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92416a2 | 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 | import OAI.Computability.UniqueGames.Repetition.CollisionLemmas
import OAI.Computability.UniqueGames.Repetition.RoundingPaddingLemmas
import OAI.Computability.UniqueGames.Repetition.RoundingThresholdLemmas
namespace OAI
section
/-! The alphabet-independent high-value bound for the actual projection
operator. Positive vectors are determinized, padded, and rounded using proved
finite constructions. No rounding or repetition hypothesis is exposed. -/
namespace UniqueGamesTheorem.Repetition.ProjectionKernel
open scoped BigOperators
open Foundations.Games
noncomputable section
variable {Q₁ Q₂ A₁ A₂ Ω : Type*}
[Fintype Q₁] [Fintype Q₂] [Fintype A₁] [Fintype A₂] [Fintype Ω]
private theorem distribution_nonempty {T : Type*} [Fintype T]
(μ : FiniteDistribution T) : Nonempty T := by
by_contra h
have : IsEmpty T := not_nonempty_iff.mp h
have hz : (∑ t, μ.weight t) = 0 := by simp
linarith [μ.normalized]
/-- Unit mass form of the Dinur--Steurer positive-vector approximation bound.
The loss is independent of both alphabets and of the auxiliary dimension. -/
theorem vectorEnergy_le_gap_rate [Nonempty A₁] [Nonempty A₂]
(K : ProjectionKernel Q₁ Q₂ A₁ A₂) (f : Ω → Q₁ → A₁ → ℝ)
(hf : ∀ ω x a, 0 ≤ f ω x a) (hmass : ∀ x, vectorMass f x ≤ 1)
{g : ℝ} (hg0 : 0 ≤ g) (hg1 : g ≤ 1)
(hvalue : K.toGame.value ≤ 1 - g) :
K.vectorEnergy f ≤ 1 - g ^ 2 / 8 := by
classical
let : Nonempty Q₂ := distribution_nonempty K.outer
let : Nonempty Q₁ := distribution_nonempty
(K.inner (Classical.choice (inferInstance : Nonempty Q₂)))
rcases isEmpty_or_nonempty Ω with hΩ | hΩ
· let : IsEmpty Ω := hΩ
have hzero : K.vectorEnergy f = 0 := by simp [vectorEnergy]
rw [hzero]
have hsq : g * g ≤ (1 : ℝ) * 1 := mul_le_mul hg1 hg1 hg0 (by norm_num)
nlinarith
· let : Nonempty Ω := hΩ
obtain ⟨labels, hdet⟩ := K.exists_vectorEnergy_le_singleLabelVector f hf
have hnonneg := vectorAmplitude_nonnegative f hf
have hrow : ∀ x, rowSquareMass (vectorAmplitude f) x ≤ 1 := by
intro x
rw [rowSquareMass_vectorAmplitude]
exact hmass x
rw [K.vectorEnergy_singleLabelVector] at hdet
by_cases hg : g = 0
· subst g
simp only [zero_pow (by decide : 2 ≠ 0), zero_div, sub_zero]
let fallback : A₁ := Classical.choice inferInstance
have hpad := pairEnergy_le_padded K.symmetricDistribution symmetricLeft
symmetricRight K.symmetricAccept labels (vectorAmplitude f) hnonneg fallback
have hone := pairEnergy_le_one_of_unit_rows K.symmetricDistribution
symmetricLeft symmetricRight K.symmetricAccept
(paddedLabel labels fallback) (paddedAmplitude (vectorAmplitude f))
(paddedAmplitude_row (vectorAmplitude f) hrow)
exact hdet.trans (hpad.trans hone)
· have hgpos : 0 < g := lt_of_le_of_ne hg0 (Ne.symm hg)
have hscore (a : Q₁ → A₁) :
partialLabelScore K.symmetricDistribution symmetricLeft symmetricRight
K.symmetricAccept a ≤ 1 - g := by
unfold partialLabelScore
rw [K.symmetric_probability_eq_assignmentEnergy]
exact (K.assignmentEnergy_le_collisionValue a).trans
(K.collisionValue_le_value.trans hvalue)
exact hdet.trans (pairEnergy_le_of_labeling_gap K.symmetricDistribution
symmetricLeft symmetricRight K.symmetricAccept labels (vectorAmplitude f)
hnonneg hrow hgpos hscore)
end
end UniqueGamesTheorem.Repetition.ProjectionKernel
end
end OAI
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