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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 | import OAI.Probability.IsingPerceptron.NamespaceSplit
namespace OAI
/-! Centering a logpartition difference using its two pressure variances. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
namespace InvariantIsing
/-- The frozen CGF of one perturbation is a scaled difference of two
pressures. Both variances suffice even though those pressures are dependent. -/
theorem centered_scaled_difference_L1_bound {Ω : Type*} [MeasurableSpace Ω]
(P : Measure Ω) [IsProbabilityMeasure P] {F G Z : Ω → ℝ} {a B : ℝ}
(hF : MemLp F 2 P) (hG : MemLp G 2 P)
(hvF : variance F P ≤ B) (hvG : variance G P ≤ B)
(he : Z =ᵐ[P] fun ω => a * (F ω - G ω)) :
MemLp Z 2 P ∧
(∫ ω, |Z ω - ∫ ω', Z ω' ∂P| ∂P) ≤ 2 * |a| * Real.sqrt B := by
have hiF := hF.integrable (by norm_num)
have hiG := hG.integrable (by norm_num)
have hZ : MemLp Z 2 P := MemLp.ae_eq he.symm ((hF.sub hG).const_mul a)
have hcF : (∫ ω, |F ω - ∫ ω', F ω' ∂P| ∂P) ≤ Real.sqrt B :=
(l1_center_le_sqrt_variance hF).trans (Real.sqrt_le_sqrt hvF)
have hcG : (∫ ω, |G ω - ∫ ω', G ω' ∂P| ∂P) ≤ Real.sqrt B :=
(l1_center_le_sqrt_variance hG).trans (Real.sqrt_le_sqrt hvG)
have hc := centered_sub_L1_bound P hiF hiG hcF hcG
have hm : (∫ ω, Z ω ∂P) = a * (∫ ω, F ω - G ω ∂P) := by
rw [integral_congr_ae he, integral_const_mul]
refine ⟨hZ, ?_⟩
rw [hm]
calc
_ = ∫ ω, |a| * |F ω - G ω - ∫ ω', F ω' - G ω' ∂P| ∂P := by
apply integral_congr_ae
filter_upwards [he] with ω hω
rw [hω, ← mul_sub, abs_mul]
_ = |a| * (∫ ω, |F ω - G ω - ∫ ω', F ω' - G ω' ∂P| ∂P) := integral_const_mul _ _
_ ≤ |a| * (Real.sqrt B + Real.sqrt B) :=
mul_le_mul_of_nonneg_left hc (abs_nonneg _)
_ = _ := by ring
end InvariantIsing
end
end OAI
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