openAI-math-dataset / lean /OAI /Probability /InvariantIsing /Spectral /ClippedEmpiricalTransform.lean
| import OAI.Probability.InvariantIsing.Spectral.CompactSpectralClipping | |
| import OAI.Probability.InvariantIsing.Spectral.ResolventLipschitz | |
| import OAI.Probability.InvariantIsing.Spectral.SpectralExcess | |
| namespace OAI | |
| /-! Uniform resolvent error for the actual finite empirical spectral clipping. -/ | |
| noncomputable section | |
| open MeasureTheory ProbabilityTheory Set | |
| namespace InvariantIsing | |
| theorem empirical_transform_clip_error {N : ℕ} (hN : 0 < N) (eig : Fin N → ℝ) | |
| {R η t : ℝ} (ht : 0 < t) | |
| (hclip : ∀ i, |spectralClip 0 R (eig i)-eig i| ≤ η) : | |
| |(∫ x, positiveResolventTest t (spectralClip 0 R x) | |
| ∂(empiricalSpectralLaw hN eig : Measure ℝ))- | |
| ∫ x, positiveResolventTest t x ∂(empiricalSpectralLaw hN eig : Measure ℝ)| ≤ η/t^2 := by | |
| let μ := (empiricalSpectralLaw hN eig : Measure ℝ) | |
| have hc : Integrable (fun x => positiveResolventTest t (spectralClip 0 R x)) μ := by | |
| apply Integrable.of_bound | |
| ((continuous_positiveResolventTest ht).comp (continuous_spectralClip 0 R)).aestronglyMeasurable (1/t) | |
| exact ae_of_all _ fun x => by | |
| change ‖positiveResolventTest t (spectralClip 0 R x)‖ ≤ 1/t | |
| rw [Real.norm_eq_abs,abs_of_nonneg (positiveResolventTest_bound ht _).1] | |
| exact (positiveResolventTest_bound ht _).2 | |
| have hi : Integrable (positiveResolventTest t) μ := by | |
| simpa only [pow_one] using positiveResolventPower_integrable ht 1 μ | |
| rw [← integral_sub hc hi,← Real.norm_eq_abs] | |
| have ha : ∀ᵐ x ∂μ, |spectralClip 0 R x-x| ≤ η := ae_finiteSpectralMeasure | |
| (fun _ => (1 : ℝ)/N) eig hclip | |
| have hb : ∀ᵐ x ∂μ, ‖positiveResolventTest t (spectralClip 0 R x)-positiveResolventTest t x‖ ≤ η/t^2 := by | |
| filter_upwards [ha] with x hx | |
| rw [Real.norm_eq_abs] | |
| exact (positiveResolventTest_sub_le ht _ _).trans | |
| (div_le_div_of_nonneg_right hx (sq_nonneg t)) | |
| simpa only [probReal_univ,mul_one] using norm_integral_le_of_norm_le_const hb | |
| theorem empirical_transform_clip_excess {N : ℕ} (hN : 0 < N) (eig : Fin N → ℝ) | |
| {R t : ℝ} (hR : 0 ≤ R) (ht : 0 < t) : | |
| |(∫ x, positiveResolventTest t (spectralClip 0 R x) | |
| ∂(empiricalSpectralLaw hN eig : Measure ℝ))- | |
| ∫ x, positiveResolventTest t x ∂(empiricalSpectralLaw hN eig : Measure ℝ)| ≤ | |
| spectralExcess eig 0 R/t^2 := by | |
| apply empirical_transform_clip_error hN eig ht | |
| intro i | |
| exact spectralClip_close hR (spectralExcess_nonneg _ _ _) (spectralExcess_bounds eig 0 R i) | |
| end InvariantIsing | |
| end | |
| end OAI | |