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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 | import OAI.Computability.PerfectCompleteness.Foundations.GlobalityLemmas
namespace OAI
section
namespace PerfectCompleteness.MatrixInverse
noncomputable section
open scoped BigOperators Classical
open UniqueGamesTheorem.Fourier.MatrixCharacters UniqueGamesTheorem.Fourier.MatrixFourier
open UniqueGamesTheorem.Fourier.MatrixNoise UniqueGamesTheorem.Fourier.MatrixRestrictions
open UniqueGamesTheorem.Fourier.MatrixLevelBridge
open UniqueGamesTheorem.Appendix.LevelInequality UniqueGamesTheorem.Appendix.RankLevelFilter
variable {E F : Type*}
[AddCommGroup E] [Module F2 E] [AddCommGroup F] [Module F2 F]
[FiniteDimensional F2 E] [FiniteDimensional F2 F]
[Finite E] [Fintype F] [Fintype (E →ₗ[F2] F2)]
[Fintype (E →ₗ[F2] F)] [Fintype (F →ₗ[F2] E)]
def noiseEnergy (f : (E →ₗ[F2] F) → ℝ) : ℝ :=
𝔼 X, f X * linearRealNoiseOperator (fun _ : F => (Fintype.card F : ℝ)⁻¹) f X
omit [Finite E] in
theorem noiseEnergy_eq (f : (E →ₗ[F2] F) → ℝ) :
noiseEnergy f = ∑ S : F →ₗ[F2] E, node (frequencyRank S) * linearCoeff f S ^ 2 := by
unfold noiseEnergy
rw [linearRealNoiseOperator_energy]
apply Finset.sum_congr rfl
intro S _
rw [linearNoiseEigenvalue_uniform, node_eq_inv_pow]
rfl
omit [Finite E] in
theorem noiseEnergy_le_cutoff (r : Nat) (f : (E →ₗ[F2] F) → ℝ) :
noiseEnergy f ≤ rankCutoffEnergy r f + node (r + 1) * (𝔼 X, f X ^ 2) := by
rw [noiseEnergy_eq]
calc
_ ≤ ∑ S : F →ₗ[F2] E,
((if frequencyRank S ≤ r then linearCoeff f S ^ 2 else 0) +
node (r + 1) * linearCoeff f S ^ 2) := by
apply Finset.sum_le_sum
intro S _
by_cases h : frequencyRank S ≤ r
· rw [ite_eq_left h]
have h₁ := mul_le_mul_of_nonneg_right (node_le_one (frequencyRank S))
(sq_nonneg (linearCoeff f S))
have h₂ := mul_nonneg (node_pos (r + 1)).le (sq_nonneg (linearCoeff f S))
linarith
· rw [ite_eq_right h, zero_add]
exact mul_le_mul_of_nonneg_right
(node_antitone (Nat.succ_le_of_lt (Nat.lt_of_not_ge h))) (sq_nonneg _)
_ = rankCutoffEnergy r f + node (r + 1) * (𝔼 X, f X ^ 2) := by
rw [Finset.sum_add_distrib, ← Finset.mul_sum, linear_parseval]
simp only [rankCutoffEnergy, Finset.sum_filter]
theorem noiseEnergy_le_of_global (r : Nat) (ρ : ℝ) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(f : (E →ₗ[F2] F) → ℝ) (hf : IsBoolean f)
(hg : IsRestrictionGlobal f r ρ) :
noiseEnergy f ≤ (levelCutoffConstant r ρ + node (r + 1)) * (𝔼 X, f X ^ 2) := by
have hcut := rankCutoffEnergy_le_of_theorem23
(UniqueGamesTheorem.Appendix.LevelTheorem.theorem23 (E := E) (F := F)) f r ρ hρ hρ1 hf hg
calc
noiseEnergy f ≤ rankCutoffEnergy r f + node (r + 1) * (𝔼 X, f X ^ 2) :=
noiseEnergy_le_cutoff r f
_ ≤ levelCutoffConstant r ρ * (𝔼 X, f X ^ 2) +
node (r + 1) * (𝔼 X, f X ^ 2) := add_le_add hcut (le_refl _)
_ = _ := (add_mul _ _ _).symm
theorem exists_dense_restriction (r : Nat) (ρ η : ℝ) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : levelCutoffConstant r ρ + node (r + 1) < η)
(f : (E →ₗ[F2] F) → ℝ) (hf : IsBoolean f)
(hpositive : 0 < 𝔼 X, f X ^ 2)
(hretention : η * (𝔼 X, f X ^ 2) ≤ noiseEnergy f) :
∃ (W : Submodule F2 E) (C' : Submodule F2 F) (T : E →ₗ[F2] F),
order W C' ≤ r ∧ ρ < (𝔼 N, restrict f W C' T N) := by
by_contra hnone
have hg : IsRestrictionGlobal f r ρ := by
apply (boolean_global_iff_density f hf r ρ).mpr
intro W C' T horder
apply le_of_not_gt
intro hdense
exact hnone ⟨W, C', T, horder, hdense⟩
have hbound := noiseEnergy_le_of_global r ρ hρ hρ1 f hf hg
have hstrict := mul_lt_mul_of_pos_right hconstants hpositive
exact (not_lt_of_ge (hretention.trans hbound)) hstrict
theorem exists_dense_restriction_of_family {Y : Type*} [Fintype Y]
(r : Nat) (ρ η : ℝ) (hρ : 0 < ρ) (hρ1 : ρ < 1)
(hconstants : levelCutoffConstant r ρ + node (r + 1) < η)
(f : Y → (E →ₗ[F2] F) → ℝ) (hf : ∀ y, IsBoolean (f y))
(hmass : (∑ y, 𝔼 X, f y X ^ 2) ≤ 1)
(hretention : η ≤ ∑ y, noiseEnergy (f y)) :
∃ (y : Y) (W : Submodule F2 E) (C' : Submodule F2 F) (T : E →ₗ[F2] F),
order W C' ≤ r ∧ ρ < (𝔼 N, restrict (f y) W C' T N) := by
by_contra hnone
have hg (y : Y) : IsRestrictionGlobal (f y) r ρ := by
apply (boolean_global_iff_density (f y) (hf y) r ρ).mpr
intro W C' T horder
apply le_of_not_gt
intro hdense
exact hnone ⟨y, W, C', T, horder, hdense⟩
have hbound : (∑ y, noiseEnergy (f y)) ≤
(levelCutoffConstant r ρ + node (r + 1)) * (∑ y, 𝔼 X, f y X ^ 2) := by
rw [Finset.mul_sum]
exact Finset.sum_le_sum fun y _ => noiseEnergy_le_of_global r ρ hρ hρ1 (f y) (hf y) (hg y)
have hnonneg : 0 ≤ levelCutoffConstant r ρ + node (r + 1) :=
add_nonneg (levelCutoffConstant_nonneg r ρ hρ.le) (node_pos _).le
have hbound' := hbound.trans (mul_le_mul_of_nonneg_left hmass hnonneg)
simp only [mul_one] at hbound'
exact (not_lt_of_ge (hretention.trans hbound')) hconstants
end
end PerfectCompleteness.MatrixInverse
end
end OAI
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