File size: 6,644 Bytes
7d9f027 | 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 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 | import Mathlib
import OAI.Probability.SKGap.Matrix.MatrixWordRegularity
namespace OAI
section
noncomputable section
namespace SKGap
open Matrix Real
open scoped BigOperators Matrix.Norms.Frobenius
variable {ι X : Type*} [Fintype ι] [DecidableEq ι] [PseudoMetricSpace X]
omit [PseudoMetricSpace X] in
lemma word_opNorm_add (M N : Matrix ι ι ℝ) : opNorm (M+N) ≤ opNorm M+opNorm N := by
unfold opNorm
rw [map_add,map_add]
exact norm_add_le _ _
omit [PseudoMetricSpace X] in
lemma real_product_four_point (a b c d e f g h : Matrix ι ι ℝ) :
‖(a*e-b*f)-(c*g-d*h)‖ ≤
‖(a-b)-(c-d)‖*opNorm e+‖c-d‖*opNorm (e-g)+
opNorm (b-d)*‖e-f‖+opNorm d*‖(e-f)-(g-h)‖ := by
have he : (a*e-b*f)-(c*g-d*h)=
((a-b)-(c-d))*e+(c-d)*(e-g)+(b-d)*(e-f)+d*((e-f)-(g-h)) := by noncomm_ring
rw [he]
calc
_ ≤ ‖((a-b)-(c-d))*e+(c-d)*(e-g)+(b-d)*(e-f)‖+‖d*((e-f)-(g-h))‖ := norm_add_le _ _
_ ≤ (‖((a-b)-(c-d))*e+(c-d)*(e-g)‖+‖(b-d)*(e-f)‖)+‖d*((e-f)-(g-h))‖ := by gcongr;exact norm_add_le _ _
_ ≤ ((‖((a-b)-(c-d))*e‖+‖(c-d)*(e-g)‖)+‖(b-d)*(e-f)‖)+‖d*((e-f)-(g-h))‖ := by gcongr;exact norm_add_le _ _
_ ≤ _ := by
gcongr
· exact frobenius_mul_le_opNorm_right _ _
· exact frobenius_mul_le_opNorm_right _ _
· exact frobenius_mul_le_opNorm _ _
· exact frobenius_mul_le_opNorm _ _
structure MatrixFactorPairBounds (B Δ : NNReal) (f g : X→Matrix ι ι ℝ) : Prop where
leftBound : ∀ x,opNorm (f x)≤B
rightBound : ∀ x,opNorm (g x)≤B
leftLip : LipschitzWith B f
rightLip : LipschitzWith B g
parameter : ∀ x,opNorm (f x-g x)≤Δ
increment : LipschitzWith Δ (fun x=>f x-g x)
def wordParameterCoeff (B : NNReal) : ℕ→NNReal
| 0 => 0
| r+1 => B^r+B*wordParameterCoeff B r
def wordMixedCoeff (B : NNReal) : ℕ→NNReal
| 0 => 0
| r+1 => B^r+B*wordParameterCoeff B r+(r:NNReal)*B^r+B*wordMixedCoeff B r
theorem matrixWord_pair_increments (F : List ((X→Matrix ι ι ℝ)×(X→Matrix ι ι ℝ)))
{B Δ : NNReal} (hF : ∀ p∈F,MatrixFactorPairBounds B Δ p.1 p.2) :
(∀ x,opNorm (matrixFactorProduct (F.map Prod.fst) x-matrixFactorProduct (F.map Prod.snd) x)≤
(wordParameterCoeff B F.length:ℝ)*Δ) ∧
LipschitzWith (wordMixedCoeff B F.length*Δ)
(fun x=>matrixFactorProduct (F.map Prod.fst) x-matrixFactorProduct (F.map Prod.snd) x) := by
induction F with
| nil =>
constructor
· intro x;simp [matrixFactorProduct,wordParameterCoeff,opNorm]
· simp [matrixFactorProduct,wordMixedCoeff]
| cons p F ih =>
have hp := hF p List.mem_cons_self
have hFt : ∀ q∈F,MatrixFactorPairBounds B Δ q.1 q.2 := fun q hq=>hF q (List.mem_cons_of_mem p hq)
have ht := ih hFt
have hbL : ∀ f∈F.map Prod.fst,∀ x,opNorm (f x)≤B := by
intro f hf;obtain ⟨q,hq,rfl⟩ := List.mem_map.mp hf;exact (hFt q hq).leftBound
have hlL : ∀ f∈F.map Prod.fst,LipschitzWith B f := by
intro f hf;obtain ⟨q,hq,rfl⟩ := List.mem_map.mp hf;exact (hFt q hq).leftLip
have htl := matrixFactorProduct_lipschitz (F.map Prod.fst) hbL hlL
constructor
· intro x
simp only [List.map_cons,matrixFactorProduct_cons,List.length_cons,wordParameterCoeff,NNReal.coe_add,NNReal.coe_mul,NNReal.coe_pow]
have he : p.1 x*matrixFactorProduct (F.map Prod.fst) x-p.2 x*matrixFactorProduct (F.map Prod.snd) x=
(p.1 x-p.2 x)*matrixFactorProduct (F.map Prod.fst) x+
p.2 x*(matrixFactorProduct (F.map Prod.fst) x-matrixFactorProduct (F.map Prod.snd) x) := by noncomm_ring
rw [he]
apply (word_opNorm_add _ _).trans
apply (add_le_add (opNorm_mul _ _) (opNorm_mul _ _)).trans
have h1 := mul_le_mul (hp.parameter x) (matrixFactorProduct_opNorm (F.map Prod.fst) hbL x)
(show 0≤opNorm (matrixFactorProduct (F.map Prod.fst) x) from norm_nonneg _) Δ.coe_nonneg
have h2 := mul_le_mul (hp.rightBound x) (ht.1 x) (show 0≤opNorm
(matrixFactorProduct (F.map Prod.fst) x-matrixFactorProduct (F.map Prod.snd) x) from norm_nonneg _) B.coe_nonneg
apply (add_le_add h1 h2).trans_eq
simp only [List.length_map]
ring
· apply LipschitzWith.of_dist_le_mul
intro x y
simp only [dist_eq_norm,List.map_cons,matrixFactorProduct_cons]
have he : (p.1 x*matrixFactorProduct (F.map Prod.fst) x-p.2 x*matrixFactorProduct (F.map Prod.snd) x)-
(p.1 y*matrixFactorProduct (F.map Prod.fst) y-p.2 y*matrixFactorProduct (F.map Prod.snd) y)=
(p.1 x*matrixFactorProduct (F.map Prod.fst) x-p.1 y*matrixFactorProduct (F.map Prod.fst) y)-
(p.2 x*matrixFactorProduct (F.map Prod.snd) x-p.2 y*matrixFactorProduct (F.map Prod.snd) y) := by abel
rw [he]
apply (real_product_four_point _ _ _ _ _ _ _ _).trans
have hi : ‖(p.1 x-p.1 y)-(p.2 x-p.2 y)‖≤(Δ:ℝ)*dist x y := by
have hh := hp.increment.dist_le_mul x y
rw [dist_eq_norm] at hh
convert! hh using 1
congr 1;abel
have hr : ‖p.2 x-p.2 y‖≤(B:ℝ)*dist x y := by simpa only [dist_eq_norm] using hp.rightLip.dist_le_mul x y
have hli : ‖matrixFactorProduct (F.map Prod.fst) x-matrixFactorProduct (F.map Prod.fst) y‖≤
((F.length:ℝ)*(B:ℝ)^F.length)*dist x y := by
simpa only [dist_eq_norm,List.length_map,NNReal.coe_mul,NNReal.coe_natCast,NNReal.coe_pow] using htl.dist_le_mul x y
have hm : ‖(matrixFactorProduct (F.map Prod.fst) x-matrixFactorProduct (F.map Prod.fst) y)-
(matrixFactorProduct (F.map Prod.snd) x-matrixFactorProduct (F.map Prod.snd) y)‖≤
((wordMixedCoeff B F.length:ℝ)*Δ)*dist x y := by
have hh := ht.2.dist_le_mul x y
rw [dist_eq_norm] at hh
convert! hh using 1
congr 1;abel
have h1 := mul_le_mul hi (matrixFactorProduct_opNorm (F.map Prod.fst) hbL x)
(show 0≤opNorm (matrixFactorProduct (F.map Prod.fst) x) from norm_nonneg _) (mul_nonneg Δ.coe_nonneg dist_nonneg)
have h2 := mul_le_mul hr (ht.1 x) (show 0≤opNorm
(matrixFactorProduct (F.map Prod.fst) x-matrixFactorProduct (F.map Prod.snd) x) from norm_nonneg _)
(mul_nonneg B.coe_nonneg dist_nonneg)
have h3 := mul_le_mul (hp.parameter y) hli (norm_nonneg _) Δ.coe_nonneg
have h4 := mul_le_mul (hp.rightBound y) hm (norm_nonneg _) B.coe_nonneg
apply (add_le_add (add_le_add (add_le_add h1 h2) h3) h4).trans_eq
simp only [List.length_map,List.length_cons,wordMixedCoeff,NNReal.coe_add,NNReal.coe_mul,
NNReal.coe_natCast,NNReal.coe_pow]
ring
end SKGap
end
end
end OAI
|