| import OAI.Probability.InvariantIsing.Arrays.NSpinTensorComparison |
| import OAI.Probability.InvariantIsing.Haar.HaarVariance |
|
|
| namespace OAI |
|
|
| /-! The Haar variance contribution for the actual tensor-enriched pressure. -/ |
|
|
| noncomputable section |
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|
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators NNReal |
|
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| namespace InvariantIsing |
|
|
| /-- Haar contributes the square of the normalized covariance rotation |
| modulus divided by volume. The Gaussian and cascade randomness have been |
| averaged in this observable, using the proved flat mean identity. -/ |
| theorem haar_tensorEnrichedPressure_variance (hpub : HaarConcentrationInput) : |
| ∃ C : ℝ, 0 < C ∧ |
| ∀ N : ℕ, 3 ≤ N → |
| ∀ μ : Measure (SpecialOrthogonal N), |
| IsProbabilityMeasure μ → μ.IsMulLeftInvariant → |
| ∀ m k : ℕ, ∀ eig c : Fin N → ℝ, |
| ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) → |
| ∀ I : Fin m → Finset (Fin N), ∀ degree : Fin k → Fin m → ℕ, |
| ∀ amplitude : Fin k → ℝ, ∀ n : ℕ, ∀ b : ℕ → ℝ, |
| CascadeExponents n b → |
| ∀ site : ℕ → ℝ≥0, ∀ monomial : ℕ → Fin k → ℝ≥0, |
| let v := fun i => tensorVarianceProfile I degree (site i) (monomial i) |
| let F := fun U : SpecialOrthogonal N => |
| tensorEnrichedPressure eig (specialRotation U) c I degree amplitude n b |
| (fun i => v (i + 1)) (v 0) |
| let L := K + 2 * (N : ℝ)⁻¹ * |
| (∑ i : Fin (n + 1), ∑ j, (monomial i j : ℝ) * amplitude j ^ 2 * ∑ a, (degree j a : ℝ)) |
| MemLp F 2 μ ∧ variance F μ ≤ C * L ^ 2 / N := by |
| obtain ⟨C, hC, hvar⟩ := haar_frobeniusLipschitz_variance hpub |
| refine ⟨C, hC, ?_⟩ |
| intro N hN μ hμ hμinv m k eig c K hK heig I degree amplitude n b hb site monomial v F L |
| let : IsProbabilityMeasure μ := hμ |
| have hNm : 0 < N := by omega |
| have hm : Measurable F := measurable_tensorEnrichedPressure eig c I degree amplitude n b _ _ |
| have hL : 0 < L := by |
| dsimp only [L] |
| exact add_pos_of_pos_of_nonneg hK (by positivity) |
| have hLip : ∀ U V, |F U - F V| ≤ L * frobeniusDistance U V := fun U V => |
| abs_tensorEnrichedPressure_sub_rotation_le hNm eig c U V K hK.le heig |
| I degree amplitude n b hb site monomial |
| exact ⟨haar_frobeniusLipschitz_memLp_two hpub N hN μ hμinv F hm L hL hLip, |
| hvar N hN μ hμ hμinv F hm L hL hLip⟩ |
|
|
| end InvariantIsing |
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| end |
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| end OAI |
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|
