File size: 3,617 Bytes
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 54 55 56 57 58 59 60 61 62 63 64 65 66 | import OAI.Probability.InvariantIsing.Gaussian.GaussianColumnDeviation
namespace OAI
/-! Conditional Gaussian quadratic fluctuation, averaged over the actual remaining columns. -/
noncomputable section
open MeasureTheory ProbabilityTheory Matrix
namespace InvariantIsing
lemma gaussianGramColumnDeviation_insert_integral {N m : ℕ} (hN : 0 < N)
{t : ℝ} (ht : 0 < t) (j : Fin (m+1)) (g : Fin m → Fin N → ℝ) :
(∫ x : Fin N → ℝ, gaussianGramColumnDeviation t (gaussianColumnArray (j.insertNth x g)) j
∂Measure.pi (fun _ : Fin N => gaussianReal 0 1)) ≤ Real.sqrt (2/((N : ℝ)*t^2)) := by
let B := gaussianGramLeaveOneOut (gaussianColumnArray (j.insertNth 0 g)) j
have hB : B.PosSemidef := gaussianGramLeaveOneOut_posSemidef _ j
have he (x : Fin N → ℝ) :
gaussianGramColumnDeviation t (gaussianColumnArray (j.insertNth x g)) j =
|(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) (WithLp.toLp 2 x)-
(positiveResolvent t B).trace/N| := by
unfold gaussianGramColumnDeviation
rw [gaussianGramColumnQuadratic_eq]
simp only [Fin.insertNth_apply_same,gaussianGramLeaveResolvent,
gaussianGramLeaveOneOut_insertNth j x 0 g,positiveResolvent,B]
simp_rw [he]
have hz : HasLaw (WithLp.toLp 2 : (Fin N → ℝ) → EuclideanSpace ℝ (Fin N))
(stdGaussian _) (Measure.pi (fun _ : Fin N => gaussianReal 0 1)) :=
⟨(PiLp.continuous_toLp 2 (fun _ : Fin N => ℝ)).measurable.aemeasurable,map_pi_eq_stdGaussian⟩
have htrans : (∫ x : Fin N → ℝ, |(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B)
(WithLp.toLp 2 x)-(positiveResolvent t B).trace/N| ∂Measure.pi (fun _ => gaussianReal 0 1)) =
∫ x, |(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x-
(positiveResolvent t B).trace/N| ∂stdGaussian _ := by
simpa only [Function.comp_def] using (hz.integral_comp
(f := fun x => |(1/(N : ℝ))*gaussianQuadraticForm (positiveResolvent t B) x-
(positiveResolvent t B).trace/N|)
(((gaussianQuadraticForm_continuous _).const_mul _).sub continuous_const).abs.aestronglyMeasurable)
rw [htrans]
exact positiveResolvent_normalized_fluctuation hN ht hB
theorem gaussianGramColumnDeviation_mean_le {N m : ℕ} (hN : 0 < N) {t : ℝ} (ht : 0 < t)
(j : Fin (m+1)) :
(∫ g : Fin (m+1) → Fin N → ℝ, gaussianGramColumnDeviation t (gaussianColumnArray g) j
∂Measure.pi (fun _ : Fin (m+1) => Measure.pi (fun _ : Fin N => gaussianReal 0 1))) ≤
Real.sqrt (2/((N : ℝ)*t^2)) := by
let μ := Measure.pi (fun _ : Fin N => gaussianReal 0 1)
let ν := Measure.pi (fun _ : Fin (m+1) => μ)
let ρ := Measure.pi (fun _ : Fin m => μ)
let e := (MeasurableEquiv.piFinSuccAbove (fun _ : Fin (m+1) => Fin N → ℝ) j).symm
let F := fun g : Fin (m+1) → Fin N → ℝ => gaussianGramColumnDeviation t (gaussianColumnArray g) j
have hm : MeasurePreserving e (μ.prod ρ) ν := (measurePreserving_piFinSuccAbove (fun _ => μ) j).symm
have hi : Integrable F ν := gaussianGramColumnDeviation_integrable hN ht j
have hip : Integrable (F ∘ e) (μ.prod ρ) := hm.hasLaw.integrable_comp hi
calc
_ = ∫ p, F (e p) ∂μ.prod ρ := (hm.integral_comp' F).symm
_ = ∫ g, ∫ x, F (e (x,g)) ∂μ ∂ρ := integral_prod_symm _ hip
_ ≤ ∫ _ : Fin m → Fin N → ℝ, Real.sqrt (2/((N : ℝ)*t^2)) ∂ρ := by
apply integral_mono hip.integral_prod_right (integrable_const _)
intro g
exact gaussianGramColumnDeviation_insert_integral hN ht j g
_ = _ := by simp
end InvariantIsing
end
end OAI
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