℝ, ∃ N : ℕ, 0 < c ∧ 0 < N ∧ |
| ∀ n, N ≤ n → ∀ a : Fin n → ℝ, (∀ i,0 ≤ a i) → (∀ i,a i ≤ A) → |
| 0 < j/(n:ℝ)*∑ i,a i → |
| augmentedGramTolerance (j/(n:ℝ)*∑ i,a i) ε < ρ/2 → |
| ∀ p : Fin 3 → EuclideanSpace ℝ (Fin n), ∀ u : EuclideanSpace ℝ (Fin n), |
| (∀ r,‖p r‖ ≤ U) → ‖u‖ ≤ U → u.ofLp⬝ᵥu.ofLp=1 → |
| ∀ x (C E : Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ), |
| C.IsHermitian → E.IsHermitian → τ*‖E‖ < ρ/2 → |
| ‖diagonalAugmentedGram a p-augmentedGram (ν := Unit) (limitingGram (b x) (e x) (s x))‖ < ρ/2 → |
| ‖C-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) |
| j ((κ x-1)*√(j*b x)/√(q x))‖ < ρ/2 → |
| (Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1)).real |
| {g | ¬(conditionalBase j a g- |
| pathDiagonal a 1*conditionalUpdateColumns j a p u g* |
| (C+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E)* |
| (conditionalUpdateColumns j a p u g)ᵀ*pathDiagonal a 1- |
| (δ*((1-sqrt j*A)^2/4)) • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef} ≤ |
| 101*exp (-c*(n:ℝ)) := by |
| obtain ⟨δ,ρ,hδ,hδ1,hρ,hcrit⟩ := compact_conditional_rank_criterion hj hj1 b e s q κ |
| hb he hs hq hk hb0 hb1 he0 hs0 hq0 hkq |
| refine ⟨δ,ρ,hδ,hδ1,hρ,?_⟩ |
| intro U hU ε τ hε hτ |
| obtain ⟨c₀,N,hc₀,hN,hgram⟩ := conditional_augmented_gram_tail hj hA hsub hU hε |
| let c₁ := τ^2/(π^2*(2*(U+1)^2)^2*j) |
| have hc₁ : 0 < c₁ := by dsimp [c₁]; positivity |
| refine ⟨min c₀ c₁,N,lt_min hc₀ hc₁,hN,?_⟩ |
| intro n hn a ha haA hB heps p u hp hu hunit x C E hC hE hEbound hdiag hcoef |
| let μ := Measure.pi (fun _ : MatrixCoordinates (Fin n) => gaussianReal 0 1) |
| let badG := {g | ¬bilinearPathGood j A a g ∨ |
| augmentedGramTolerance (j/(n:ℝ)*∑ i,a i) ε < |
| ‖actualAugmentedGram j a p u g-diagonalAugmentedGram a p‖} |
| let badS := {g | τ ≤ |matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u|} |
| have hnpos := hN.trans_le hn |
| have hgprob : μ.real badG ≤ 99*exp (-c₀*(n:ℝ)) := hgram n hn a ha haA hB p u hp hu hunit |
| have hsprob : μ.real badS ≤ 2*exp (-c₁*(n:ℝ)) := by |
| have hh := goe_scalar_tail hnpos hj hU hτ.le u hu |
| convert hh using 1 |
| congr 2 |
| dsimp [c₁] |
| ring |
| have hsubsets : {g | ¬(conditionalBase j a g- |
| pathDiagonal a 1*conditionalUpdateColumns j a p u g* |
| (C+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E)* |
| (conditionalUpdateColumns j a p u g)ᵀ*pathDiagonal a 1- |
| (δ*((1-sqrt j*A)^2/4)) • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef} ⊆ badG ∪ badS := by |
| intro g hg |
| by_contra hbad |
| have hgG : g ∉ badG := fun hh => hbad (Or.inl hh) |
| have hgS : g ∉ badS := fun hh => hbad (Or.inr hh) |
| simp only [badG,mem_ofPred_eq,not_or,not_lt] at hgG |
| have hgS' : |matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u| < τ := lt_of_not_ge hgS |
| let D := pathDiagonal a 1 |
| let S := conditionalBase j a g |
| let V := conditionalUpdateColumns j a p u g |
| let Cg := C+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E |
| let lo := (1-sqrt j*A)^2/4 |
| have hmargin : (S-lo • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef := |
| bilinearPathGood_margin a g (not_not.mp hgG.1) |
| have hlo : 0 ≤ lo := by dsimp [lo]; positivity |
| have hSpos : S.PosDef := by |
| have hh := hmargin.add_posSemidef ((Matrix.PosSemidef.one : (1 : Matrix (Fin n) (Fin n) ℝ).PosSemidef).smul hlo) |
| convert hh using 1; abel |
| have hD : D.IsHermitian := isHermitian_diagonal_iff.mpr |
| (fun i => isSelfAdjoint_iff.mpr (star_trivial _)) |
| have hCg : Cg.IsHermitian := hC.add (hE.smul (R := ℝ) |
| (isSelfAdjoint_iff.mpr (star_trivial _))) |
| have hG : ‖Vᵀ*D*S⁻¹*D*V-augmentedGram (ν := Unit) (limitingGram (b x) (e x) (s x))‖ < ρ := by |
| have hh := norm_sub_le_norm_sub_add_norm_sub (actualAugmentedGram j a p u g) (diagonalAugmentedGram a p) |
| (augmentedGram (ν := Unit) (limitingGram (b x) (e x) (s x))) |
| have heq : Vᵀ*D*S⁻¹*D*V=actualAugmentedGram j a p u g := by |
| simp only [V,D,S,conditionalUpdateColumns,conditionalBase,actualAugmentedGram, |
| exactG,Fintype.card_fin,Matrix.mul_assoc] |
| rw [heq] |
| exact hh.trans_lt (by linarith [hgG.2]) |
| have hCgclose : ‖Cg-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) |
| j ((κ x-1)*√(j*b x)/√(q x))‖ < ρ := by |
| have hn : ‖matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E‖ ≤ τ*‖E‖ := by |
| rw [norm_smul,Real.norm_eq_abs] |
| exact mul_le_mul_of_nonneg_right hgS'.le (norm_nonneg _) |
| have hh := norm_add_le (C-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) |
| j ((κ x-1)*√(j*b x)/√(q x))) (matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E) |
| have heq : Cg-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) |
| j ((κ x-1)*√(j*b x)/√(q x))= |
| (C-conditionalCoefficient j ((2*j*q x+2*(j*b x)*(κ x-1))/q x) |
| j ((κ x-1)*√(j*b x)/√(q x)))+matrixBilinear (goeMatrix (j/(n:ℝ)) g) u u • E := by |
| dsimp [Cg]; abel |
| rw [heq] |
| exact hh.trans_lt (by linarith) |
| exact hg (hcrit x (Fin n) S D V Cg lo hSpos hD hCg hmargin.posSemidef hG hCgclose) |
| apply (measureReal_mono hsubsets (measure_ne_top _ _)).trans |
| apply (measureReal_union_le badG badS).trans |
| apply (add_le_add hgprob hsprob).trans |
| have he₀ : exp (-c₀*(n:ℝ)) ≤ exp (-min c₀ c₁*(n:ℝ)) := by |
| apply exp_le_exp.mpr; simpa only [neg_mul] using neg_le_neg (mul_le_mul_of_nonneg_right (min_le_left c₀ c₁) (Nat.cast_nonneg n : (0:ℝ) ≤ n)) |
| have he₁ : exp (-c₁*(n:ℝ)) ≤ exp (-min c₀ c₁*(n:ℝ)) := by |
| apply exp_le_exp.mpr; simpa only [neg_mul] using neg_le_neg (mul_le_mul_of_nonneg_right (min_le_right c₀ c₁) (Nat.cast_nonneg n : (0:ℝ) ≤ n)) |
| nlinarith |
| end SKGap |
| end |
| end |
|
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| end OAI |
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