| 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 |
|
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| end OAI |
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