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2f887db | 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 | 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
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