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8e4d533 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 | import OAI.Geometry.Convex.GeneralMahler.Linear.SpectralIBP
namespace OAI
/-! §06. Commuting normalization controls weighted norm. -/
noncomputable section
open Set Filter MeasureTheory MeasureTheory.Measure Matrix Real Metric
open scoped Topology NNReal ENNReal RealInnerProductSpace MatrixOrder Matrix.Norms.L2Operator
namespace GeneralMahler
open HMode Profile Layers
variable {m:ℕ}
lemma sum_hsn (X Y:Mat m) : hsN (X+Y)≤125/100*hsN X+5*hsN Y := by
simp only [hsN_eq]
rw [mats_s2,mats_s2,← mats_add]
apply mats_le; intro i
rw [← Finset.sum_add_distrib]
apply Finset.sum_le_sum; intro j _
rw [Matrix.add_apply]
nlinarith [sq_nonneg (X i j-4*Y i j)]
lemma H_J_frame {X W:Mat m} (u:Frm m) (l:Fin m→ℝ) (h:u.loc W=diagonal l):
hsN (jprod W X)=mats (fun i=>∑ j,((l i+l j)/2)^2*(u.loc X i j)^2) := by
rw [← hsN_loc (jprod W X) u,hsN_eq]; unfold jprod
rw [u.loc_smul,u.loc_add,u.loc_mul,u.loc_mul,h]
congr 1; ext i; apply Finset.sum_congr rfl; intro j _
simp only [Matrix.smul_apply,Matrix.add_apply,smul_eq_mul,Matrix.diagonal_mul,Matrix.mul_diagonal]
ring
lemma Jgap_lower {W:Mat m} (hw:W.IsHermitian) (hu: scalar m mstar ≤ W) (X:Mat m) :
mstar^2*hsN X≤ hsN (jprod W X) := by
let u := Frm.eu hw
let f := u.eig W
have hi : u.loc W=diagonal f := Frm.eu_d hw
have hp (i:Fin m) : mstar≤f i := by
have he : f i∈ spectrum ℝ W := by
unfold f u; rw [hw.spectrum_real_eq_range_eigenvalues,Frm.eu_eig]; exact ⟨_,rfl⟩
exact (spectrum_order W hw (l:=mstar) (r:=mstar)).1.mp hu _ he
rw [H_J_frame u f hi,← hsN_loc X u,hsN_eq,mats_s2]
apply mats_le; intro i; apply Finset.sum_le_sum; intro j _
apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
have hv := hp i; have hf := hp j; unfold mstar at *
nlinarith
lemma Jdev_upper {W:Mat m} (hW:W∈specBox m tmin tmax) (X:Mat m) :
hsN (jprod (W-scalar m tc) X) ≤ Profile.tr^2*hsN X := by
let u := Frm.eu hW.1
let f := u.eig W
have hi : u.loc W=diagonal f := Frm.eu_d hW.1
have hh : u.loc (W-scalar m tc)=diagonal (fun i=>f i-tc) := by
rw [u.loc_sub,hi,u.loc_s]
ext i j; by_cases he:i=j
· subst j; simp [scalar]
simp [scalar,he]
have hp (i:Fin m) : tmin≤f i ∧ f i≤tmax := by
have he : f i∈ spectrum ℝ W := by
unfold f u; rw [hW.1.spectrum_real_eq_range_eigenvalues,Frm.eu_eig]; exact ⟨_,rfl⟩
have h := spectrum_order W hW.1 (l:=tmin) (r:=tmax)
exact ⟨h.1.mp hW.2.1 _ he,h.2.mp hW.2.2 _ he⟩
rw [H_J_frame u _ hh,← hsN_loc X u,hsN_eq,mats_s2]
apply mats_le; intro i; apply Finset.sum_le_sum; intro j _
apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
have hv := hp i; have hf := hp j
unfold tmin tmax tc Profile.tr at *; nlinarith [hv.1,hv.2,hf.1,hf.2]
lemma commute_update {C K T:Mat m} (h:updEq C K T=0) (M:Mat m):
jprod (updGap C T) (cmu M T)=cmu M (K+tc • C)+jprod (T-scalar m tc) (cmu M C) := by
have he : K= -(lam • (T*T)+jprod C T) := by
unfold updEq at h
apply eq_neg_of_add_eq_zero_left
rwa [add_comm]
rw [he]
dsimp only [jprod,updGap,cmu]
simp only [add_mul,mul_add, mul_sub,sub_mul,neg_mul,mul_neg,
smul_mul_assoc,mul_smul_comm,scalar,one_mul,mul_one,mul_assoc]
module
theorem Comm_bound {C K T:Mat m} (hc:C.IsHermitian)
(hT:T∈specBox m tmin tmax)
(he:updEq C K T=0) (hH:scalar m mstar ≤ updGap C T) (M:Mat m) :
mstar^2*hsN (cmu M T) ≤
(125/100)*hsN (cmu M (K+tc • C))+ (5*Profile.tr^2)*hsN (cmu M C) := by
have hh : (updGap C T).IsHermitian :=
hc.neg.sub (hT.1.smul (by simp [IsSelfAdjoint]))
have hi := Jgap_lower hh hH (cmu M T)
rw [commute_update he] at hi
apply (hi.trans (sum_hsn _ _)).trans
apply add_le_add le_rfl; rw [mul_assoc]
exact mul_le_mul_of_nonneg_left (Jdev_upper hT _) (by norm_num)
end GeneralMahler
end
end OAI
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