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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 | import OAI.Probability.FreeIsing.Trees.ResponseRowEnergy
namespace OAI
noncomputable section
open MeasureTheory Set
namespace Problem367.ResponseRows
section Deterministic
variable {Ω ι : Type*}
theorem norm_tree_coefficient_le
{d : ℕ} {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1)
(s : TreeVertex d → Finset ι) (X : TreeVertex d → ι → Ω → ℝ)
(v : TreeVertex d) (ω : Ω) :
‖coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω‖ ≤
theta ^ v.1.length := by
have hp := prod_responseAttenuation_pos (s v) (fun i => X v i ω) hθ
have hle := prod_attenuation_le_one (s v) (fun i => X v i ω) hθ
have hw : 0 ≤ theta ^ v.1.length := pow_nonneg hθ0 _
rw [coefficient, pathProduct, Real.norm_eq_abs, abs_mul, abs_of_nonneg hw,
abs_of_pos hp]
exact mul_le_of_le_one_right hw hle
theorem summable_tree_coefficient_sq
{d : ℕ} {theta : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1)
(hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1)
(s : TreeVertex d → Finset ι) (X : TreeVertex d → ι → Ω → ℝ) (ω : Ω) :
Summable (fun v : TreeVertex d =>
(coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω) ^ 2) := by
simpa only [Real.norm_eq_abs, sq_abs] using
summable_norm_sq_of_radial_bound d hsub
(fun v => coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω)
(fun v => norm_tree_coefficient_le hθ0 hθ s X v ω)
theorem summable_regularPathWeight {d : ℕ} {theta : ℝ}
(hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) :
Summable (regularPathWeight d theta) := by
have hs := summable_regularPathWeight_mul_length_sq hsub
apply Summable.of_nonneg_of_le (regularPathWeight_nonneg d theta) _ hs
intro n
have hr : (1 : ℝ) ≤ ((n + 1 : ℕ) : ℝ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le n)
have hr2 : (1 : ℝ) ≤ ((n + 1 : ℕ) : ℝ) ^ 2 := by nlinarith
simpa only [mul_one] using
mul_le_mul_of_nonneg_left hr2 (regularPathWeight_nonneg d theta n)
theorem summable_regularPathWeight_sqrt_length_sq
{d : ℕ} {theta A : ℝ} (hA : 0 ≤ A)
(hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) :
Summable (fun n => regularPathWeight d theta n *
Real.sqrt (A * ((n + 1 : ℕ) : ℝ) ^ 2)) := by
have hs := (summable_regularPathWeight_mul_length_sq hsub).mul_left (Real.sqrt A)
apply Summable.of_nonneg_of_le (fun n => mul_nonneg
(regularPathWeight_nonneg d theta n) (Real.sqrt_nonneg _)) _ hs
intro n
have hr : (1 : ℝ) ≤ ((n + 1 : ℕ) : ℝ) := by exact_mod_cast Nat.succ_le_succ (Nat.zero_le n)
rw [Real.sqrt_mul hA, Real.sqrt_sq (Nat.cast_nonneg _)]
have hr2 : ((n + 1 : ℕ) : ℝ) ≤ ((n + 1 : ℕ) : ℝ) ^ 2 := by nlinarith
calc
regularPathWeight d theta n * (Real.sqrt A * ((n + 1 : ℕ) : ℝ)) ≤
regularPathWeight d theta n * (Real.sqrt A * ((n + 1 : ℕ) : ℝ) ^ 2) := by
exact mul_le_mul_of_nonneg_left
(mul_le_mul_of_nonneg_left hr2 (Real.sqrt_nonneg A))
(regularPathWeight_nonneg d theta n)
_ = _ := by ring
end Deterministic
section Probability
variable {Ω : Type*} [MeasurableSpace Ω] {μ : Measure Ω}
theorem integral_tree_density_le_subcritical
{d : ℕ} {ι : Type*} (s : TreeVertex d → Finset ι)
(hcard : ∀ v, (s v).card = v.1.length)
{theta C : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1)
(hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) (hC : 0 ≤ C)
{X Y : TreeVertex d → ι → Ω → ℝ} {H H' : Ω → ℝ}
(hX : ∀ v i, i ∈ s v → Measurable (X v i))
(hY : ∀ v i, i ∈ s v → Measurable (Y v i))
(hInt : ∀ v i, i ∈ s v → Integrable (fun ω => (X v i ω - Y v i ω) ^ 2) μ)
(hH : Measurable H) (hH' : Measurable H')
(hU : MemLp (fun ω => H ω - H' ω) 2 μ)
(hlog : ∀ v i, i ∈ s v →
(∫ ω, (Real.log (responseAttenuation theta (X v i ω)) -
Real.log (responseAttenuation theta (Y v i ω))) ^ 2 ∂μ) ≤
C * (∫ ω, (H ω - H' ω) ^ 2 ∂μ)) :
let c := coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X
let c' := coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s Y
Integrable (DiscrepancyEnergy.density c c' (fun ω => H ω - H' ω)
(fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω))) μ ∧
(∫ ω, DiscrepancyEnergy.density c c' (fun ω => H ω - H' ω)
(fun ω => Real.tanh (H ω)) (fun ω => Real.tanh (H' ω)) ω ∂μ) ≤
subcriticalPathConstant d theta (2 * C) * (∫ ω, (H ω - H' ω) ^ 2 ∂μ) := by
dsimp only
let V := ∫ ω, (H ω - H' ω) ^ 2 ∂μ
let Q := fun n : ℕ => (2 * C) * ((n + 1 : ℕ) : ℝ) ^ 2 * V
let L := fun _ : ℕ => V
have hV : 0 ≤ V := integral_nonneg (fun ω => sq_nonneg _)
have hQ0 (n : ℕ) : 0 ≤ Q n := by dsimp [Q]; positivity
have hL0 (n : ℕ) : 0 ≤ L n := hV
have hQ (n : ℕ) (v : TreeVertex d) (hv : v.1.length = n + 1) :
pathNoiseEnergy μ theta (s v) (X v) (Y v) ≤ Q n := by
have h := integral_weighted_path_log_sq_le (s v) hθ (hX v) (hY v) (hInt v) (hlog v)
rw [hcard v, hv] at h
change _ ≤ _ at h
calc
pathNoiseEnergy μ theta (s v) (X v) (Y v) ≤
2 * ((n + 1 : ℕ) : ℝ) ^ 2 * (C * V) := h
_ = Q n := by dsimp [Q]; ring
have hL (n : ℕ) (v : TreeVertex d) (_hv : v.1.length = n + 1) :
pathLinearEnergy μ theta (s v) (X v) (fun ω => H ω - H' ω) ≤ L n :=
(integral_mul_prod_responseAttenuation_sq_le (s v) hθ (hX v) hU).2
have hQs : Summable (fun n => regularPathWeight d theta n * Q n) := by
convert! (summable_regularPathWeight_mul_length_sq hsub).mul_left (2 * C * V) using 1
funext n
dsimp only [Q]
ring
have hLs : Summable (fun n => regularPathWeight d theta n * L n) :=
(summable_regularPathWeight hsub).mul_right V
have hRs : Summable (fun n => regularPathWeight d theta n * Real.sqrt (Q n)) := by
convert! summable_regularPathWeight_sqrt_length_sq (A := 2 * C * V)
(by positivity) hsub using 1
funext n
congr 2
dsimp only [Q]
ring
have hrows : ∀ᵐ ω ∂μ,
Summable (fun v => (coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s X v ω) ^ 2) ∧
Summable (fun v => (coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s Y v ω) ^ 2) := by
filter_upwards [] with ω
exact ⟨summable_tree_coefficient_sq hθ0 hθ hsub s X ω,
summable_tree_coefficient_sq hθ0 hθ hsub s Y ω⟩
have hb := integral_tree_density_le_shell_energies s hcard hθ hX hY hInt hH hH' hU
hQ0 hL0 hQ hL hQs hLs hRs hrows
refine ⟨hb.1, ?_⟩
exact (subcritical_noise_drift_bound hsub (by positivity : 0 ≤ 2 * C) hV
hQ0 hL0 (fun n => le_rfl) (fun n => le_rfl) hb.2).2
theorem eq_zero_of_subcritical_response_variance_identity
{d : ℕ} {ι : Type*} (s : TreeVertex d → Finset ι)
(hcard : ∀ v, (s v).card = v.1.length)
{theta C T : ℝ} (hθ0 : 0 ≤ theta) (hθ : theta ^ 2 ≤ 1)
(hsub : ((d - 1 : ℕ) : ℝ) * theta ^ 2 < 1) (hC : 0 ≤ C)
{X Y : ℝ → TreeVertex d → ι → Ω → ℝ} {H H' : ℝ → Ω → ℝ}
(hX : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v → Measurable (X t v i))
(hY : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v → Measurable (Y t v i))
(hInt : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v →
Integrable (fun ω => (X t v i ω - Y t v i ω) ^ 2) μ)
(hH : ∀ t ∈ Icc 0 T, Measurable (H t))
(hH' : ∀ t ∈ Icc 0 T, Measurable (H' t))
(hU : ∀ t ∈ Icc 0 T, MemLp (fun ω => H t ω - H' t ω) 2 μ)
(hlog : ∀ t ∈ Icc 0 T, ∀ v i, i ∈ s v →
(∫ ω, (Real.log (responseAttenuation theta (X t v i ω)) -
Real.log (responseAttenuation theta (Y t v i ω))) ^ 2 ∂μ) ≤
C * (∫ ω, (H t ω - H' t ω) ^ 2 ∂μ))
(hAC : AbsolutelyContinuousOnInterval (fun t => ∫ ω, (H t ω - H' t ω) ^ 2 ∂μ) 0 T)
(hzero : (∫ ω, (H 0 ω - H' 0 ω) ^ 2 ∂μ) = 0)
(hderiv : ∀ᵐ t : ℝ, t ∈ Icc 0 T →
deriv (fun r => ∫ ω, (H r ω - H' r ω) ^ 2 ∂μ) t =
∫ ω, DiscrepancyEnergy.density
(coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s (X t))
(coefficient theta (fun v : TreeVertex d => theta ^ v.1.length) s (Y t))
(fun ω => H t ω - H' t ω)
(fun ω => Real.tanh (H t ω)) (fun ω => Real.tanh (H' t ω)) ω ∂μ) :
∀ t ∈ Icc 0 T, (∫ ω, (H t ω - H' t ω) ^ 2 ∂μ) = 0 := by
apply eq_zero_of_ae_deriv_le_mul_self
(C := subcriticalPathConstant d theta (2 * C)) hAC hzero
(fun t _ => integral_nonneg (fun ω => sq_nonneg _))
filter_upwards [hderiv] with t ht hmem
rw [ht hmem]
exact (integral_tree_density_le_subcritical s hcard hθ0 hθ hsub hC
(hX t hmem) (hY t hmem) (hInt t hmem) (hH t hmem) (hH' t hmem)
(hU t hmem) (hlog t hmem)).2
end Probability
end Problem367.ResponseRows
end
end OAI
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