| 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 |
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|
| end |
|
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| end OAI |
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