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f61e547 | 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 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 | import OAI.Probability.FreeIsing.Conditioning.ShiftedLoss
import OAI.Probability.FreeIsing.Energy.LinearTiltVariance
import OAI.Probability.FreeIsing.Estimates.MessageFunction
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory
namespace Problem367
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω} [IsProbabilityMeasure μ]
lemma variance_cavityMessage_linearSignalTilt
{theta eta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hη : |eta| ≤ 1)
{Y : Ω → ℝ} (hY : Measurable Y) (T : Ω ≃ᵐ Ω)
(hT : MeasurePreserving T μ μ) (hYodd : ∀ᵐ w ∂μ, Y (T w) = -Y w) :
(1 - theta ^ 2) * theta ^ 2 * (∫ w, Real.tanh (Y w) ^ 2 ∂μ) ≤
variance (fun w => cavityMessage theta (Y w))
(linearSignalTilt μ (fun w => Real.tanh (Y w)) (eta * theta)) := by
have hθsq : theta ^ 2 ≤ 1 := by nlinarith
have hq : Measurable (fun w => Real.tanh (Y w)) :=
lipschitzWith_tanh_response.continuous.measurable.comp hY
have hqb : ∀ᵐ w ∂μ, |Real.tanh (Y w)| ≤ 1 :=
Filter.Eventually.of_forall fun w => (Real.abs_tanh_lt_one _).le
have ha : |eta * theta| ≤ 1 := by
rw [abs_mul, abs_of_nonneg hθ0]
calc
|eta| * theta ≤ 1 * theta := mul_le_mul_of_nonneg_right hη hθ0
_ ≤ 1 := by simpa using hθ1.le
have hgMeas : Measurable (fun w => cavityMessage theta (Y w)) :=
(lipschitzWith_cavityMessage hθ0 hθ1.le).continuous.measurable.comp hY
have hg : MemLp (fun w => cavityMessage theta (Y w)) 2 μ :=
MemLp.of_bound hgMeas.aestronglyMeasurable (Real.artanh theta)
(Filter.Eventually.of_forall fun w => by
simpa only [Real.norm_eq_abs] using cavityMessage_abs_le_artanh hθ0 hθ1 (Y w))
apply variance_linearSignalTilt_ge_signal_moment hq hqb ha hg T hT
· filter_upwards [hYodd] with w hw
rw [hw, Real.tanh_neg]
· filter_upwards [hYodd] with w hw
rw [hw, cavityMessage_neg hθsq]
· have hηsq := (sq_le_one_iff_abs_le_one eta).mpr hη
nlinarith [mul_le_mul_of_nonneg_right hηsq (sq_nonneg theta)]
· exact hθsq
· filter_upwards [] with w
have h := cavityMessage_abs_ge_tanh hθsq (Y w)
have hs := (sq_le_sq₀ (mul_nonneg (abs_nonneg theta) (abs_nonneg _))
(abs_nonneg _)).mpr h
simpa only [mul_pow, sq_abs] using hs
def sideSignalLossCoefficient (theta : ℝ) : ℝ :=
shiftedLossCoefficient theta (Real.artanh theta) * (1 - theta ^ 2) * theta ^ 2
lemma sideSignalLossCoefficient_pos {theta : ℝ} (hθ0 : 0 < theta) (hθ1 : theta < 1) :
0 < sideSignalLossCoefficient theta := by
have hθsq : theta ^ 2 < 1 := by nlinarith
exact mul_pos (mul_pos (shiftedLossCoefficient_pos hθsq) (sub_pos.mpr hθsq))
(sq_pos_of_pos hθ0)
lemma integral_tilted_cavity_attenuation_le
{theta eta : ℝ} (hθ0 : 0 ≤ theta) (hθ1 : theta < 1) (hη : |eta| ≤ 1)
{Y : Ω → ℝ} (hY : Measurable Y) (T : Ω ≃ᵐ Ω)
(hT : MeasurePreserving T μ μ) (hYodd : ∀ᵐ w ∂μ, Y (T w) = -Y w) (z : ℝ) :
(∫ w, responseAttenuation theta (z + cavityMessage theta (Y w)) ^ 2
∂linearSignalTilt μ (fun w => Real.tanh (Y w)) (eta * theta)) ≤
1 - sideSignalLossCoefficient theta * (∫ w, Real.tanh (Y w) ^ 2 ∂μ) := by
have hθsq : theta ^ 2 < 1 := by nlinarith
have hq : Measurable (fun w => Real.tanh (Y w)) :=
lipschitzWith_tanh_response.continuous.measurable.comp hY
have hqb : ∀ᵐ w ∂μ, |Real.tanh (Y w)| ≤ 1 :=
Filter.Eventually.of_forall fun w => (Real.abs_tanh_lt_one _).le
have ha : |eta * theta| ≤ 1 := by
rw [abs_mul, abs_of_nonneg hθ0]
calc
|eta| * theta ≤ 1 * theta := mul_le_mul_of_nonneg_right hη hθ0
_ ≤ 1 := by simpa using hθ1.le
have hqodd : ∀ᵐ w ∂μ, Real.tanh (Y (T w)) = -Real.tanh (Y w) := by
filter_upwards [hYodd] with w hw
rw [hw, Real.tanh_neg]
let := linearSignalTilt_probability hq hqb ha
(integral_eq_zero_of_ae_sign_reversal T hT hqodd)
have hgMeas : Measurable (fun w => cavityMessage theta (Y w)) :=
(lipschitzWith_cavityMessage hθ0 hθ1.le).continuous.measurable.comp hY
have hloss := integral_shifted_attenuation_loss_explicit
(μ := linearSignalTilt μ (fun w => Real.tanh (Y w)) (eta * theta)) hθsq.le hgMeas
(Filter.Eventually.of_forall fun w => cavityMessage_abs_le_artanh hθ0 hθ1 (Y w)) z
have hvariance := variance_cavityMessage_linearSignalTilt hθ0 hθ1 hη hY T hT hYodd
have hmul := mul_le_mul_of_nonneg_left hvariance
(shiftedLossCoefficient_pos (beta := Real.artanh theta) hθsq).le
unfold sideSignalLossCoefficient
nlinarith [hloss]
end Problem367
end
end OAI
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