| import Mathlib |
| import OAI.Probability.SKGap.Matrix.MatrixEntryCLM |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| namespace SKGap |
| open Matrix Real |
| open scoped BigOperators Matrix.Norms.Frobenius |
| variable {ι : Type*} [Fintype ι] [DecidableEq ι] |
|
|
| local instance : Fact (1≤(4:ENNReal)) := ⟨by norm_num⟩ |
| def offDiagonalVector (M : Matrix ι ι ℝ) : PiLp 4 (fun _ : ι×ι=>ℝ) := |
| WithLp.toLp 4 (fun p=>if p.1=p.2 then 0 else M p.1 p.2) |
| def diagonalSeminorm (M : Matrix ι ι ℝ) : ℝ := ‖diagonalVector M‖ |
| def offDiagonalSeminorm (M : Matrix ι ι ℝ) : ℝ := ‖offDiagonalVector M‖ |
|
|
| omit [DecidableEq ι] in |
| lemma diagonalSeminorm_sq (M : Matrix ι ι ℝ) : diagonalSeminorm M^2=∑ i,(M i i)^2 := by |
| simp only [diagonalSeminorm,EuclideanSpace.norm_sq_eq,diagonalVector,Real.norm_eq_abs,sq_abs] |
|
|
| lemma offDiagonalSeminorm_fourth (M : Matrix ι ι ℝ) : |
| offDiagonalSeminorm M^4=∑ p : ι×ι,if p.1=p.2 then 0 else (M p.1 p.2)^4 := by |
| rw [offDiagonalSeminorm,PiLp.norm_eq_of_nat (p:=4) 4 (by norm_num)] |
| rw [←Real.rpow_natCast,←Real.rpow_mul (Finset.sum_nonneg (fun _ _=>pow_nonneg (norm_nonneg _) _))] |
| norm_num only [Nat.cast_ofNat,one_div,mul_inv_cancel₀ (by norm_num : (4:ℝ)≠0),Real.rpow_one] |
| apply Finset.sum_congr rfl |
| intro p _ |
| simp only [offDiagonalVector,WithLp.ofLp_toLp] |
| split_ifs |
| · simp |
| · rw [Real.norm_eq_abs] |
| calc |
| |M p.1 p.2|^4=(|M p.1 p.2|^2)^2 := by ring |
| _ = ((M p.1 p.2)^2)^2 := by rw [sq_abs] |
| _ = (M p.1 p.2)^4 := by ring |
|
|
| lemma diagonalSeminorm_le (M : Matrix ι ι ℝ) : diagonalSeminorm M≤‖M‖ := by |
| apply le_of_pow_le_pow_left₀ (by decide : 2≠0) (norm_nonneg _) |
| rw [diagonalSeminorm_sq,frobenius_sq] |
| exact Finset.sum_le_sum (fun i _=>Finset.single_le_sum (fun k _=>sq_nonneg (M i k)) (Finset.mem_univ i)) |
|
|
| lemma offDiagonalSeminorm_le (M : Matrix ι ι ℝ) : offDiagonalSeminorm M≤‖M‖ := by |
| let a : ι×ι→ℝ := fun p=>if p.1=p.2 then 0 else (M p.1 p.2)^2 |
| have ha : ∀ p,0≤a p := by intro p;dsimp [a];split_ifs <;> positivity |
| have hs : (∑ p,a p)≤‖M‖^2 := by |
| rw [frobenius_sq] |
| rw [show (∑ i,∑ k,(M i k)^2) = ∑ p : ι×ι,(M p.1 p.2)^2 from (Fintype.sum_prod_type fun p : ι×ι=>(M p.1 p.2)^2).symm] |
| apply Finset.sum_le_sum |
| intro p _ |
| dsimp [a] |
| split_ifs |
| · exact sq_nonneg _ |
| · exact le_rfl |
| apply le_of_pow_le_pow_left₀ (by decide : 4≠0) (norm_nonneg _) |
| rw [offDiagonalSeminorm_fourth] |
| calc |
| _ = ∑ p,a p^2 := by |
| apply Finset.sum_congr rfl |
| intro p _ |
| dsimp [a] |
| split_ifs <;> ring |
| _ ≤ (∑ p,a p)^2 := Finset.sum_sq_le_sq_sum_of_nonneg (fun p _=>ha p) |
| _ ≤ (‖M‖^2)^2 := pow_le_pow_left₀ (Finset.sum_nonneg (fun p _=>ha p)) hs 2 |
| _ = ‖M‖^4 := by ring |
|
|
| omit [Fintype ι] [DecidableEq ι] in |
| @[simp] lemma diagonalVector_sub (M N : Matrix ι ι ℝ) : diagonalVector (M-N)=diagonalVector M-diagonalVector N := by rfl |
| omit [Fintype ι] [DecidableEq ι] in |
| @[simp] lemma diagonalVector_add (M N : Matrix ι ι ℝ) : diagonalVector (M+N)=diagonalVector M+diagonalVector N := by rfl |
| omit [Fintype ι] in |
| @[simp] lemma offDiagonalVector_sub (M N : Matrix ι ι ℝ) : offDiagonalVector (M-N)=offDiagonalVector M-offDiagonalVector N := by |
| ext p |
| simp only [offDiagonalVector,WithLp.ofLp_toLp,PiLp.sub_apply] |
| split_ifs <;> simp |
| omit [Fintype ι] in |
| @[simp] lemma offDiagonalVector_add (M N : Matrix ι ι ℝ) : offDiagonalVector (M+N)=offDiagonalVector M+offDiagonalVector N := by |
| ext p |
| simp only [offDiagonalVector,WithLp.ofLp_toLp,PiLp.add_apply] |
| split_ifs <;> simp |
|
|
| lemma diagonalSeminorm_lipschitz : LipschitzWith 1 (diagonalSeminorm (ι:=ι)) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro M N |
| simpa only [NNReal.coe_one,one_mul,dist_eq_norm,Real.norm_eq_abs,diagonalSeminorm,←diagonalVector_sub] using! |
| (abs_norm_sub_norm_le (diagonalVector M) (diagonalVector N)).trans (diagonalSeminorm_le (M-N)) |
| lemma offDiagonalSeminorm_lipschitz : LipschitzWith 1 (offDiagonalSeminorm (ι:=ι)) := by |
| apply LipschitzWith.of_dist_le_mul |
| intro M N |
| have hh : ‖offDiagonalVector M-offDiagonalVector N‖≤‖M-N‖ := by |
| simpa only [←offDiagonalVector_sub,offDiagonalSeminorm] using! offDiagonalSeminorm_le (M-N) |
| simpa only [NNReal.coe_one,one_mul,dist_eq_norm,Real.norm_eq_abs,offDiagonalSeminorm] using! |
| (abs_norm_sub_norm_le (offDiagonalVector M) (offDiagonalVector N)).trans hh |
| omit [DecidableEq ι] in |
| lemma diagonalSeminorm_add (M N : Matrix ι ι ℝ) : |
| diagonalSeminorm (M+N)≤diagonalSeminorm M+diagonalSeminorm N := by |
| simpa only [diagonalSeminorm,diagonalVector_add] using norm_add_le (diagonalVector M) (diagonalVector N) |
| lemma offDiagonalSeminorm_add (M N : Matrix ι ι ℝ) : |
| offDiagonalSeminorm (M+N)≤offDiagonalSeminorm M+offDiagonalSeminorm N := by |
| simpa only [offDiagonalSeminorm,offDiagonalVector_add] using norm_add_le (offDiagonalVector M) (offDiagonalVector N) |
| end SKGap |
| end |
| end |
|
|
| end OAI |
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