| import OAI.Probability.InvariantIsing.Core.FiniteSpectrumStepUpper |
| import OAI.Probability.InvariantIsing.Core.TrialGridApproximation |
|
|
| namespace OAI |
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| /-! Manuscript Proposition up:bound, in the equivalent eventual-epsilon |
| form against every trial of finite entropy. -/ |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron Set Filter |
| open scoped BigOperators Topology |
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| namespace InvariantIsing |
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|
| theorem finiteSpectrum_meanPressure_upper |
| (hhaar : HaarConcentrationInput) (hgauss : GaussianLipschitzVarianceInput) |
| (N : ℕ → ℕ) (hN : ∀ k, 3 ≤ N k) (hNlim : Tendsto N atTop atTop) (m : ℕ) |
| (μ : (k : ℕ) → Measure (SpecialOrthogonal (N k))) [∀ k, IsProbabilityMeasure (μ k)] |
| (hμinv : ∀ k, (μ k).IsMulLeftInvariant) |
| (eig : (k : ℕ) → Fin (N k) → ℝ) |
| (K : ℝ) (hK : 0 < K) (heig : ∀ k i, |eig k i| ≤ K) |
| (I : (k : ℕ) → Fin m → Finset (Fin (N k))) |
| (hdis : ∀ k, Set.PairwiseDisjoint (Set.univ : Set (Fin m)) (I k)) |
| (hcover : ∀ k, Finset.univ.biUnion (I k) = Finset.univ) |
| (lam : Fin m → ℝ) (hlam : ∀ k a i, i ∈ I k a → eig k i = lam a) |
| (ρ : Fin m → ℝ) (hρpos : ∀ a, 0 < ρ a) (hρsum : ∑ a, ρ a = 1) |
| (hρ : Tendsto (fun k a => ((I k a).card : ℝ) / N k) atTop (𝓝 ρ)) |
| (p : OverlapPath) (S : ℝ) (hS : entropyFunctional p ≤ (S : EReal)) : |
| ∀ ε : ℝ, 0 < ε → ∀ᶠ k in atTop, |
| (∫ U : SpecialOrthogonal (N k), |
| rotatedPressure (eig k) (specialRotation U) (fun _ => 0) ∂μ k) ≤ |
| S + spectralFunctional (finiteR ρ lam hρpos hρsum) p + ε := by |
| intro ε hε |
| have hhalf : 0 < ε / 2 := half_pos hε |
| have hgrid := lowerGridPath_spectral_tendsto ρ lam hρpos hρsum p |
| obtain ⟨n, hn⟩ := (hgrid.eventually (gt_mem_nhds |
| (show spectralFunctional (finiteR ρ lam hρpos hρsum) p < |
| spectralFunctional (finiteR ρ lam hρpos hρsum) p + ε / 2 by linarith))).exists |
| have hup := finiteSpectrum_meanPressure_step_upper hhaar hgauss N hN hNlim m n |
| μ hμinv eig K hK heig I hdis hcover lam hlam ρ hρpos hρsum hρ |
| (uniformCut n) (uniformCut_strict n) (uniformCut_zero n) (uniformCut_last n) |
| (lowerGridPath p n) |
| (fun i => (1 - 1 / (n + 1 : ℕ)) * p ((i.val : ℝ) / (n + 1 : ℕ))) |
| (lowerGridPath_on_cell p n) (1 / (n + 1 : ℕ)) (by positivity) |
| (ae_of_all _ (lowerGridPath_below_one p n)) S |
| ((lowerGridPath_entropy_le p n).trans hS) (ε / 2) hhalf |
| filter_upwards [hup] with k hk |
| linarith |
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|
| end InvariantIsing |
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| end |
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| end OAI |
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|
