openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /EmpiricalClippingDistance.lean
| import OAI.Probability.InvariantIsing.Core.LevyMapDistance | |
| import OAI.Probability.InvariantIsing.Spectral.CompactSpectralClipping | |
| import OAI.Probability.InvariantIsing.Spectral.SpectralExcess | |
| namespace OAI | |
| /-! The finite spectral clipping moves the empirical law by at most its edge excess. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory | |
| namespace InvariantIsing | |
| theorem empirical_clipping_distance {N : ℕ} (hN : 0 < N) (eig : Fin N → ℝ) | |
| {R : ℝ} (hR : 0 ≤ R) : | |
| dist (LevyProkhorov.ofMeasure (empiricalSpectralLaw hN eig)) | |
| (LevyProkhorov.ofMeasure ((empiricalSpectralLaw hN eig).map (spectralClip 0 R))) ≤ | |
| spectralExcess eig 0 R := by | |
| let μ := empiricalSpectralLaw hN eig | |
| have hh : ∀ᵐ x ∂(μ : Measure ℝ), dist (id x) (spectralClip 0 R x) ≤ spectralExcess eig 0 R := by | |
| apply ae_finiteSpectralMeasure | |
| intro i | |
| rw [Real.dist_eq,abs_sub_comm] | |
| exact spectralClip_close hR (spectralExcess_nonneg _ _ _) (spectralExcess_bounds eig 0 R i) | |
| have h := levyProkhorov_map_dist_le_ae (μ : Measure ℝ) id (spectralClip 0 R) | |
| measurable_id (continuous_spectralClip 0 R).measurable (spectralExcess_nonneg _ _ _) hh | |
| rw [Measure.map_id] at h | |
| exact h | |
| end InvariantIsing | |
| end | |
| end OAI | |