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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 | import OAI.Probability.InvariantIsing.Arrays.NSpinTensorFrozenCGF
import OAI.Probability.InvariantIsing.Arrays.NSpinTensorHaarFluctuation
namespace OAI
/-! The actual flat spin/leaf pressure and marked-cascade pressure have the same law. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal ENNReal
namespace InvariantIsing
def tensorFlatToNoise {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(p : LabeledTree n × (ℕ → ℝ)) :
(SpinTensorIndex I degree → ℝ) × NoiseTree (SpinTensorIndex I degree → ℝ) n :=
let q := tensorFlatToCoordinates I degree n v p
(q.1, labeledNoiseJoin (SpinTensorIndex I degree → ℝ) n
(q.2.1, markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2.2))
theorem tensorFlatToNoise_measurePreserving {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (b : ℕ → ℝ)
(v : ℕ → SpinTensorIndex I degree → ℝ≥0) :
MeasurePreserving (tensorFlatToNoise I degree n v)
((labeledCascadeLaw n b : Measure (LabeledTree n)).prod gaussianCoordinates)
(tensorRootTreeLaw I degree n b (fun i => v (i + 1)) (v 0)) := by
have hp : MeasurePreserving (fun q : TensorCoordinateData I degree n =>
labeledNoiseJoin (SpinTensorIndex I degree → ℝ) n
(q.1, markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2))
(tensorCoordinateLaw I degree n b (fun i => v (i + 1)))
(tensorCascadeLaw I degree n b (fun i => v (i + 1))) :=
⟨by fun_prop, labeledNoiseCoordinates_law (SpinTensorIndex I degree → ℝ) n b
(fun i => tensorGaussianLaw I degree (v (i + 1)))⟩
have h := ((MeasurePreserving.id (tensorGaussianLaw I degree (v 0) :
Measure (SpinTensorIndex I degree → ℝ))).prod hp).comp
(tensorFlatToCoordinates_measurePreserving I degree n b v)
convert h using 1
· rfl
· rfl
private lemma spinTensorEnergy_labeled {N m k : ℕ} (U : Rotation N)
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ) (amplitude : Fin k → ℝ)
(n : ℕ) (z : SpinTensorIndex I degree → ℝ)
(G : ForestVertex n → SpinTensorIndex I degree → ℝ) (leaf : LabeledLeaf n) (σ : Spin N) :
spinTensorEnergy U I degree amplitude
(labeledEnergy n (markForestOfCoords (SpinTensorIndex I degree → ℝ) n G) leaf z) σ =
spinTensorEnergy U I degree amplitude z σ +
∑ i : Fin n, spinTensorEnergy U I degree amplitude (G (edgeAt n leaf i)) σ := by
unfold spinTensorEnergy
simp only [labeledEnergy_edge_sum, add_mul, Finset.sum_add_distrib]
congr 1
simp_rw [Finset.sum_mul]
rw [Finset.sum_comm]
private lemma tensorCoordinateLog_eq_terminal {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) (z : SpinTensorIndex I degree → ℝ) (q : TensorCoordinateData I degree n) :
Real.log (∫ x, Real.exp (tensorLabeledEnergy eig U c I degree amplitude n z
(tensorEnergyCoordinates U I degree amplitude n q) x)
∂tensorLabeledReference N n (tensorEnergyCoordinates U I degree amplitude n q)) =
Real.log (∫ leaf, Real.exp (spinTensorTerminal eig U c I degree amplitude
(labeledEnergy n (markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2) leaf z))
∂labeledLeafLaw n q.1) := by
let H := tensorSpinBaseEnergy eig U c I degree amplitude z
let G := markForestOfCoords (Spin N → ℝ) n
(fun w => spinTensorEnergy U I degree amplitude (q.2 w))
have he (x : Spin N × LabeledLeaf n) : tensorLabeledEnergy eig U c I degree amplitude n z
(tensorEnergyCoordinates U I degree amplitude n q) x = labeledEnergy n G x.2 H x.1 := by
rw [labeledEnergy_edge_sum]
rfl
simp_rw [he]
change labeledEnergyLog n (uniformSpinPrior N : Measure (Spin N)) H (q.1, G) = _
rw [labeledEnergyLog_eq_leaf]
apply congrArg Real.log
apply integral_congr_ae
refine ae_of_all _ fun leaf => ?_
apply congrArg Real.exp
rw [finiteLogIntegral_uniformSpinPrior]
unfold spinTensorTerminal
congr 1
funext σ
rw [spinTensorEnergy_labeled, labeledEnergy_edge_sum]
change (rotatedEnergy eig U σ + fieldEnergy c σ + spinTensorEnergy U I degree amplitude z σ) + _ = _
ring
/-- On every good marked tree the coordinate partition is exactly the
actual marked-cascade partition, uniformly in its rotation and root mark. -/
theorem tensorCoordinateLog_eq_noise {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) (z : SpinTensorIndex I degree → ℝ) (q : TensorCoordinateData I degree n)
(hg : GoodNoiseTree (SpinTensorIndex I degree → ℝ) n
(labeledNoiseJoin (SpinTensorIndex I degree → ℝ) n
(q.1, markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2)))
(ht : 0 < noiseTreeTotal (SpinTensorIndex I degree → ℝ) n
(labeledNoiseJoin (SpinTensorIndex I degree → ℝ) n
(q.1, markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2)) ∧
noiseTreeTotal (SpinTensorIndex I degree → ℝ) n
(labeledNoiseJoin (SpinTensorIndex I degree → ℝ) n
(q.1, markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2)) < ∞) :
Real.log (∫ x, Real.exp (tensorLabeledEnergy eig U c I degree amplitude n z
(tensorEnergyCoordinates U I degree amplitude n q) x)
∂tensorLabeledReference N n (tensorEnergyCoordinates U I degree amplitude n q)) =
tensorCascadeLog eig U c I degree amplitude n z
(labeledNoiseJoin (SpinTensorIndex I degree → ℝ) n
(q.1, markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2)) := by
rw [tensorCoordinateLog_eq_terminal]
have he := labeled_terminal_integral n q.1
(markForestOfCoords (SpinTensorIndex I degree → ℝ) n q.2) hg ht
(X := fun _ => spinTensorTerminal eig U c I degree amplitude) (u := fun _ p => p.1 + p.2)
(fun _ => measurable_spinTensorTerminal eig U c I degree amplitude)
(fun _ => measurable_fst.add measurable_snd) z
simp only [labeledEnergy_terminal] at he
exact congrArg Real.log he
/-- The good-tree event is independent of the terminal energy, so this
identity holds simultaneously for every rotation on one full-measure set. -/
theorem tensorFlatLog_eq_noise_ae {N m k : ℕ} (eig 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) :
∀ᵐ p : LabeledTree n × (ℕ → ℝ)
∂(labeledCascadeLaw n b : Measure (LabeledTree n)).prod gaussianCoordinates,
∀ U : Rotation N, tensorFlatLog eig U c I degree amplitude n v p =
tensorCascadeLog eig U c I degree amplitude n (tensorFlatToNoise I degree n v p).1
(tensorFlatToNoise I degree n v p).2 := by
let M := (SpinTensorIndex I degree → ℝ)
let ν := fun i => tensorGaussianLaw I degree (v (i + 1))
let : IsProbabilityMeasure (tensorCascadeLaw I degree n b (fun i => v (i + 1))) :=
(noiseCascadeLaw M n b ν).prop
have hp := tensorFlatToNoise_measurePreserving I degree n b v
have hg : ∀ᵐ T ∂tensorCascadeLaw I degree n b (fun i => v (i + 1)), GoodNoiseTree M n T :=
noiseCascade_good M n b hb ν
have ht : ∀ᵐ T ∂tensorCascadeLaw I degree n b (fun i => v (i + 1)),
0 < noiseTreeTotal M n T ∧ noiseTreeTotal M n T < ∞ := by
apply ae_of_ae_map (p := fun T : RawTree n => 0 < rawTreeTotal n T ∧ rawTreeTotal n T < ∞)
(measurable_noiseTreeForget M n).aemeasurable
change ∀ᵐ T : RawTree n ∂(noiseCascadeLaw M n b ν : Measure (NoiseTree M n)).map
(noiseTreeForget M n), 0 < rawTreeTotal n T ∧ rawTreeTotal n T < ∞
rw [noiseCascadeLaw_forget]
exact (rawCascade_total_moments n b hb).1
have hproj := measurePreserving_snd (μ := (tensorGaussianLaw I degree (v 0) : Measure M))
(ν := tensorCascadeLaw I degree n b (fun i => v (i + 1)))
have hg' := hp.quasiMeasurePreserving.ae (hproj.quasiMeasurePreserving.ae hg)
have ht' := hp.quasiMeasurePreserving.ae (hproj.quasiMeasurePreserving.ae ht)
filter_upwards [hg', ht'] with p hgp htp
intro U
let q := tensorFlatToCoordinates I degree n v p
have he : tensorFlatLog eig U c I degree amplitude n v p =
Real.log (∫ x, Real.exp (tensorLabeledEnergy eig U c I degree amplitude n q.1
(tensorEnergyCoordinates U I degree amplitude n q.2) x)
∂tensorLabeledReference N n (tensorEnergyCoordinates U I degree amplitude n q.2)) := by
apply congrArg Real.log
apply integral_congr_ae
exact ae_of_all _ fun x => congrArg Real.exp
(tensorLabeledEnergy_flat_eq eig U c I degree amplitude n v p x).symm
exact he.trans (tensorCoordinateLog_eq_noise eig U c I degree amplitude n q.1 q.2 hgp htp)
def tensorFlatToDisorder {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0)
(p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ)) :
SpecialOrthogonal N × ((SpinTensorIndex I degree → ℝ) × NoiseTree (SpinTensorIndex I degree → ℝ) n) :=
(p.1.1, tensorFlatToNoise I degree n v (p.1.2, p.2))
theorem tensorFlatToDisorder_measurePreserving {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ]
(I : Fin m → Finset (Fin N)) (degree : Fin k → Fin m → ℕ)
(n : ℕ) (b : ℕ → ℝ) (v : ℕ → SpinTensorIndex I degree → ℝ≥0) :
MeasurePreserving (tensorFlatToDisorder I degree n v)
((μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))).prod gaussianCoordinates)
(μ.prod (tensorRootTreeLaw I degree n b (fun i => v (i + 1)) (v 0))) := by
have hp := ((MeasurePreserving.id μ).prod
(tensorFlatToNoise_measurePreserving I degree n b v)).comp
(measurePreserving_prodAssoc μ (labeledCascadeLaw n b : Measure (LabeledTree n)) gaussianCoordinates)
convert hp using 1
rfl
/-- Exact almost-everywhere pressure identification with the actual
three-disorder model, suitable for transporting moments and variance. -/
theorem tensorFlatPressure_eq_disorder_ae {N m k : ℕ}
(μ : Measure (SpecialOrthogonal N)) [IsProbabilityMeasure μ] (eig 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) :
(fun p : (SpecialOrthogonal N × LabeledTree n) × (ℕ → ℝ) =>
(N : ℝ)⁻¹ * tensorFlatLog eig (specialRotation p.1.1) c I degree amplitude n v (p.1.2, p.2)) =ᵐ[
(μ.prod (labeledCascadeLaw n b : Measure (LabeledTree n))).prod gaussianCoordinates]
(tensorDisorderPressure eig c I degree amplitude n ∘ tensorFlatToDisorder I degree n v) := by
have hproj := (measurePreserving_snd (μ := μ)
(ν := (labeledCascadeLaw n b : Measure (LabeledTree n)).prod gaussianCoordinates)).comp
(measurePreserving_prodAssoc μ (labeledCascadeLaw n b : Measure (LabeledTree n)) gaussianCoordinates)
have he := hproj.quasiMeasurePreserving.ae (tensorFlatLog_eq_noise_ae eig c I degree amplitude n b v hb)
filter_upwards [he] with p hp
exact congrArg ((N : ℝ)⁻¹ * ·) (hp (specialRotation p.1.1))
end InvariantIsing
end
end OAI
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