| import OAI.Combinatorics.FiniteCircle.Translations |
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| namespace OAI |
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| namespace FiniteCircle |
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| open scoped BigOperators List |
| open scoped ComplexConjugate |
| open RCLike Module |
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| noncomputable def upperRates {α X : Type*} (word : X → List α) (f : X → RealL1) : Set ℝ := |
| {r | ∃ x y : X, x ≠ y ∧ r = ‖f x-f y‖/(editDistance (word x) (word y):ℝ)} |
| noncomputable def lowerRates {α X : Type*} (word : X → List α) (f : X → RealL1) : Set ℝ := |
| {r | ∃ x y : X, x ≠ y ∧ r = (editDistance (word x) (word y):ℝ)/‖f x-f y‖} |
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| theorem upperRates_finite {α X : Type*} [Finite X] (word : X → List α) (f : X → RealL1) : |
| (upperRates word f).Finite := by |
| apply (Set.finite_range (fun z : X×X => ‖f z.1-f z.2‖/(editDistance (word z.1) (word z.2):ℝ))).subset |
| rintro r ⟨x,y,_,rfl⟩ |
| exact ⟨(x,y),rfl⟩ |
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| theorem lowerRates_finite {α X : Type*} [Finite X] (word : X → List α) (f : X → RealL1) : |
| (lowerRates word f).Finite := by |
| apply (Set.finite_range (fun z : X×X => (editDistance (word z.1) (word z.2):ℝ)/‖f z.1-f z.2‖)).subset |
| rintro r ⟨x,y,_,rfl⟩ |
| exact ⟨(x,y),rfl⟩ |
|
|
| theorem distortion_nonneg {α X : Type*} (word : X → List α) (f : X → RealL1) : |
| 0 ≤ distortion word f := by |
| apply mul_nonneg |
| · apply Real.sSup_nonneg |
| rintro r ⟨x,y,_,rfl⟩ |
| positivity |
| · apply Real.sSup_nonneg |
| rintro r ⟨x,y,_,rfl⟩ |
| positivity |
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| theorem finite_l1_embedding (X : Type*) [Fintype X] : ∃ f : X → RealL1, Function.Injective f := by |
| classical |
| let e : X → ℕ := fun x => (Fintype.equivFin X x).val |
| have he : Function.Injective e := fun x y h => (Fintype.equivFin X).injective (Fin.ext h) |
| refine ⟨fun x => lp.single 1 (e x) (1:ℝ), ?_⟩ |
| intro x y h |
| by_contra hxy |
| have hne : e x ≠ e y := fun h => hxy (he h) |
| have ht := congrArg (fun f : RealL1 => f (e x)) h |
| simp only [lp.single_apply_self, lp.single_apply, Pi.single_eq_of_ne hne] at ht |
| norm_num at ht |
|
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| theorem optimalDistortion_nonneg {α X : Type*} (word : X → List α) : 0 ≤ optimalDistortion word := by |
| apply Real.sInf_nonneg |
| rintro D ⟨f,_,rfl⟩ |
| exact distortion_nonneg word f |
|
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| theorem optimalDistortion_lower {α X : Type*} [Fintype X] (word : X → List α) (D : ℝ) |
| (hD : ∀ f : X → RealL1, Function.Injective f → D ≤ distortion word f) : |
| D ≤ optimalDistortion word := by |
| apply le_csInf |
| · obtain ⟨f,hf⟩ := finite_l1_embedding X |
| exact ⟨distortion word f,f,hf,rfl⟩ |
| · rintro b ⟨f,hf,rfl⟩ |
| exact hD f hf |
|
|
| theorem optimalDistortion_le {α X : Type*} (word : X → List α) |
| (f : X → RealL1) (hf : Function.Injective f) : |
| optimalDistortion word ≤ distortion word f := by |
| apply csInf_le |
| · refine ⟨0,?_⟩ |
| rintro b ⟨g,_,rfl⟩ |
| exact distortion_nonneg word g |
| · exact ⟨f,hf,rfl⟩ |
|
|
| theorem distortion_control {α X : Type*} [Finite X] [Nontrivial X] |
| (word : X → List α) (hw : Function.Injective word) |
| (f : X → RealL1) (hf : Function.Injective f) : |
| ∃ A B : ℝ, 0<A ∧ 0<B ∧ distortion word f=A*B ∧ |
| (∀ x y, ‖f x-f y‖ ≤ A*(editDistance (word x) (word y):ℝ)) ∧ |
| (∀ x y, (editDistance (word x) (word y):ℝ) ≤ B*‖f x-f y‖) := by |
| let A := sSup (upperRates word f) |
| let B := sSup (lowerRates word f) |
| have hED (x y : X) (hxy : x ≠ y) : (0:ℝ)<editDistance (word x) (word y) := by |
| exact_mod_cast Nat.pos_of_ne_zero (fun he => hxy (hw (editDistance_eq_zero.mp he))) |
| have hn (x y : X) (hxy : x ≠ y) : 0<‖f x-f y‖ := norm_pos_iff.mpr (sub_ne_zero.mpr (fun he => hxy (hf he))) |
| obtain ⟨x,y,hxy⟩ := exists_pair_ne X |
| have ha : ‖f x-f y‖/(editDistance (word x) (word y):ℝ) ≤ A := |
| le_csSup (upperRates_finite word f).bddAbove ⟨x,y,hxy,rfl⟩ |
| have hb : (editDistance (word x) (word y):ℝ)/‖f x-f y‖ ≤ B := |
| le_csSup (lowerRates_finite word f).bddAbove ⟨x,y,hxy,rfl⟩ |
| refine ⟨A,B,(div_pos (hn x y hxy) (hED x y hxy)).trans_le ha, |
| (div_pos (hED x y hxy) (hn x y hxy)).trans_le hb,rfl,?_,?_⟩ |
| · intro x y |
| by_cases hxy : x=y |
| · subst y; simp |
| · exact (div_le_iff₀ (hED x y hxy)).mp (le_csSup (upperRates_finite word f).bddAbove ⟨x,y,hxy,rfl⟩) |
| · intro x y |
| by_cases hxy : x=y |
| · subst y; simp |
| · exact (div_le_iff₀ (hn x y hxy)).mp (le_csSup (lowerRates_finite word f).bddAbove ⟨x,y,hxy,rfl⟩) |
|
|
| theorem distortion_upper {α X : Type*} [Nontrivial X] |
| (word : X → List α) (hw : Function.Injective word) |
| (f : X → RealL1) (A B : ℝ) (hA : 0≤A) |
| (hu : ∀ x y, ‖f x-f y‖ ≤ A*(editDistance (word x) (word y):ℝ)) |
| (hl : ∀ x y, (editDistance (word x) (word y):ℝ) ≤ B*‖f x-f y‖) : |
| Function.Injective f ∧ distortion word f ≤ A*B := by |
| have hf : Function.Injective f := by |
| intro x y he |
| have hwxy := hl x y |
| rw [he,sub_self,norm_zero,mul_zero] at hwxy |
| have ht : editDistance (word x) (word y)=0 := by exact_mod_cast (le_antisymm hwxy (by positivity)) |
| exact hw (editDistance_eq_zero.mp ht) |
| refine ⟨hf,?_⟩ |
| obtain ⟨x,y,hxy⟩ := exists_pair_ne X |
| have hune : (upperRates word f).Nonempty := ⟨_,x,y,hxy,rfl⟩ |
| have hlne : (lowerRates word f).Nonempty := ⟨_,x,y,hxy,rfl⟩ |
| have hU : sSup (upperRates word f) ≤ A := by |
| apply csSup_le hune |
| rintro r ⟨x,y,hxy,rfl⟩ |
| have hed : (0:ℝ)<editDistance (word x) (word y) := by |
| exact_mod_cast Nat.pos_of_ne_zero (fun he => hxy (hw (editDistance_eq_zero.mp he))) |
| exact (div_le_iff₀ hed).mpr (hu x y) |
| have hL : sSup (lowerRates word f) ≤ B := by |
| apply csSup_le hlne |
| rintro r ⟨x,y,hxy,rfl⟩ |
| have hn : 0<‖f x-f y‖ := norm_pos_iff.mpr (sub_ne_zero.mpr (fun he => hxy (hf he))) |
| exact (div_le_iff₀ hn).mpr (hl x y) |
| apply mul_le_mul hU hL |
| · apply Real.sSup_nonneg |
| rintro r ⟨x,y,_,rfl⟩ |
| positivity |
| · exact hA |
|
|
| noncomputable def finiteL1 (M : ℕ) (v : Fin M → ℝ) : RealL1 := |
| ∑ i, lp.single 1 i.val (v i) |
|
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| theorem finiteL1_sub (M : ℕ) (v w : Fin M → ℝ) : |
| finiteL1 M v-finiteL1 M w=finiteL1 M (v-w) := by |
| unfold finiteL1 |
| rw [← Finset.sum_sub_distrib] |
| apply Finset.sum_congr rfl |
| intro i _ |
| exact (lp.single_sub (E:=fun _ : ℕ=>ℝ) 1 i.val (v i) (w i)).symm |
|
|
| theorem finiteL1_norm (M : ℕ) (v : Fin M → ℝ) : ‖finiteL1 M v‖=∑ i, |v i| := by |
| classical |
| let g : ℕ → ℝ := fun i => if hi : i<M then v ⟨i,hi⟩ else 0 |
| have he : finiteL1 M v = ∑ i ∈ Finset.range M, lp.single 1 i (g i) := by |
| unfold finiteL1 |
| rw [← Fin.sum_univ_eq_sum_range] |
| apply Finset.sum_congr rfl |
| intro i _ |
| simp [g] |
| rw [he] |
| have ht := lp.norm_sum_single (E:=fun _ : ℕ=>ℝ) (p:=1) (by norm_num) g (Finset.range M) |
| simpa [g,← Fin.sum_univ_eq_sum_range] using ht |
|
|
| theorem finiteL1_norm_sub (M : ℕ) (v w : Fin M → ℝ) : |
| ‖finiteL1 M v-finiteL1 M w‖=∑ i, |v i-w i| := by |
| rw [finiteL1_sub,finiteL1_norm] |
| rfl |
|
|
| theorem dyadic_expected_budget {I : Type*} [Fintype I] [DecidableEq I] |
| {M : ℕ} [NeZero M] (f : (I → ZMod M) → RealL1) |
| (A u L copies : ℝ) (hcL : L=(M:ℝ)*Fintype.card I*copies) |
| (hrsingle : ∀ (i : I) (j : Fin (Nat.log 2 M+1)), |
| (𝔼 v, ‖f (v+Pi.single i ((2^j.val:ℕ):ZMod M))-f v‖) ≤ A*(u*2*copies*(2:ℝ)^j.val)) |
| : (∑ i, ∑ j : Fin (Nat.log 2 M+1), (M:ℝ)/(2:ℝ)^j.val * |
| (𝔼 v, ‖f (v+Pi.single i ((2^j.val:ℕ):ZMod M))-f v‖)) ≤ |
| A*u*2*L*(Nat.log 2 M+1:ℕ) := by |
| classical |
| calc |
| _ ≤ ∑ i : I, ∑ j : Fin (Nat.log 2 M+1), (M:ℝ)/(2:ℝ)^j.val*(A*(u*2*copies*(2:ℝ)^j.val)) := by |
| apply Finset.sum_le_sum |
| intro i _ |
| apply Finset.sum_le_sum |
| intro j _ |
| exact mul_le_mul_of_nonneg_left (hrsingle i j) (by positivity) |
| _ = ∑ _i : I, ∑ _j : Fin (Nat.log 2 M+1), (M:ℝ)*A*u*2*copies := by |
| apply Finset.sum_congr rfl |
| intro i _ |
| apply Finset.sum_congr rfl |
| intro j _ |
| field_simp |
| _ = _ := by simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fin,nsmul_eq_mul]; rw [hcL]; ring |
|
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| theorem fourier_edit_distortion {I α : Type*} [Fintype I] [DecidableEq I] |
| {M : ℕ} [NeZero M] (H : AddSubgroup (I → ZMod M)) [Fintype H] |
| (w : I → ZMod M) (B : ℝ) (hB : 0<B) |
| (hb : ∀ χ : AddChar (I → ZMod M) ℂ, χ.compAddMonoidHom H.subtype=0 → χ w≠1 → |
| B ≤ ∑ i, ((coordinateFrequency χ i).valMinAbs.natAbs:ℝ)) |
| (W : (I → ZMod M) → List α) (h R L copies u δ : ℝ) (hL : 0<L) |
| (hcL : L=(M:ℝ)*Fintype.card I*copies) |
| (hcheap : ∀ v (c : H), (editDistance (W v) (W (v+c)):ℝ) ≤ u*2*h*L/R) |
| (hsingle : ∀ v i (j : Fin (Nat.log 2 M+1)), |
| (editDistance (W v) (W (v+Pi.single i ((2^j.val:ℕ):ZMod M))):ℝ) ≤ u*2*copies*(2:ℝ)^j.val) |
| (hlong : ∀ v, δ*L ≤ (editDistance (W v) (W (v+w)):ℝ)) |
| (f : (I → ZMod M) → RealL1) (A A' : ℝ) (hA : 0≤A) (hA' : 0≤A') |
| (hu : ∀ v z, ‖f (v+z)-f v‖ ≤ A*(editDistance (W v) (W (v+z)):ℝ)) |
| (hl : ∀ v z, (editDistance (W v) (W (v+z)):ℝ) ≤ A'*‖f (v+z)-f v‖) : |
| δ ≤ (A*A')*u*(4*h/R+2*(Nat.log 2 M+1:ℕ)/B) := by |
| classical |
| have hleft : δ*L ≤ A'*(𝔼 v, ‖f (v+w)-f v‖) := by |
| rw [Finset.mul_expect] |
| apply Finset.le_expect Finset.univ_nonempty |
| intro v _ |
| exact (hlong v).trans (hl v w) |
| have hrcheap : (𝔼 c : H, 𝔼 v, ‖f (v+c)-f v‖) ≤ A*(u*2*h*L/R) := by |
| apply Finset.expect_le Finset.univ_nonempty |
| intro c _ |
| apply Finset.expect_le Finset.univ_nonempty |
| intro v _ |
| exact (hu v c).trans (mul_le_mul_of_nonneg_left (hcheap v c) hA) |
| have hrsingle (i : I) (j : Fin (Nat.log 2 M+1)) : |
| (𝔼 v, ‖f (v+Pi.single i ((2^j.val:ℕ):ZMod M))-f v‖) ≤ A*(u*2*copies*(2:ℝ)^j.val) := by |
| apply Finset.expect_le Finset.univ_nonempty |
| intro v _ |
| exact (hu v _).trans (mul_le_mul_of_nonneg_left (hsingle v i j) hA) |
| have hsum := dyadic_expected_budget f A u L copies hcL hrsingle |
| have hspec := weighted_l1_inequality H w B hB hb f |
| have hbd := add_le_add (mul_le_mul_of_nonneg_left hrcheap (by norm_num : (0:ℝ)≤2)) |
| (mul_le_mul_of_nonneg_left hsum (le_of_lt (one_div_pos.mpr hB))) |
| have hright := hspec.trans hbd |
| have heq : 2*(A*(u*2*h*L/R))+(1/B)*(A*u*2*L*(Nat.log 2 M+1:ℕ)) = |
| A*u*L*(4*h/R+2*(Nat.log 2 M+1:ℕ)/B) := by ring |
| rw [heq] at hright |
| have ht := hleft.trans (mul_le_mul_of_nonneg_left hright hA') |
| have he : A'*(A*u*L*(4*h/R+2*(Nat.log 2 M+1:ℕ)/B)) = |
| ((A*A')*u*(4*h/R+2*(Nat.log 2 M+1:ℕ)/B))*L := by ring |
| rw [he] at ht |
| exact le_of_mul_le_mul_right ht hL |
| |
| theorem range_distortion_lower {I α : Type*} [Fintype I] [DecidableEq I] |
| {M : ℕ} [NeZero M] (H : AddSubgroup (I → ZMod M)) [Fintype H] |
| (w : I → ZMod M) (B : ℝ) (hB : 0<B) |
| (hb : ∀ χ : AddChar (I → ZMod M) ℂ, χ.compAddMonoidHom H.subtype=0 → χ w≠1 → |
| B ≤ ∑ i, ((coordinateFrequency χ i).valMinAbs.natAbs:ℝ)) |
| (W : (I → ZMod M) → List α) (h R L copies u δ D : ℝ) (hL : 0<L) (hδ : 0<δ) |
| (hcL : L=(M:ℝ)*Fintype.card I*copies) |
| (hcheap : ∀ v (c : H), (editDistance (W v) (W (v+c)):ℝ) ≤ u*2*h*L/R) |
| (hsingle : ∀ v i (j : Fin (Nat.log 2 M+1)), |
| (editDistance (W v) (W (v+Pi.single i ((2^j.val:ℕ):ZMod M))):ℝ) ≤ u*2*copies*(2:ℝ)^j.val) |
| (hlong : ∀ v, δ*L ≤ (editDistance (W v) (W (v+w)):ℝ)) |
| (hnum : ∀ t : ℝ, 0≤t → δ ≤ t*u*(4*h/R+2*(Nat.log 2 M+1:ℕ)/B) → D≤t) : |
| D ≤ optimalDistortion (fun x : Set.range W => x.val) := by |
| classical |
| let X := Set.range W |
| let : Fintype X := (Set.finite_range W).fintype |
| have hw0 : W 0 ≠ W w := by |
| intro he |
| have ht := hlong 0 |
| simp only [zero_add, ← he,editDistance_self,Nat.cast_zero] at ht |
| exact (mul_pos hδ hL).not_ge ht |
| let : Nontrivial X := ⟨⟨⟨W 0,⟨0,rfl⟩⟩,⟨W w,⟨w,rfl⟩⟩,fun he => hw0 (congrArg Subtype.val he)⟩⟩ |
| apply optimalDistortion_lower |
| intro f hf |
| obtain ⟨A,A',hA,hA',hD,hu,hl⟩ := distortion_control (fun x : X => x.val) Subtype.val_injective f hf |
| let F := fun v => f (⟨W v,⟨v,rfl⟩⟩ : X) |
| have hcontrolu (v z : I → ZMod M) : ‖F (v+z)-F v‖ ≤ A*(editDistance (W v) (W (v+z)):ℝ) := by |
| rw [norm_sub_rev] |
| exact hu ⟨W v,⟨v,rfl⟩⟩ ⟨W (v+z),⟨v+z,rfl⟩⟩ |
| have hcontroll (v z : I → ZMod M) : (editDistance (W v) (W (v+z)):ℝ) ≤ A'*‖F (v+z)-F v‖ := by |
| rw [norm_sub_rev] |
| exact hl ⟨W v,⟨v,rfl⟩⟩ ⟨W (v+z),⟨v+z,rfl⟩⟩ |
| have ht := fourier_edit_distortion H w B hB hb W h R L copies u δ hL hcL hcheap hsingle hlong |
| F A A' hA.le hA'.le hcontrolu hcontroll |
| rw [hD] |
| exact hnum (A*A') (mul_nonneg hA.le hA'.le) ht |
|
|
| end FiniteCircle |
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| end OAI |
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