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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 | import OAI.Probability.InvariantIsing.Fields.CascadeMarking
import OAI.Probability.InvariantIsing.Haar.Rotation
import OAI.Probability.IsingPerceptron.EnrichedFieldCap
namespace OAI
/-!
Gaussian cascade marking for the actual finite Ising partition function at
a fixed orthogonal rotation. The Haar variable is fixed in this result.
-/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace InvariantIsing
def rotatedFieldTerminal {N : ℕ} (eig : Fin N → ℝ) (U : Rotation N)
(c z : Fin N → ℝ) : ℝ :=
logPartition (fun σ => rotatedEnergy eig U σ + fieldEnergy (c + z) σ)
lemma measurable_rotatedFieldTerminal {N : ℕ} (eig : Fin N → ℝ) (U : Rotation N)
(c : Fin N → ℝ) : Measurable (rotatedFieldTerminal eig U c) := by
unfold rotatedFieldTerminal logPartition fieldEnergy
fun_prop
lemma abs_rotatedFieldTerminal_sub_le {N : ℕ} (eig : Fin N → ℝ) (U : Rotation N)
(c z w : Fin N → ℝ) :
|rotatedFieldTerminal eig U c z - rotatedFieldTerminal eig U c w| ≤
(N : ℝ) * ‖z - w‖ := by
have he (σ : Spin N) :
|(rotatedEnergy eig U σ + fieldEnergy (c + z) σ) -
(rotatedEnergy eig U σ + fieldEnergy (c + w) σ)| ≤ (N : ℝ) * ‖z - w‖ := by
rw [add_sub_add_left_eq_sub]
calc
_ ≤ ∑ i, |(c + z) i - (c + w) i| := abs_fieldEnergy_sub_le _ _ σ
_ = ∑ i, |z i - w i| := by simp only [Pi.add_apply, add_sub_add_left_eq_sub]
_ ≤ ∑ _ : Fin N, ‖z - w‖ := Finset.sum_le_sum fun i _ => by
simpa only [Pi.sub_apply, Real.norm_eq_abs] using norm_le_pi_norm (z - w) i
_ = (N : ℝ) * ‖z - w‖ := by simp
exact abs_logPartition_sub_le _ _ _ he
lemma rotatedFieldTerminal_linearGrowth {N : ℕ} (eig : Fin N → ℝ) (U : Rotation N)
(c : Fin N → ℝ) : IsingPerceptron.HasLinearGrowth (rotatedFieldTerminal eig U c) := by
refine ⟨|rotatedFieldTerminal eig U c 0|, N, abs_nonneg _, Nat.cast_nonneg _, fun z => ?_⟩
have hl := abs_rotatedFieldTerminal_sub_le eig U c z 0
rw [sub_zero] at hl
have ht := abs_add_le (rotatedFieldTerminal eig U c z - rotatedFieldTerminal eig U c 0)
(rotatedFieldTerminal eig U c 0)
rw [sub_add_cancel] at ht
linarith
def vectorGaussianLaw (N : ℕ) (v : ℝ≥0) : ProbabilityMeasure (Fin N → ℝ) :=
⟨Measure.pi (fun _ : Fin N => gaussianReal 0 v), inferInstance⟩
lemma vectorGaussianLaw_moments {N : ℕ} (hN : 0 < N) (v : ℝ≥0) :
IsingPerceptron.ExponentialNormMoments (vectorGaussianLaw N v : Measure (Fin N → ℝ)) := by
let : Nonempty (Fin N) := ⟨⟨0, hN⟩⟩
have hm := IsingPerceptron.finite_gaussian_exponentialNormMoments
(P := Measure.pi (fun _ : Fin N => gaussianReal 0 v))
(X := fun z : Fin N → ℝ => z) measurable_id (fun _ => 0) (fun _ => v)
(fun i => (measurePreserving_eval (fun _ : Fin N => gaussianReal 0 v) i).map_eq)
change IsingPerceptron.ExponentialNormMoments
((Measure.pi (fun _ : Fin N => gaussianReal 0 v)).map id) at hm
rw [Measure.map_id] at hm
exact hm
def rotatedCascadeLaw (N n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) :
Measure (IsingPerceptron.NoiseTree (Fin N → ℝ) n) :=
IsingPerceptron.noiseCascadeLaw (Fin N → ℝ) n b (fun i => vectorGaussianLaw N (v i))
def rotatedCascadeLog {N : ℕ} (n : ℕ) (eig : Fin N → ℝ) (U : Rotation N)
(c z : Fin N → ℝ) (T : IsingPerceptron.NoiseTree (Fin N → ℝ) n) : ℝ :=
Real.log (∫ leaf, Real.exp (IsingPerceptron.noiseLeafTerminal n
(fun _ => rotatedFieldTerminal eig U c) (fun _ p => p.1 + p.2) z leaf)
∂IsingPerceptron.noiseLeafKernel (Fin N → ℝ) n T)
def rotatedCascadeValue {N : ℕ} (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0)
(eig : Fin N → ℝ) (U : Rotation N) (c z : Fin N → ℝ) : ℝ :=
IsingPerceptron.cascadeRecursion n b (fun i => vectorGaussianLaw N (v i))
(fun _ p => p.1 + p.2) (rotatedFieldTerminal eig U c) z
/-- The published Gaussian marking recursion specialized to the actual
finite rotated Ising terminal, with the Haar rotation held fixed. -/
theorem rotatedCascadeLog_recursion {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ)
(v : ℕ → ℝ≥0) (hb : IsingPerceptron.CascadeExponents n b)
(eig : Fin N → ℝ) (U : Rotation N) (c z : Fin N → ℝ) :
Integrable (rotatedCascadeLog n eig U c z) (rotatedCascadeLaw N n b v) ∧
(∫ T, rotatedCascadeLog n eig U c z T ∂rotatedCascadeLaw N n b v) =
rotatedCascadeValue n b v eig U c z := by
have h := IsingPerceptron.linearGrowth_noiseCascade_recursion n b hb
(fun i => vectorGaussianLaw N (v i)) (fun i _ => vectorGaussianLaw_moments hN (v i))
(measurable_rotatedFieldTerminal eig U c) (rotatedFieldTerminal_linearGrowth eig U c) z
have he := IsingPerceptron.noise_leaf_log_eq_ratio n b hb
(fun i => vectorGaussianLaw N (v i))
(fun _ => measurable_fst.add measurable_snd) (measurable_rotatedFieldTerminal eig U c) z
refine ⟨h.2.1.congr (Filter.EventuallyEq.symm he), ?_⟩
exact (integral_congr_ae he).trans h.2.2.1
lemma integrable_rotatedCascadeValue {N : ℕ} (hN : 0 < N) (n : ℕ) (b : ℕ → ℝ)
(v : ℕ → ℝ≥0) (hb : IsingPerceptron.CascadeExponents n b)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ) (root : ℝ≥0) :
Integrable (rotatedCascadeValue n b v eig U c)
(vectorGaussianLaw N root : Measure (Fin N → ℝ)) := by
have hm := IsingPerceptron.measurable_cascadeRecursion n b
(fun i => vectorGaussianLaw N (v i))
(fun _ => measurable_fst.add measurable_snd) (measurable_rotatedFieldTerminal eig U c)
change Measurable (rotatedCascadeValue n b v eig U c) at hm
have hg := IsingPerceptron.cascadeRecursion_linearGrowth n b
(fun i => vectorGaussianLaw N (v i)) (fun i _ => vectorGaussianLaw_moments hN (v i))
(measurable_rotatedFieldTerminal eig U c) (rotatedFieldTerminal_linearGrowth eig U c)
(fun i hi => (hb.1 i hi).1)
obtain ⟨C, L, _, hL, hbound⟩ := hg
have hi := (integrable_const C).add ((vectorGaussianLaw_moments hN root 1).const_mul L)
apply hi.mono' hm.aestronglyMeasurable
exact ae_of_all _ fun z => by
change |rotatedCascadeValue n b v eig U c z| ≤ C + L * Real.exp (1 * ‖z‖)
rw [one_mul]
exact (hbound z).trans (add_le_add le_rfl
(mul_le_mul_of_nonneg_left
((le_add_of_nonneg_right zero_le_one).trans (Real.add_one_le_exp ‖z‖)) hL))
def rotatedEnrichedPressure {N : ℕ} (n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0)
(root : ℝ≥0) (eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ) : ℝ :=
(N : ℝ)⁻¹ * ∫ z, rotatedCascadeValue n b v eig U c z
∂(vectorGaussianLaw N root : Measure (Fin N → ℝ))
/-- The fixed-rotation enriched pressure is the actual iterated Gaussian
root and cascade expectation of the finite Ising log partition. -/
theorem rotatedEnrichedPressure_eq_cascade_expectation {N : ℕ} (hN : 0 < N)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → ℝ≥0) (root : ℝ≥0)
(hb : IsingPerceptron.CascadeExponents n b)
(eig : Fin N → ℝ) (U : Rotation N) (c : Fin N → ℝ) :
rotatedEnrichedPressure n b v root eig U c =
(N : ℝ)⁻¹ * ∫ z, (∫ T, rotatedCascadeLog n eig U c z T
∂rotatedCascadeLaw N n b v) ∂(vectorGaussianLaw N root : Measure (Fin N → ℝ)) := by
unfold rotatedEnrichedPressure
congr 1
apply integral_congr_ae
exact ae_of_all _ fun z => (rotatedCascadeLog_recursion hN n b v hb eig U c z).2.symm
end InvariantIsing
end
end OAI
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