| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Convolution |
| import OAI.Computability.MinUncut.Analysis.GaussianSmallBall |
|
|
| namespace OAI |
|
|
| section |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Inner |
| open BinaryFourier RowNoise GaussianBudget |
| attribute [local instance] Classical.propDecidable |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
| variable {m n : ℕ} |
|
|
| def evaluationProof (a : A) : FoldedProof A where |
| answer P := P a |
| folded _ := rfl |
|
|
| def codeVector (z : Code m n) : Point m n → ℝ := |
| fun x => (Real.sqrt (n^m:ℕ))⁻¹ * BinaryFourier.sign (z.val x) |
|
|
| lemma codeVector_norm (hn : 0<n) (z : Code m n) : ‖vector (codeVector z)‖=1 := by |
| have hn' : (0:ℝ)<(n^m:ℕ) := by positivity |
| have he : ‖vector (codeVector z)‖^2=1 := by |
| simp only [EuclideanSpace.real_norm_sq_eq,vector,codeVector,mul_pow,sign_sq,mul_one, |
| Finset.sum_const,Finset.card_univ,Fintype.card_fun,Fintype.card_fin,nsmul_eq_mul,inv_pow] |
| rw [Real.sq_sqrt hn'.le,mul_inv_cancel₀ hn'.ne'] |
| nlinarith [norm_nonneg (vector (codeVector z))] |
|
|
| lemma score_codeVector (z : Code m n) (c : Point m n → ℝ) : |
| score (codeVector z) c=(Real.sqrt (n^m:ℕ))⁻¹ * ∑ x, c x*BinaryFourier.sign (z.val x) := by |
| simp only [score,codeVector,mul_assoc,← Finset.mul_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
|
|
| def labelFunctional (a : A) (x : Point m n) : Module.Dual F₂ (FaceArray A m n) where |
| toFun B := ∑ i : Fin m, B ⟨i,face x i⟩ a |
| map_add' _ _ := by simp [Finset.sum_add_distrib] |
| map_smul' _ _ := by simp [Finset.mul_sum] |
|
|
| omit [Fintype A] in |
| lemma character_label (a : A) (x : Point m n) (B : FaceArray A m n) : |
| character (labelFunctional a x) B=BinaryFourier.sign ((labelCode B a).val x) := rfl |
|
|
| lemma labelFunctional_degree (a : A) (x : Point m n) : degree (labelFunctional a x) ≤ m := by |
| classical |
| have hs : mask (labelFunctional a x) ⊆ Finset.univ.image (fun i : Fin m => (⟨i,face x i⟩ : Row m n)) := by |
| intro r hr |
| by_contra h |
| have hneq : ∀ i : Fin m, (⟨i,face x i⟩ : Row m n)≠r := by |
| intro i hi |
| exact h (Finset.mem_image.mpr ⟨i,Finset.mem_univ _,hi⟩) |
| have hz : frequency (labelFunctional a x) r=0 := by |
| ext b |
| simp only [frequency,LinearMap.comp_apply,labelFunctional,LinearMap.coe_mk,AddHom.coe_mk, |
| LinearMap.single_apply] |
| change (∑ i : Fin m, (Pi.single r b : FaceArray A m n) ⟨i,face x i⟩ a)=0 |
| apply Finset.sum_eq_zero |
| intro i _ |
| rw [Pi.single_eq_of_ne (hneq i)] |
| rfl |
| exact (Finset.mem_filter.mp hr).2 hz |
| exact (Finset.card_le_card hs).trans ((Finset.card_image_le).trans_eq (by simp)) |
|
|
| lemma character_noise_distance {R W : Type*} [Fintype R] [DecidableEq R] |
| [Fintype W] [DecidableEq W] [AddCommGroup W] [Module F₂ W] |
| (a : ℝ) (α : Module.Dual F₂ (R → W)) (B : R → W) : |
| (𝔼 C, density a B C*(character α B-character α C)^2)=2*(1-(1-a)^degree α) := by |
| have he (C : R → W) : density a B C*(character α B-character α C)^2= |
| 2*density a B C-2*character α B*(density a B C*character α C) := by |
| have hb := character_sq α B |
| have hc := character_sq α C |
| nlinarith only [congrArg (fun t => density a B C*t) hb,congrArg (fun t => density a B C*t) hc] |
| simp_rw [he,Finset.expect_sub_distrib,← Finset.mul_expect,density_mean,mul_one] |
| change 2-2*character α B*noise a (character α) B=_ |
| rw [noise_character] |
| have hb := character_sq α B |
| nlinarith only [congrArg (fun t => (1-a)^degree α*t) hb] |
|
|
| lemma character_noise_distance_le {R W : Type*} [Fintype R] [DecidableEq R] |
| [Fintype W] [DecidableEq W] [AddCommGroup W] [Module F₂ W] |
| {ρ : ℝ} (hρ1 : ρ≤1) (α : Module.Dual F₂ (R → W)) (B : R → W) : |
| (𝔼 C, density ρ B C*(character α B-character α C)^2)≤2*(degree α:ℝ)*ρ := by |
| rw [character_noise_distance] |
| have h := one_add_mul_le_pow (by linarith : -2 ≤ -ρ) (degree α) |
| simp only [mul_neg,← sub_eq_add_neg] at h |
| linarith |
|
|
| end MinUncut.Inner |
|
|
| end |
| end |
|
|
| end OAI |
|
|
