| 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] |
|
|
| / |
| 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 |
|
|
