| import OAI.Probability.InvariantIsing.Arrays.TensorPerturbationVariance |
| import OAI.Probability.InvariantIsing.Arrays.MonomialEnumeration |
| import OAI.Probability.InvariantIsing.Spectral.SpectralDiagonalPerturbation |
|
|
| namespace OAI |
|
|
| /-! Uniform variance for both terms of the actual finite perturbation. -/ |
|
|
| noncomputable section |
|
|
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators NNReal |
|
|
| namespace InvariantIsing |
|
|
| /-- The explicit complete monomial enumeration removes the degree-cap |
| assumption. The diagonal perturbation is the exact eigenvalue shift proved |
| in `rotatedEnergy_diagonalPerturbation`. -/ |
| theorem fullPerturbationPressure_variance (hhaar : HaarConcentrationInput) |
| (hgauss : GaussianLipschitzVarianceInput) : |
| ∃ C : ℝ, 0 < C ∧ |
| ∀ N : ℕ, 3 ≤ N → |
| ∀ μ : Measure (SpecialOrthogonal N), IsProbabilityMeasure μ → μ.IsMulLeftInvariant → |
| ∀ m : ℕ, ∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) → |
| ∀ I : Fin m → Finset (Fin N), ∀ u : Fin N → ℝ, (∀ j, |u j| ≤ 2) → |
| ∀ v : Fin m → ℝ, (∀ a, |v a| ≤ 2) → ∀ t : ℝ, |t| ≤ 1 → |
| ∀ n : ℕ, ∀ b : ℕ → ℝ, CascadeExponents n b → |
| ∀ h : ℕ → ℝ, Monotone h → 0 ≤ h 0 → ∀ H : ℝ, h n ≤ H → |
| let degree := fun j : Fin N => enumeratedSpectralDegree m j |
| let treeDegree := fun j : Fin N => enumeratedTreeDegree m j |
| let profile := tensorPathProfile I degree n treeDegree h |
| let P := tensorRootTreeLaw I degree n b (fun i => profile (i + 1)) (profile 0) |
| let F := tensorDisorderPressure (diagonalPerturbedEigenvalues eig I v t) c I degree |
| (tensorPerturbationAmplitude N u) n |
| MemLp F 2 (μ.prod P) ∧ variance F (μ.prod P) ≤ |
| (4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2 |
| ∂(rawCascadeLaw n b : Measure (RawTree n))) + H + 4 + |
| C * (K + 4 * m + 8) ^ 2) / N := by |
| obtain ⟨C, hC, hv⟩ := tensorPerturbationPressure_variance hhaar hgauss |
| refine ⟨C, hC, ?_⟩ |
| intro N hN μ hμ hμinv m eig c K hK heig I u hu v hvbound t ht n b hb h hh h0 H hH |
| degree treeDegree profile P F |
| have hNm : 0 < N := by omega |
| have hK' : 0 < K + 4 * m := by positivity |
| have heig' : ∀ i, |diagonalPerturbedEigenvalues eig I v t i| ≤ K + 4 * m := |
| diagonalPerturbedEigenvalues_abs_le hNm eig I v hvbound t ht K heig |
| have hd : ∀ j : Fin N, (∑ a, (degree j a : ℝ)) ≤ (1 : ℝ) * ((j : ℝ) + 1) := by |
| intro j |
| simpa only [one_mul] using enumeratedSpectralDegree_real_sum_le m j |
| simpa only [mul_one] using hv N hN μ hμ hμinv m |
| (diagonalPerturbedEigenvalues eig I v t) c (K + 4 * m) hK' heig' I degree u hu |
| 1 (by norm_num) hd n treeDegree b hb h hh h0 H hH |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
|
|
