| import Mathlib |
| import OAI.Probability.SKGap.Matrix.MatrixWhitening |
| import OAI.Probability.SKGap.Localization.BilinearUnion |
| import OAI.Probability.SKGap.Localization.GramUniformRobust |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real |
| open scoped BigOperators |
| variable {ι : Type*} |
|
|
| def conditionalColumns (X M G N : ι → ℝ) : Matrix ι ((Fin 3) ⊕ Unit) ℝ := |
| fun i k => Sum.elim (fun l => ![X i,M i,G i] l) (fun _ => N i) k |
|
|
| def conditionalCoupling (c : ℝ) : Matrix (Fin 3) Unit ℝ := fun k _ => ![0,c,0] k |
|
|
| def conditionalCoefficient (j cM cG cN : ℝ) : |
| Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ := |
| fromBlocks !![j,0,0;0,cM,cG;0,cG,0] (conditionalCoupling cN) |
| (conditionalCoupling cN)ᵀ 0 |
|
|
| lemma conditionalCoefficient_hermitian (j cM cG cN : ℝ) : |
| (conditionalCoefficient j cM cG cN).IsHermitian := by |
| ext i k |
| cases i with |
| | inl i => cases k with |
| | inl k => fin_cases i <;> fin_cases k <;> simp [conditionalCoefficient,Matrix.conjTranspose_apply] |
| | inr k => simp [conditionalCoefficient,Matrix.conjTranspose_apply] |
| | inr i => cases k <;> simp [conditionalCoefficient,Matrix.conjTranspose_apply] |
|
|
| lemma conditional_rank_expansion (X M G N : ι → ℝ) (j cM cG cN : ℝ) : |
| conditionalColumns X M G N*conditionalCoefficient j cM cG cN* |
| (conditionalColumns X M G N)ᵀ = |
| j • vecMulVec X X+cM • vecMulVec M M+ |
| cG • (vecMulVec M G+vecMulVec G M)+cN • (vecMulVec M N+vecMulVec N M) := by |
| ext i k |
| simp [Matrix.mul_apply,Fintype.sum_sum_type,Fin.sum_univ_succ, |
| conditionalColumns,conditionalCoefficient,conditionalCoupling,vecMulVec_apply] |
| ring |
|
|
| lemma conditional_schur_coefficient {j b s q κ : ℝ} (hb : 0 < b) (hj : 0 < j) |
| (hs : 0 < s) (hq : 0 < q) (hk : κ^2=1-j*q/s) : |
| (2*j*q+2*(j*b)*(κ-1))/q = j*(2-j*b/s)-((κ-1)*√(j*b)/√q)^2 := by |
| have hB : 0 < j*b := mul_pos hj hb |
| have hq0 : √q ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hq) |
| rw [div_pow,mul_pow,Real.sq_sqrt hB.le,Real.sq_sqrt hq.le] |
| field_simp |
| have hk' : κ^2*s=s-j*q := by |
| calc |
| _ = (1-j*q/s)*s := by rw [hk] |
| _ = _ := by field_simp |
| nlinarith [hk'] |
|
|
| lemma conditionalCoefficient_at_equality {j b s q κ : ℝ} (hb : 0 < b) (hj : 0 < j) |
| (hs : 0 < s) (hq : 0 < q) (hk : κ^2=1-j*q/s) : |
| conditionalCoefficient j ((2*j*q+2*(j*b)*(κ-1))/q) j ((κ-1)*√(j*b)/√q)= |
| augmentedCoefficient (rankCoefficient j b s) (conditionalCoupling ((κ-1)*√(j*b)/√q)) := by |
| have h := conditional_schur_coefficient hb hj hs hq hk |
| ext i k |
| cases i with |
| | inl i => cases k with |
| | inl k => fin_cases i <;> fin_cases k <;> |
| simp [conditionalCoefficient,augmentedCoefficient,rankCoefficient,conditionalCoupling, |
| Matrix.mul_apply,h,pow_two] |
| | inr k => simp [conditionalCoefficient,augmentedCoefficient] |
| | inr i => cases k <;> simp [conditionalCoefficient,augmentedCoefficient] |
|
|
| theorem conditional_rank_change_basis (X U Z V : ι → ℝ) (j cU cZ cV : ℝ) |
| {q s B : ℝ} (hq : 0 < q) (hs : 0 < s) (hB : 0 < B) : |
| let Q := conditionalColumns X (√q • U) ((√s)⁻¹ • Z) ((√B)⁻¹ • (V-B • U)) |
| Q*conditionalCoefficient j ((cU+2*B*cV)/q) (cZ*√s/√q) (cV*√B/√q)*Qᵀ = |
| j • vecMulVec X X+cU • vecMulVec U U+ |
| cZ • (vecMulVec U Z+vecMulVec Z U)+cV • (vecMulVec U V+vecMulVec V U) := by |
| dsimp only |
| rw [conditional_rank_expansion] |
| have hq0 : √q ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hq) |
| have hs0 : √s ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hs) |
| have hB0 : √B ≠ 0 := ne_of_gt (Real.sqrt_pos.2 hB) |
| ext i k |
| simp only [Matrix.add_apply,Matrix.smul_apply,vecMulVec_apply,Pi.smul_apply,Pi.sub_apply, |
| smul_eq_mul] |
| field_simp |
| rw [Real.sq_sqrt hq.le]; ring |
| end SKGap |
| end |
| end |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Set Real |
| open scoped MatrixOrder Matrix.Norms.Frobenius |
|
|
| theorem compact_conditional_rank_criterion {α : Type*} [TopologicalSpace α] [CompactSpace α] |
| {j : ℝ} (hj : 0 < j) (hj1 : j < 1) (b e s q κ : α → ℝ) |
| (hb : Continuous b) (he : Continuous e) (hs : Continuous s) |
| (hq : Continuous q) (hκ : 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) |
| (hκsq : ∀ x, κ x^2=1-j*q x/s x) : |
| ∃ δ ρ : ℝ, 0 < δ ∧ δ < 1 ∧ 0 < ρ ∧ |
| ∀ x (ι : Type) [Fintype ι] [DecidableEq ι] |
| (S D : Matrix ι ι ℝ) (V : Matrix ι ((Fin 3) ⊕ Unit) ℝ) |
| (C : Matrix ((Fin 3) ⊕ Unit) ((Fin 3) ⊕ Unit) ℝ) (c : ℝ), |
| S.PosDef → D.IsHermitian → C.IsHermitian → |
| (S-c • (1 : Matrix ι ι ℝ)).PosSemidef → |
| ‖Vᵀ*D*S⁻¹*D*V-augmentedGram (ν := Unit) (limitingGram (b x) (e x) (s x))‖ < ρ → |
| ‖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 → |
| (S-D*V*C*Vᵀ*D-(δ*c) • (1 : Matrix ι ι ℝ)).PosDef := by |
| let coupling := fun x => conditionalCoupling ((κ x-1)*√(j*b x)/√(q x)) |
| have hcoupling : Continuous coupling := by |
| unfold coupling conditionalCoupling |
| fun_prop (disch := intro x; exact ne_of_gt (Real.sqrt_pos.2 (hq0 x))) |
| obtain ⟨δ,ρ,hδ,hδ1,hρ,hcrit⟩ := compact_limiting_augmented_gram_robust hj hj1 |
| b e s coupling hb he hs hcoupling hb1 he0 (fun x => (hs0 x).ne') |
| refine ⟨δ,ρ,hδ,hδ1,hρ,?_⟩ |
| intro x ι _ _ S D V C c hS hD hC hc hg hcc |
| obtain ⟨R,hR,hRR,hu,hgram⟩ := positive_whitening_gram hS D hD V |
| have hcc' : ‖C-augmentedCoefficient (rankCoefficient j (b x) (s x)) (coupling x)‖ < ρ := by |
| rw [conditionalCoefficient_at_equality (hb0 x) hj (hs0 x) (hq0 x) (hκsq x)] at hcc |
| exact hcc |
| have hh := hcrit x ι (R⁻¹*D*V) C hC (by rw [hgram]; exact hg) hcc' |
| have hp := positive_unwhitening hR hD hRR hu V C δ hh |
| have hp' := hp.add_posSemidef (hc.smul hδ.le) |
| convert hp' using 1 |
| rw [smul_sub,smul_smul,sub_smul,one_smul] |
| abel |
| end SKGap |
| end |
| end |
| |
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real Set |
| open scoped BigOperators MatrixOrder Matrix.Norms.Frobenius |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
| |
| omit [DecidableEq ι] in |
| lemma quadratic_lift_real (M : Matrix ι ι ℝ) (v : EuclideanSpace ℝ ι) : |
| ComplexSpectral.quadratic (RealComplex.liftMatrix M) (RealComplex.liftVec v)=rayleigh M v := by |
| simp [ComplexSpectral.quadratic,RealComplex.liftMatrix,RealComplex.liftVec, |
| rayleigh,dotProduct,mulVec] |
| |
| lemma complex_lowerRayleigh_margin {M : Matrix ι ι ℝ} (hM : M.IsHermitian) |
| {c : ℝ} (hc : c < ComplexSpectral.lowerRayleigh (RealComplex.liftMatrix M)) : |
| (M-c • (1 : Matrix ι ι ℝ)).PosDef := by |
| apply Matrix.PosDef.of_dotProduct_mulVec_pos (hM.sub |
| (Matrix.isHermitian_one.smul (R := ℝ) (isSelfAdjoint_iff.mpr (star_trivial c)))) |
| intro x hx |
| apply rayleigh_pos_of_unit (M := M-c • (1 : Matrix ι ι ℝ)) (fun v hv => ?_) |
| (v := WithLp.toLp 2 x) (fun hz => hx (congrArg WithLp.ofLp hz)) |
| have hv' : ‖v‖=1 := hv |
| have hh := ComplexSpectral.lowerRayleigh_le (RealComplex.liftMatrix M) |
| (x := RealComplex.liftVec v) (by |
| simp only [ComplexSpectral.unitSphere,Metric.mem_sphere,dist_zero_right, |
| RealComplex.norm_liftVec,hv']) |
| rw [quadratic_lift_real] at hh |
| have hsq : v.ofLp⬝ᵥv.ofLp=1 := by |
| have hn := EuclideanSpace.real_norm_sq_eq v |
| rw [hv'] at hn |
| simpa only [dotProduct,pow_two,one_mul] using hn.symm |
| change 0 < v.ofLp⬝ᵥ((M-c • (1 : Matrix ι ι ℝ))*ᵥv.ofLp) |
| rw [Matrix.sub_mulVec,Matrix.smul_mulVec,Matrix.one_mulVec,dotProduct_sub, |
| dotProduct_smul,smul_eq_mul,hsq,mul_one] |
| exact sub_pos.mpr (lt_of_lt_of_le hc hh) |
|
|
| theorem bilinearPathGood_margin {n : ℕ} {j A : ℝ} (a : Fin n → ℝ) |
| (g : MatrixCoordinates (Fin n) → ℝ) (hg : bilinearPathGood j A a g) : |
| ((1-pathDiagonal a 1*(goeMatrix (j/(n:ℝ)) g- |
| ((j/(n:ℝ))*∑ i,a i) • 1)*pathDiagonal a 1)- |
| ((1-sqrt j*A)^2/4) • (1 : Matrix (Fin n) (Fin n) ℝ)).PosDef := by |
| have hh := hg 1 ⟨by norm_num,le_rfl⟩ |
| change _ ∧ _ ∧ _ at hh |
| have he : pathShift 1 ((j/(n:ℝ))*∑ i,a i)=((j/(n:ℝ))*∑ i,a i) • |
| (1 : Matrix (Fin n) (Fin n) ℝ) := by |
| ext i k |
| simp [pathShift,Matrix.diagonal,Matrix.one_apply] |
| rw [he] at hh |
| apply complex_lowerRayleigh_margin ?_ hh.2.1 |
| apply Matrix.isHermitian_one.sub |
| have hD : (pathDiagonal a 1).IsHermitian := by |
| exact isHermitian_diagonal_iff.mpr (fun i => isSelfAdjoint_iff.mpr (star_trivial _)) |
| have hW : (goeMatrix (j/(n:ℝ)) g).IsHermitian := by |
| change (goeMatrix (j/(n:ℝ)) g)ᴴ=goeMatrix (j/(n:ℝ)) g |
| simpa only [Matrix.conjTranspose_eq_transpose_of_trivial] using goeMatrix_transpose (j/(n:ℝ)) g |
| change (pathDiagonal a 1*(goeMatrix (j/(n:ℝ)) g-((j/(n:ℝ))*∑ i,a i) • 1)*pathDiagonal a 1).IsHermitian |
| unfold Matrix.IsHermitian |
| simp only [Matrix.conjTranspose_mul,Matrix.conjTranspose_sub,Matrix.conjTranspose_smul, |
| Matrix.conjTranspose_one,hD.eq,hW.eq,star_trivial,Matrix.mul_assoc] |
| end SKGap |
| end |
| end |
|
|
| end OAI |
|
|
