| import OAI.Analysis.ForestSpace.Diamonds |
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| namespace OAI |
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| open scoped BigOperators |
| open Finset |
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| namespace ForestSpace |
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| noncomputable section |
| attribute [local instance] Classical.propDecidable |
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| namespace Diamond |
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| noncomputable section |
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| section Distortion |
| variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℝ E] |
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| / |
| def LensProperty (E : Type) [NormedAddCommGroup E] [NormedSpace ℝ E] : Prop := |
| ∀ (R ε : ℝ), 0 < R → 0 < ε → ∀ (x : E) (y : ℕ → E), |
| (∀ j, ‖x+y j‖ ≤ R) → (∀ j, ‖x-y j‖ ≤ R) → |
| (∀ i j, i ≠ j → ε ≤ ‖y i-y j‖) → ε^2/48 ≤ R^2-‖x‖^2 |
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| theorem parent_step (hLens : LensProperty E) (R ℓ : ℝ) (hR : 0 < R) (hℓ : 0 < ℓ) |
| (a b : E) (w : ℕ → E) |
| (hp : ∀ j, ‖w j-a‖/ℓ ≤ R) (hm : ∀ j, ‖b-w j‖/ℓ ≤ R) |
| (hs : ∀ i j, i ≠ j → 2*ℓ ≤ ‖w i-w j‖) : |
| (‖b-a‖/(2*ℓ))^2+1/12 ≤ R^2 := by |
| let x : E := (2*ℓ)⁻¹ • (b-a) |
| let y : ℕ → E := fun j => ℓ⁻¹ • (w j-(1/2 : ℝ) • (a+b)) |
| have hc : (2*ℓ)⁻¹ = (1/2 : ℝ)*ℓ⁻¹ := by field_simp |
| have hplus : ∀ j, x+y j = ℓ⁻¹ • (w j-a) := by |
| intro j |
| dsimp [x,y] |
| rw [hc] |
| module |
| have hminus : ∀ j, x-y j = ℓ⁻¹ • (b-w j) := by |
| intro j |
| dsimp [x,y] |
| rw [hc] |
| module |
| have hdn : ∀ i j, ‖y i-y j‖ = ‖w i-w j‖/ℓ := by |
| intro i j |
| have he : y i-y j = ℓ⁻¹ • (w i-w j) := by dsimp [y]; module |
| rw [he,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hℓ.le)] |
| ring |
| have hh := hLens R 2 hR (by norm_num) x y |
| (by |
| intro j |
| rw [hplus,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hℓ.le)] |
| simpa only [div_eq_mul_inv,mul_comm] using hp j) |
| (by |
| intro j |
| rw [hminus,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hℓ.le)] |
| simpa only [div_eq_mul_inv,mul_comm] using hm j) |
| (by intro i j hij; rw [hdn]; exact (le_div_iff₀ hℓ).mpr (hs i j hij)) |
| have hx : ‖x‖ = ‖b-a‖/(2*ℓ) := by |
| dsimp [x] |
| rw [norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr (mul_nonneg (by norm_num) hℓ.le))] |
| ring |
| rw [hx] at hh |
| norm_num at hh |
| linarith |
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| / |
| of any supremum of edge stretches. -/ |
| theorem edge_sq_loss (hLens : LensProperty E) (A : Network) (hA : A.graph.Connected) |
| (n : ℕ) (f : (A.iterate n).V → E) (C : ℝ) (hC : 0 ≤ C) |
| (hl : ∀ u v, ((A.iterate n).graph.dist u v : ℝ) ≤ ‖f u-f v‖) |
| (hu : ∀ u v, ‖f u-f v‖ ≤ C*((A.iterate n).graph.dist u v : ℝ)) |
| (e : A.E) : |
| (‖f (A.old n (A.right e))-f (A.old n (A.left e))‖/(2:ℝ)^n)^2+(n:ℝ)/12 ≤ C^2 := by |
| induction n generalizing A with |
| | zero => |
| have hh := hu (A.right e) (A.left e) |
| have he : A.graph.dist (A.right e) (A.left e) = 1 := |
| SimpleGraph.dist_eq_one_iff_adj.mpr (A.edge_adj e).symm |
| change ‖f (A.right e)-f (A.left e)‖ ≤ C*(A.graph.dist (A.right e) (A.left e) : ℝ) at hh |
| rw [he,Nat.cast_one,mul_one] at hh |
| simpa using (sq_le_sq₀ (norm_nonneg _) hC).mpr hh |
| | succ n ih => |
| let ℓ : ℝ := (2:ℝ)^n |
| have hℓ : 0 < ℓ := pow_pos (by norm_num) _ |
| let a := f (A.refine.old n (Sum.inl (A.left e))) |
| let b := f (A.refine.old n (Sum.inl (A.right e))) |
| let w : ℕ → E := fun j => f (A.refine.old n (Sum.inr (e,j))) |
| have hchild : ∀ c : A.refine.E, |
| (‖f (A.refine.old n (A.refine.right c))-f (A.refine.old n (A.refine.left c))‖/ℓ)^2+ |
| (n:ℝ)/12 ≤ C^2 := by |
| intro c |
| exact ih A.refine (A.refine_connected hA) f hl hu c |
| have hrad : 1 ≤ C^2-(n:ℝ)/12 := by |
| have hb := hchild (e,0,false) |
| have hd := A.refine.old_edge_dist (A.refine_connected hA) n (e,0,false) |
| have hlo := hl (A.refine.old n (Sum.inl (A.left e))) (A.refine.old n (Sum.inr (e,0))) |
| have hd' : ((A.refine.iterate n).graph.dist |
| (A.refine.old n (Sum.inl (A.left e))) (A.refine.old n (Sum.inr (e,0))) : ℝ) = ℓ := by |
| dsimp [ℓ] |
| exact_mod_cast hd |
| change ((A.refine.iterate n).graph.dist |
| (A.refine.old n (Sum.inl (A.left e))) (A.refine.old n (Sum.inr (e,0))) : ℝ) ≤ |
| ‖a-w 0‖ at hlo |
| rw [hd',norm_sub_rev] at hlo |
| have hc : 1 ≤ ‖w 0-a‖/ℓ := (le_div_iff₀ hℓ).mpr (by simpa using hlo) |
| change (‖w 0-a‖/ℓ)^2+(n:ℝ)/12 ≤ C^2 at hb |
| nlinarith |
| let R : ℝ := Real.sqrt (C^2-(n:ℝ)/12) |
| have hR : 0 < R := Real.sqrt_pos.mpr (by linarith) |
| have hR2 : R^2 = C^2-(n:ℝ)/12 := Real.sq_sqrt (by linarith) |
| have hp : ∀ j, ‖w j-a‖/ℓ ≤ R := by |
| intro j |
| have hh := hchild (e,j,false) |
| change (‖w j-a‖/ℓ)^2+(n:ℝ)/12 ≤ C^2 at hh |
| exact Real.le_sqrt_of_sq_le (by linarith) |
| have hm : ∀ j, ‖b-w j‖/ℓ ≤ R := by |
| intro j |
| have hh := hchild (e,j,true) |
| change (‖b-w j‖/ℓ)^2+(n:ℝ)/12 ≤ C^2 at hh |
| exact Real.le_sqrt_of_sq_le (by linarith) |
| have hs : ∀ i j, i ≠ j → 2*ℓ ≤ ‖w i-w j‖ := by |
| intro i j hij |
| have hh := hl (A.refine.old n (Sum.inr (e,i))) (A.refine.old n (Sum.inr (e,j))) |
| rw [A.old_middle_dist hA n e i j hij] at hh |
| have he : ((2^(n+1) : ℕ) : ℝ) = 2*ℓ := by |
| dsimp [ℓ] |
| norm_cast |
| rw [pow_succ,mul_comm] |
| exact he ▸ hh |
| have hh := parent_step hLens R ℓ hR hℓ a b w hp hm hs |
| rw [hR2] at hh |
| have he : (2:ℝ)^(n+1) = 2*ℓ := by dsimp [ℓ]; rw [pow_succ,mul_comm] |
| change (‖b-a‖/(2:ℝ)^(n+1))^2+((n+1:ℕ):ℝ)/12 ≤ C^2 |
| rw [he,Nat.cast_add,Nat.cast_one] |
| linarith |
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|
| / |
| def EmbeddingBound (k : ℕ) (f : (diamond k).V → E) (C : ℝ) : Prop := |
| ∃ s : ℝ, 0 < s ∧ ∀ u v, |
| s*distance k u v ≤ ‖f u-f v‖ ∧ ‖f u-f v‖ ≤ C*s*distance k u v |
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| theorem distortion_lower (hLens : LensProperty E) (k : ℕ) |
| (f : (diamond k).V → E) (C : ℝ) (hf : EmbeddingBound k f C) : |
| Real.sqrt (1+(k:ℝ)/12) ≤ C := by |
| obtain ⟨s,hs,hf⟩ := hf |
| let g : (diamond k).V → E := fun v => s⁻¹ • f v |
| have hgn : ∀ u v, ‖g u-g v‖ = ‖f u-f v‖/s := by |
| intro u v |
| dsimp [g] |
| rw [← smul_sub,norm_smul,Real.norm_of_nonneg (inv_nonneg.mpr hs.le)] |
| ring |
| have hl : ∀ u v, ((base.iterate k).graph.dist u v : ℝ) ≤ ‖g u-g v‖ := by |
| intro u v |
| rw [hgn] |
| exact (le_div_iff₀ hs).mpr (by simpa only [distance,mul_comm] using (hf u v).1) |
| have hu : ∀ u v, ‖g u-g v‖ ≤ C*((base.iterate k).graph.dist u v : ℝ) := by |
| intro u v |
| rw [hgn] |
| exact (div_le_iff₀ hs).mpr (by simpa only [distance,mul_assoc,mul_comm,mul_left_comm] using (hf u v).2) |
| let a := base.old k (base.left ()) |
| let b := base.old k (base.right ()) |
| have hd : ((base.iterate k).graph.dist a b : ℝ) = (2:ℝ)^k := by |
| exact_mod_cast (base.old_edge_dist base_connected k ()) |
| have hc : 1 ≤ C := by |
| have hlo := hl a b |
| have hup := hu a b |
| rw [hd] at hlo hup |
| have hpow : 0 < (2:ℝ)^k := pow_pos (by norm_num) _ |
| nlinarith |
| have hbound := edge_sq_loss hLens base base_connected k g C (by linarith) hl hu () |
| have hlo : 1 ≤ ‖g b-g a‖/(2:ℝ)^k := by |
| have hh := hl a b |
| rw [hd,norm_sub_rev] at hh |
| exact (le_div_iff₀ (pow_pos (by norm_num : (0:ℝ)<2) k)).mpr (by simpa using hh) |
| change (‖g b-g a‖/(2:ℝ)^k)^2+(k:ℝ)/12 ≤ C^2 at hbound |
| apply Real.sqrt_le_iff.mpr |
| exact ⟨by linarith,by nlinarith⟩ |
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| end Distortion |
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| end |
| end Diamond |
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| / |
| bi-Lipschitz convention and the actual shortest-path distance on D_k. -/ |
| theorem main_diamond_distortion (k : ℕ) |
| (f : (Diamond.diamond k).V → FullDual Vertex) (C : ℝ) |
| (hf : Diamond.EmbeddingBound k f C) : |
| Real.sqrt (1+(k:ℝ)/12) ≤ C := |
| Diamond.distortion_lower main_lens k f C hf |
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| end |
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| end ForestSpace |
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| end OAI |
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