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b317718 | 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 | 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
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
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
end
end OAI
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