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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 | import OAI.Probability.InvariantIsing.Arrays.NSpinTensorRetained
import OAI.Probability.IsingPerceptron.NormalizedRestrictionPi
import OAI.Probability.IsingPerceptron.SamplingIndependent
namespace OAI
/-!
The finite Ising tensor application of the conditional marked-path law.
The conditional marked-path law is the finite two-path specialization of
Panchenko--Talagrand, C. R. Acad. Sci. Paris I 345 (2007), 653--656,
Section 3, Lemma 3.1, equation (8), p.655, and Corollary 3.2, equation (9),
p.656. The mark spaces are complete separable metric spaces with their Borel
sigma algebras; ancestor conditional distributions may be arbitrary and
the measurable terminal need only satisfy `E exp X < infinity`.
The Gaussian-vector terminal satisfies this exponential integrability
hypothesis.
The law describes the joint branching and marks, including their
conditional ancestor distributions.
Averaging the fixed-root law gives the random-root law.
Forest edge `i` has shared mark exactly when `i < commonDepth`.
-/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
/-- The published marking theorem's exponential integrability hypothesis
holds for the actual finite Ising terminal on a finite Gaussian path. -/
theorem tensorFinitePath_exp_integrable {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(z : SpinTensorIndex I degree → ℝ) :
Integrable (fun w : Fin n → SpinTensorIndex I degree → ℝ =>
Real.exp (spinTensorTerminal eig U c I degree amplitude (z + ∑ i, w i)))
(Measure.pi (fun i : Fin n =>
(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ)))) := by
obtain ⟨C, L, _, hL, hbound⟩ := spinTensorTerminal_linearGrowth eig U c I degree amplitude
have hprod : Integrable (fun w : Fin n → SpinTensorIndex I degree → ℝ =>
∏ i, Real.exp (L * ‖w i‖))
(Measure.pi (fun i : Fin n =>
(tensorGaussianLaw I degree (v i) : Measure (SpinTensorIndex I degree → ℝ)))) :=
Integrable.fintype_prod (f := fun (i : Fin n) (a : SpinTensorIndex I degree → ℝ) =>
Real.exp (L * ‖a‖)) fun i => tensorGaussianLaw_moments hN I degree (v i) L
apply (hprod.const_mul (Real.exp (C + L * ‖z‖))).mono'
(((measurable_spinTensorTerminal eig U c I degree amplitude).comp
(measurable_const.add (Finset.measurable_sum _ fun i _ => measurable_pi_apply i))).exp.aestronglyMeasurable)
apply ae_of_all
intro w
rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _), ← Real.exp_sum, ← Real.exp_add]
apply Real.exp_le_exp.mpr
calc
_ ≤ |spinTensorTerminal eig U c I degree amplitude (z + ∑ i, w i)| := le_abs_self _
_ ≤ C + L * ‖z + ∑ i, w i‖ := hbound _
_ ≤ C + L * (‖z‖ + ∑ i, ‖w i‖) := by
apply add_le_add_right (mul_le_mul_of_nonneg_left _ hL)
exact (norm_add_le _ _).trans (add_le_add_right (norm_sum_le _ _) _)
_ = C + L * ‖z‖ + ∑ i, L * ‖w i‖ := by rw [mul_add, Finset.mul_sum]; ring
private lemma tensor_abs_integral_bound {A : Type*} [MeasurableSpace A]
{μ : Measure A} [IsProbabilityMeasure μ] (f : A → ℝ) {C : ℝ}
(hf : ∀ a, |f a| ≤ C) : |∫ a, f a ∂μ| ≤ C := by
have hh : ∀ᵐ a ∂μ, ‖f a‖ ≤ C := ae_of_all _ fun a => by
simpa only [Real.norm_eq_abs] using hf a
simpa only [Real.norm_eq_abs, probReal_univ, mul_one] using
norm_integral_le_of_norm_le_const (μ := μ) (f := f) hh
/-- Iterated conditional expectation of two Gaussian increment paths. Before
their branching depth the two paths receive the same tilted increment;
afterwards their next increments are conditionally independent. -/
def tensorAncestorPairPathMean {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) (d : ℕ) :
(q : ℕ) → ℕ → (SpinTensorIndex I degree → ℝ) → (SpinTensorIndex I degree → ℝ) →
((Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) → ℝ) → ℝ
| 0, _, _, _, F => F Fin.elim0
| q + 1, i, z₁, z₂, F =>
if i < d then
∫ a, tensorAncestorPairPathMean eig U c I degree amplitude n b v d q (i + 1)
(z₁ + a) (z₂ + a) (fun w => F (Fin.cons (a, a) w))
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₁
else
∫ a₁, ∫ a₂, tensorAncestorPairPathMean eig U c I degree amplitude n b v d q (i + 1)
(z₁ + a₁) (z₂ + a₂) (fun w => F (Fin.cons (a₁, a₂) w))
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₂
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₁
theorem tensorAncestorPairPathMean_abs_le {N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b) (d q i : ℕ)
(z₁ z₂ : SpinTensorIndex I degree → ℝ)
(F : (Fin q → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) → ℝ)
{C : ℝ} (hF : ∀ w, |F w| ≤ C) :
|tensorAncestorPairPathMean eig U c I degree amplitude n b v d q i z₁ z₂ F| ≤ C := by
induction q generalizing i z₁ z₂ with
| zero => exact hF _
| succ q ih =>
have := tensorAncestorMarkKernel_markov hN eig U c I degree amplitude n b v hb i
dsimp only [tensorAncestorPairPathMean]
split_ifs
· apply tensor_abs_integral_bound
intro a
exact ih (i + 1) (z₁ + a) (z₂ + a)
(fun w => F (Fin.cons (a, a) w)) (fun w => hF _)
· have hinner (a₁ : SpinTensorIndex I degree → ℝ) :
|∫ a₂, tensorAncestorPairPathMean eig U c I degree amplitude n b v d q (i + 1)
(z₁ + a₁) (z₂ + a₂) (fun w => F (Fin.cons (a₁, a₂) w))
∂tensorAncestorMarkKernel eig U c I degree amplitude n b v i z₂| ≤ C := by
apply tensor_abs_integral_bound
intro a₂
exact ih (i + 1) (z₁ + a₁) (z₂ + a₂)
(fun w => F (Fin.cons (a₁, a₂) w)) (fun w => hF _)
exact tensor_abs_integral_bound _ hinner
/-- The actual tilted two-leaf Gaussian path expectation, with the spin
residual integrated in the finite Ising terminal. -/
def tensorTiltedPairPathMean {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(z : SpinTensorIndex I degree → ℝ)
(Φ : ℕ × (Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) → ℝ) : ℝ :=
∫ p : TensorCoordinateData I degree n,
referenceReplicaMean (labeledLeafLaw n p.1)
(fun α => spinTensorTerminal eig U c I degree amplitude
(labeledEnergy n (markForestOfCoords (SpinTensorIndex I degree → ℝ) n p.2) α z))
(fun σ : Fin 2 → LabeledLeaf n => Φ
(labeledCommonDepth n (σ 0) (σ 1),
fun i => (p.2 (edgeAt n (σ 0) i), p.2 (edgeAt n (σ 1) i))))
∂tensorCoordinateLaw I degree n b v
/-- The exact finite tensor specialization of the published joint marked
path law. The unperturbed cascade supplies the shared-prefix depth and
the normalized ancestor kernels supply the conditional marks. -/
def TensorAncestorPairLaw {N m k : ℕ}
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) : Prop :=
∀ z : SpinTensorIndex I degree → ℝ,
∀ Φ : ℕ × (Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ)) → ℝ,
Measurable Φ → (∃ C : ℝ, ∀ p, |Φ p| ≤ C) →
tensorTiltedPairPathMean eig U c I degree amplitude n b v z Φ =
∫ α : ℕ → LabeledLeaf n,
tensorAncestorPairPathMean eig U c I degree amplitude n b v
(labeledCommonDepth n (α 0) (α 1)) n 0 z z
(fun w => Φ (labeledCommonDepth n (α 0) (α 1), w))
∂cascadeReplicaLaw n b
/-- Conditional two-path marking law for finite hierarchical tensor fields. -/
def PanchenkoTalagrandTensorPairInput : Prop :=
∀ N m k : ℕ, 0 < N →
∀ (eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0),
CascadeExponents n b → TensorAncestorPairLaw eig U c I degree amplitude n b v
/-- The independent root Gaussian is averaged after the fixed-root path
identity. Root level zero and forest levels one through `n` agree with
the flat tensor coefficient convention. -/
theorem tensorRootedPairPath_identity (hpub : PanchenkoTalagrandTensorPairInput)
{N m k : ℕ} (hN : 0 < N)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(hb : CascadeExponents n b)
(Φ : (SpinTensorIndex I degree → ℝ) ×
(ℕ × (Fin n → (SpinTensorIndex I degree → ℝ) × (SpinTensorIndex I degree → ℝ))) → ℝ)
(hΦ : Measurable Φ) (hΦb : ∃ C : ℝ, ∀ p, |Φ p| ≤ C) :
(∫ z, tensorTiltedPairPathMean eig U c I degree amplitude n b (fun i => v (i + 1)) z
(fun p => Φ (z, p))
∂(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ))) =
∫ z, ∫ α : ℕ → LabeledLeaf n,
tensorAncestorPairPathMean eig U c I degree amplitude n b (fun i => v (i + 1))
(labeledCommonDepth n (α 0) (α 1)) n 0 z z
(fun w => Φ (z, labeledCommonDepth n (α 0) (α 1), w))
∂cascadeReplicaLaw n b
∂(tensorGaussianLaw I degree (v 0) : Measure (SpinTensorIndex I degree → ℝ)) := by
apply integral_congr_ae
apply ae_of_all
intro z
apply hpub N m k hN eig U c I degree amplitude n b (fun i => v (i + 1)) hb z
· exact hΦ.comp (measurable_const.prodMk measurable_id)
· obtain ⟨C, hC⟩ := hΦb
exact ⟨C, fun p => hC (z, p)⟩
end InvariantIsing
end
end OAI
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