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a89d282 | 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 | 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
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