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b317718 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 | import OAI.Probability.FreeIsing.Gaussian.BinaryFusion
namespace OAI
noncomputable section
open MeasureTheory
namespace Problem367
theorem ae_binaryFusion_bounds {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
(hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1) :
∀ᵐ p ∂mu.prod nu, |p.1| < 1 ∧ |p.2| < 1 := by
have hu' : ∀ᵐ p ∂mu.prod nu, |p.1| < 1 :=
measurePreserving_fst.quasiMeasurePreserving.tendsto_ae hu
have hv' : ∀ᵐ p ∂mu.prod nu, |p.2| < 1 :=
measurePreserving_snd.quasiMeasurePreserving.tendsto_ae hv
exact hu'.and hv'
theorem integrable_binaryFusion_integrand {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
(hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1) :
Integrable (fun p : ℝ × ℝ => (p.1 + p.2) ^ 2 / (1 + p.1 * p.2))
(mu.prod nu) := by
refine (integrable_const (2 : ℝ)).mono' ?_ ?_
· apply Measurable.aestronglyMeasurable
fun_prop
filter_upwards [ae_binaryFusion_bounds hu hv] with p hp
rw [Real.norm_eq_abs, abs_of_nonneg
(div_nonneg (sq_nonneg _) (binaryFusion_denominator_pos hp.1 hp.2).le)]
exact binaryFusion_integrand_le_two hp.1 hp.2
theorem integrable_binaryFusionLoss {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
(hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1) :
Integrable (fun p : ℝ × ℝ => binaryFusionLoss p.1 p.2) (mu.prod nu) := by
refine (integrable_const (2 : ℝ)).mono' ?_ ?_
· unfold binaryFusionLoss
apply Measurable.aestronglyMeasurable
fun_prop
· filter_upwards [ae_binaryFusion_bounds hu hv] with p hp
rw [Real.norm_eq_abs, abs_of_nonneg (binaryFusionLoss_nonneg hp.1 hp.2)]
have h := binaryFusionLoss_le_sum_sq hp.1 hp.2
have hu' := (sq_lt_one_iff_abs_lt_one p.1).mpr hp.1
have hv' := (sq_lt_one_iff_abs_lt_one p.2).mpr hp.2
linarith
theorem integral_binaryFusion_symmetric_square {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
(hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1)
(hsym : Measure.map (fun v : ℝ => -v) nu = nu) :
(∫ p : ℝ × ℝ, (p.1 + p.2) ^ 2 / (1 + p.1 * p.2) ∂mu.prod nu) =
(∫ u : ℝ, u ^ 2 ∂mu) + (∫ v : ℝ, v ^ 2 ∂nu) -
∫ p : ℝ × ℝ, binaryFusionLoss p.1 p.2 ∂mu.prod nu := by
let R : (ℝ × ℝ) ≃ᵐ (ℝ × ℝ) :=
(MeasurableEquiv.refl ℝ).prodCongr (MeasurableEquiv.neg ℝ)
have hR : MeasurePreserving R (mu.prod nu) (mu.prod nu) :=
(MeasurePreserving.id mu).prod ⟨measurable_neg, hsym⟩
have hi := integrable_binaryFusion_integrand hu hv
have hin : Integrable (fun p : ℝ × ℝ =>
(p.1 - p.2) ^ 2 / (1 - p.1 * p.2)) (mu.prod nu) := by
have h := (hR.integrable_comp hi.aestronglyMeasurable).mpr hi
change Integrable (fun p : ℝ × ℝ =>
(p.1 + -p.2) ^ 2 / (1 + p.1 * -p.2)) (mu.prod nu) at h
simpa [sub_eq_add_neg] using h
have heq : (∫ p : ℝ × ℝ, (p.1 - p.2) ^ 2 / (1 - p.1 * p.2) ∂mu.prod nu) =
∫ p : ℝ × ℝ, (p.1 + p.2) ^ 2 / (1 + p.1 * p.2) ∂mu.prod nu := by
have h := hR.integral_comp'
(fun p : ℝ × ℝ => (p.1 + p.2) ^ 2 / (1 + p.1 * p.2))
change (∫ p : ℝ × ℝ, (p.1 + -p.2) ^ 2 / (1 + p.1 * -p.2) ∂mu.prod nu) = _ at h
simpa [sub_eq_add_neg] using h
have hiu : Integrable (fun p : ℝ × ℝ => p.1 ^ 2) (mu.prod nu) := by
refine (integrable_const (1 : ℝ)).mono' (by fun_prop) ?_
filter_upwards [ae_binaryFusion_bounds hu hv] with p hp
simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg p.1)] using
((sq_lt_one_iff_abs_lt_one p.1).mpr hp.1).le
have hiv : Integrable (fun p : ℝ × ℝ => p.2 ^ 2) (mu.prod nu) := by
refine (integrable_const (1 : ℝ)).mono' (by fun_prop) ?_
filter_upwards [ae_binaryFusion_bounds hu hv] with p hp
simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg p.2)] using
((sq_lt_one_iff_abs_lt_one p.2).mpr hp.2).le
have hl := integrable_binaryFusionLoss hu hv
have hiuv : Integrable (fun p : ℝ × ℝ => p.1 ^ 2 + p.2 ^ 2) (mu.prod nu) := hiu.add hiv
have hsum : (∫ p : ℝ × ℝ, p.1 ^ 2 + p.2 ^ 2 ∂mu.prod nu) =
(∫ p : ℝ × ℝ, p.1 ^ 2 ∂mu.prod nu) +
∫ p : ℝ × ℝ, p.2 ^ 2 ∂mu.prod nu := integral_add hiu hiv
have hscalar :
(∫ p : ℝ × ℝ,
((p.1 + p.2) ^ 2 / (1 + p.1 * p.2) +
(p.1 - p.2) ^ 2 / (1 - p.1 * p.2)) / 2 ∂mu.prod nu) =
∫ p : ℝ × ℝ, p.1 ^ 2 + p.2 ^ 2 - binaryFusionLoss p.1 p.2 ∂mu.prod nu := by
apply integral_congr_ae
filter_upwards [ae_binaryFusion_bounds hu hv] with p hp
exact binaryFusion_symmetric_square hp.1 hp.2
rw [integral_div, integral_add hi hin, heq,
integral_sub hiuv hl, hsum,
integral_fun_fst (μ := mu) (ν := nu) (fun u : ℝ => u ^ 2),
integral_fun_snd (μ := mu) (ν := nu) (fun v : ℝ => v ^ 2)] at hscalar
simp only [probReal_univ, one_smul] at hscalar
linarith
theorem integral_binaryFusionLoss_nonneg {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
(hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1) :
0 ≤ ∫ p : ℝ × ℝ, binaryFusionLoss p.1 p.2 ∂mu.prod nu := by
apply integral_nonneg_of_ae
filter_upwards [ae_binaryFusion_bounds hu hv] with p hp
exact binaryFusionLoss_nonneg hp.1 hp.2
theorem integral_binaryFusion_le_sum_sq {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu]
(hu : ∀ᵐ u ∂mu, |u| < 1) (hv : ∀ᵐ v ∂nu, |v| < 1)
(hsym : Measure.map (fun v : ℝ => -v) nu = nu) :
(∫ p : ℝ × ℝ, (p.1 + p.2) ^ 2 / (1 + p.1 * p.2) ∂mu.prod nu) ≤
(∫ u : ℝ, u ^ 2 ∂mu) + (∫ v : ℝ, v ^ 2 ∂nu) := by
rw [integral_binaryFusion_symmetric_square hu hv hsym]
exact sub_le_self _ (integral_binaryFusionLoss_nonneg hu hv)
private theorem integrable_sq_of_ae_abs_lt_one {mu : Measure ℝ}
[IsProbabilityMeasure mu] (hu : ∀ᵐ u ∂mu, |u| < 1) :
Integrable (fun u : ℝ => u ^ 2) mu := by
refine (integrable_const (1 : ℝ)).mono' (by fun_prop) ?_
filter_upwards [hu] with u hu
simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg u)] using
((sq_lt_one_iff_abs_lt_one u).mpr hu).le
theorem integral_binaryFusionLoss_lower_bound {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu] {theta : ℝ}
(hu : ∀ᵐ u ∂mu, |u| ≤ theta) (hv : ∀ᵐ v ∂nu, |v| ≤ theta)
(htheta : theta < 1) :
2 * (1 - theta ^ 2) * (∫ u : ℝ, u ^ 2 ∂mu) * (∫ v : ℝ, v ^ 2 ∂nu) ≤
∫ p : ℝ × ℝ, binaryFusionLoss p.1 p.2 ∂mu.prod nu := by
have hu1 : ∀ᵐ u ∂mu, |u| < 1 := hu.mono (fun _ h => lt_of_le_of_lt h htheta)
have hv1 : ∀ᵐ v ∂nu, |v| < 1 := hv.mono (fun _ h => lt_of_le_of_lt h htheta)
have hiu := integrable_sq_of_ae_abs_lt_one hu1
have hiv := integrable_sq_of_ae_abs_lt_one hv1
have hip := hiu.mul_prod hiv
have hl := integrable_binaryFusionLoss hu1 hv1
have hu' : ∀ᵐ p ∂mu.prod nu, |p.1| ≤ theta :=
measurePreserving_fst.quasiMeasurePreserving.tendsto_ae hu
have hv' : ∀ᵐ p ∂mu.prod nu, |p.2| ≤ theta :=
measurePreserving_snd.quasiMeasurePreserving.tendsto_ae hv
have hb : (∫ p : ℝ × ℝ, (2 * (1 - theta ^ 2)) * (p.1 ^ 2 * p.2 ^ 2) ∂mu.prod nu) ≤
∫ p : ℝ × ℝ, binaryFusionLoss p.1 p.2 ∂mu.prod nu := by
apply integral_mono_ae (hip.const_mul _) hl
filter_upwards [hu', hv'] with p hpu hpv
simpa only [mul_assoc] using binaryFusionLoss_lower_bound hpu hpv htheta
rw [integral_const_mul,
integral_prod_mul (μ := mu) (ν := nu) (fun u : ℝ => u ^ 2) (fun v : ℝ => v ^ 2)] at hb
simpa only [mul_assoc] using hb
theorem integral_binaryFusion_le_sum_sq_sub_loss {mu nu : Measure ℝ}
[IsProbabilityMeasure mu] [IsProbabilityMeasure nu] {theta : ℝ}
(hu : ∀ᵐ u ∂mu, |u| ≤ theta) (hv : ∀ᵐ v ∂nu, |v| ≤ theta)
(htheta : theta < 1) (hsym : Measure.map (fun v : ℝ => -v) nu = nu) :
(∫ p : ℝ × ℝ, (p.1 + p.2) ^ 2 / (1 + p.1 * p.2) ∂mu.prod nu) ≤
(∫ u : ℝ, u ^ 2 ∂mu) + (∫ v : ℝ, v ^ 2 ∂nu) -
2 * (1 - theta ^ 2) * (∫ u : ℝ, u ^ 2 ∂mu) * (∫ v : ℝ, v ^ 2 ∂nu) := by
have hu1 : ∀ᵐ u ∂mu, |u| < 1 := hu.mono (fun _ h => lt_of_le_of_lt h htheta)
have hv1 : ∀ᵐ v ∂nu, |v| < 1 := hv.mono (fun _ h => lt_of_le_of_lt h htheta)
rw [integral_binaryFusion_symmetric_square hu1 hv1 hsym]
exact sub_le_sub_left (integral_binaryFusionLoss_lower_bound hu hv htheta) _
end Problem367
end
end OAI
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