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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 | import OAI.Probability.InvariantIsing.Pressure.RandomCoefficientGG
namespace OAI
/-! Control of the finite spectral GG diagonal error by its actual L1 deviation. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators
namespace InvariantIsing
variable {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X]
[Countable X] [MeasurableSingletonClass X]
{P : Measure Ω} [IsProbabilityMeasure P] {ν : Ω → Measure X}
[∀ ω, IsProbabilityMeasure (ν ω)]
lemma randomCoefficient_bounded_observable_covariance (hν : Measurable ν)
(A : Ω → X → ℕ →₀ ℝ)
(hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => cylinderField (A p.1.1 p.2) p.1.2))
{B : ℝ} (hA : ∀ ω x, (A ω x).sum (fun _ c => c ^ 2) ≤ B)
(t b : ℝ) (Y : Ω → X → ℝ) (hYm : Measurable (Function.uncurry Y))
{K : ℝ} (hK : 0 ≤ K) (hYb : ∀ ω x, |Y ω x| ≤ K)
{r : ℕ} (D : Ω → (Fin (r + 1) → X) → ℝ)
(hDm : Measurable (Function.uncurry D)) {c : ℝ} (hc : 0 ≤ c)
(hD : ∀ ω σ, |D ω σ| ≤ c) :
let E := fun z : Ω × (ℕ → ℝ) => ∫ x, |Y z.1 x - b|
∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)
Integrable E (P.prod gaussianCoordinates) ∧
|randomCoefficientAverage P ν A t (fun ω σ => Y ω (σ 0) * D ω σ) -
b * randomCoefficientAverage P ν A t D| ≤
c * ∫ z, E z ∂P.prod gaussianCoordinates := by
have hYm' : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => Y p.1.1 p.2) :=
hYm.comp (measurable_fst.fst.prodMk measurable_snd)
have he := randomCoefficient_all_exp_ae (P := P) hν A hAm hA
have hdev (ω : Ω) (x : X) : |Y ω x - b| ≤ K + |b| :=
(abs_sub _ _).trans (add_le_add (hYb ω x) le_rfl)
have hEm := measurable_random_tilted_integral (ν := fun z : Ω × (ℕ → ℝ) => ν z.1)
(hν.comp measurable_fst) (hAm.const_mul t) (hYm'.sub_const b |>.abs)
have hEi : Integrable (fun z : Ω × (ℕ → ℝ) => ∫ x, |Y z.1 x - b|
∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2))
(P.prod gaussianCoordinates) := by
apply (integrable_const (K + |b|)).mono' hEm.aestronglyMeasurable
filter_upwards [he] with z hz
have : IsProbabilityMeasure ((ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) :=
isProbabilityMeasure_tilted (hz t)
rw [Real.norm_of_nonneg (integral_nonneg (fun _ => abs_nonneg _))]
calc
_ ≤ ∫ _x, K + |b| ∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2) :=
integral_mono
(integrable_of_measurable_abs_le (measurable_of_countable _)
(fun x => by simpa only [abs_abs] using hdev z.1 x))
(integrable_const _) (hdev z.1)
_ = _ := by simp
have hYi : ∀ᵐ z : Ω × (ℕ → ℝ) ∂P.prod gaussianCoordinates,
Integrable (Y z.1) ((ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) := by
filter_upwards [he] with z hz
have : IsProbabilityMeasure ((ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)) :=
isProbabilityMeasure_tilted (hz t)
exact integrable_of_measurable_abs_le (measurable_of_countable _) (hYb z.1)
have ht : Measurable (fun p : (Ω × (ℕ → ℝ)) × X =>
cylinderField (t • A p.1.1 p.2) p.1.2) := by
simpa only [cylinderField_smul] using hAm.const_mul t
have hYDm : Measurable (Function.uncurry (fun ω (σ : Fin (r + 1) → X) =>
Y ω (σ 0) * D ω σ)) := by
have hp : Measurable (fun p : Ω × (Fin (r + 1) → X) => (p.1, p.2 0)) :=
measurable_fst.prodMk ((measurable_pi_apply 0).comp measurable_snd)
exact (hYm.comp hp).mul hDm
have hiU := integrable_randomCoefficient_mean (P := P) hν (fun ω x => t • A ω x) ht
(fun ω (σ : Fin (r + 1) → X) => Y ω (σ 0) * D ω σ) hYDm (mul_nonneg hK hc)
(fun ω σ => by rw [abs_mul]; exact mul_le_mul (hYb ω _) (hD ω σ) (abs_nonneg _) hK)
have hiV := integrable_randomCoefficient_mean (P := P) hν (fun ω x => t • A ω x) ht D hDm hc hD
simp only [cylinderField_smul] at hiU hiV
refine ⟨hEi, ?_⟩
exact random_replica_energy_covariance_bound (ν := fun z : Ω × (ℕ → ℝ) => ν z.1)
(H := fun p => t * cylinderField (A p.1.1 p.2) p.1.2) (Y := fun p => Y p.1.1 p.2)
(fun p => D p.1.1 p.2)
(hDm.comp (measurable_fst.fst.prodMk measurable_snd)) hc (fun z => hD z.1) b
(he.mono fun _ hz => hz t) hYi hiU hiV hEi
/-- No constant self-overlap is imposed at finite N. Its actual Gibbs L1
fluctuation controls precisely the extra covariance in the GG estimate. -/
theorem randomCoefficient_diagonal_covariance_le (hν : Measurable ν)
(A : Ω → X → ℕ →₀ ℝ)
(hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => cylinderField (A p.1.1 p.2) p.1.2))
(hcross : Measurable (fun p : Ω × (X × X) => cylinderCross (A p.1 p.2.1) (A p.1 p.2.2)))
{B : ℝ} (hA : ∀ ω x, (A ω x).sum (fun _ c => c ^ 2) ≤ B)
(t d : ℝ) {r : ℕ} (D : Ω → (Fin (r + 1) → X) → ℝ)
(hDm : Measurable (Function.uncurry D)) {c : ℝ} (hc : 0 ≤ c)
(hD : ∀ ω σ, |D ω σ| ≤ c) :
|randomCoefficientAverage P ν A t
(fun ω σ => D ω σ * cylinderCross (A ω (σ 0)) (A ω (σ 0))) -
randomCoefficientAverage P ν A t D * randomCoefficientAverage P ν A t
(fun ω (σ : Fin 1 → X) => cylinderCross (A ω (σ 0)) (A ω (σ 0)))| ≤
2 * c * (∫ z : Ω × (ℕ → ℝ), ∫ x, |cylinderCross (A z.1 x) (A z.1 x) - d|
∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)
∂P.prod gaussianCoordinates) := by
let Y := fun (ω : Ω) (x : X) => cylinderCross (A ω x) (A ω x)
have hYm : Measurable (Function.uncurry Y) := by
have hp : Measurable (fun p : Ω × X => (p.1, (p.2, p.2))) :=
measurable_fst.prodMk (measurable_snd.prodMk measurable_snd)
exact hcross.comp hp
have hYb (ω : Ω) (x : X) : |Y ω x| ≤ |B| := cylinderCross_bound (A ω) (hA ω) x x
let E := ∫ z : Ω × (ℕ → ℝ), ∫ x, |Y z.1 x - d|
∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)
∂P.prod gaussianCoordinates
let V := randomCoefficientAverage P ν A t D
let L := randomCoefficientAverage P ν A t (fun ω σ => D ω σ * Y ω (σ 0))
let L₀ := randomCoefficientAverage P ν A t (fun ω (σ : Fin 1 → X) => Y ω (σ 0))
have hL : |L - d * V| ≤ c * E := by
have h := (randomCoefficient_bounded_observable_covariance (P := P) hν A hAm hA t d
Y hYm (abs_nonneg B) hYb D hDm hc hD).2
have heq : (fun ω σ => Y ω (σ 0) * D ω σ) = (fun ω σ => D ω σ * Y ω (σ 0)) := by
funext ω σ
exact mul_comm _ _
rw [heq] at h
exact h
have hL₀ : |L₀ - d| ≤ E := by
have h := (randomCoefficient_bounded_observable_covariance (P := P) (r := 0)
hν A hAm hA t d Y hYm (abs_nonneg B) hYb
(fun (_ : Ω) (_ : Fin 1 → X) => 1) measurable_const (c := 1) (by norm_num)
(fun _ _ => by norm_num)).2
simpa only [mul_one, randomCoefficientAverage_one hν A hAm hA, one_mul] using h
have hV : |V| ≤ c := randomCoefficientAverage_abs_le hν A hAm t D hDm hc hD
change |L - V * L₀| ≤ 2 * c * E
calc
_ = |(L - d * V) - V * (L₀ - d)| := by congr 1; ring
_ ≤ |L - d * V| + |V| * |L₀ - d| := by
simpa only [abs_mul] using (abs_sub (L - d * V) (V * (L₀ - d)))
_ ≤ c * E + c * E := add_le_add hL (mul_le_mul hV hL₀ (abs_nonneg _) hc)
_ = _ := by ring
/-- Quantitative GG with both errors expressed as actual Gibbs L1 deviations.
The energy fluctuation is divided by the perturbation scale when passing to
the limiting GG identities; the diagonal fluctuation is not. -/
theorem randomCoefficient_gaussian_gg_L1_bound (hν : Measurable ν)
(A : Ω → X → ℕ →₀ ℝ)
(hAm : Measurable (fun p : (Ω × (ℕ → ℝ)) × X => cylinderField (A p.1.1 p.2) p.1.2))
(hcross : Measurable (fun p : Ω × (X × X) => cylinderCross (A p.1 p.2.1) (A p.1 p.2.2)))
{B : ℝ} (hA : ∀ ω x, (A ω x).sum (fun _ c => c ^ 2) ≤ B)
(t b d : ℝ) {r : ℕ} (D : Ω → (Fin (r + 1) → X) → ℝ)
(hDm : Measurable (Function.uncurry D)) {c : ℝ} (hc : 0 ≤ c)
(hD : ∀ ω σ, |D ω σ| ≤ c)
(hE : Integrable (fun z : Ω × (ℕ → ℝ) => ∫ x,
|cylinderField (A z.1 x) z.2 - b|
∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2))
(P.prod gaussianCoordinates)) :
|t| * |(r + 1 : ℕ) * randomCoefficientAverage P ν A t
(fun ω (τ : Fin (r + 1 + 1) → X) => D ω (Fin.tail τ) *
cylinderCross (A ω ((Fin.tail τ) 0)) (A ω (τ 0))) -
randomCoefficientAverage P ν A t D * randomCoefficientAverage P ν A t
(fun ω (τ : Fin 2 → X) => cylinderCross (A ω (τ 1)) (A ω (τ 0))) -
randomCoefficientAverage P ν A t
(fun ω σ => D ω σ * ∑ l : Fin r, cylinderCross (A ω (σ 0)) (A ω (σ l.succ)))| ≤
2 * c * ((∫ z : Ω × (ℕ → ℝ), ∫ x, |cylinderField (A z.1 x) z.2 - b|
∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)
∂P.prod gaussianCoordinates) +
|t| * (∫ z : Ω × (ℕ → ℝ), ∫ x, |cylinderCross (A z.1 x) (A z.1 x) - d|
∂(ν z.1).tilted (fun x => t * cylinderField (A z.1 x) z.2)
∂P.prod gaussianCoordinates)) := by
have hg := randomCoefficient_gaussian_gg_bound (P := P) hν A hAm hcross hA t b D hDm hc hD hE
have hd := randomCoefficient_diagonal_covariance_le (P := P) hν A hAm hcross hA t d D hDm hc hD
refine hg.trans ((add_le_add le_rfl (mul_le_mul_of_nonneg_left hd (abs_nonneg t))).trans_eq ?_)
ring
end InvariantIsing
end
end OAI
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