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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 | import OAI.Probability.InvariantIsing.Arrays.NSpinTensorHaar
import OAI.Probability.InvariantIsing.Arrays.NSpinTensorFluctuation
namespace OAI
/-! Full Haar, Gaussian-root, and cascade fluctuations for the actual finite tensor model. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
/-- Independent common Gaussian root and marked cascade. -/
def tensorRootTreeLaw {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) (root : SpinTensorIndex I degree → ℝ≥0) :
Measure ((SpinTensorIndex I degree → ℝ) × NoiseTree (SpinTensorIndex I degree → ℝ) n) :=
(tensorGaussianLaw I degree root : Measure (SpinTensorIndex I degree → ℝ)).prod
(tensorCascadeLaw I degree n b v)
instance tensorRootTreeLaw_probability {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) (root : SpinTensorIndex I degree → ℝ≥0) :
IsProbabilityMeasure (tensorRootTreeLaw I degree n b v root) := by
let : IsProbabilityMeasure (tensorCascadeLaw I degree n b v) :=
(noiseCascadeLaw (SpinTensorIndex I degree → ℝ) n b
(fun i => tensorGaussianLaw I degree (v i))).prop
unfold tensorRootTreeLaw
infer_instance
/-- The normalized spin/leaf log partition retaining all three disorders. -/
def tensorDisorderPressure {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)
(p : SpecialOrthogonal N × ((SpinTensorIndex I degree → ℝ) ×
NoiseTree (SpinTensorIndex I degree → ℝ) n)) : ℝ :=
(N : ℝ)⁻¹ * tensorCascadeLog eig (specialRotation p.1) c I degree amplitude n p.2.1 p.2.2
private lemma measurable_tensorLeafState {N m k : ℕ} (I : Fin m → Finset (Fin N))
(degree : Fin k → Fin m → ℕ) (n : ℕ) :
Measurable (fun p : (SpinTensorIndex I degree → ℝ) × NoiseLeaf (SpinTensorIndex I degree → ℝ) n =>
tensorLeafState I degree n p.1 p.2) := by
induction n with
| zero => exact measurable_fst
| succ n ih =>
exact ih.comp ((measurable_fst.add measurable_snd.snd.fst).prodMk measurable_snd.snd.snd)
lemma measurable_tensorDisorderPressure {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) :
Measurable (tensorDisorderPressure eig c I degree amplitude n) := by
let Ω := SpecialOrthogonal N × ((SpinTensorIndex I degree → ℝ) ×
NoiseTree (SpinTensorIndex I degree → ℝ) n)
have hs : Measurable (fun p : Ω × NoiseLeaf (SpinTensorIndex I degree → ℝ) n =>
(p.1.1, tensorLeafState I degree n p.1.2.1 p.2)) :=
measurable_fst.fst.prodMk ((measurable_tensorLeafState I degree n).comp
(measurable_fst.snd.fst.prodMk measurable_snd))
have ht := ((measurable_spinTensorTerminal_joint eig c I degree amplitude).comp hs).exp
have hi := (ht.stronglyMeasurable.integral_kernel_prod_right'
(κ := (noiseLeafKernel (SpinTensorIndex I degree → ℝ) n).comap
(fun p : Ω => p.2.2) measurable_snd.snd)).measurable.log
have hlog : Measurable (fun p : Ω => tensorCascadeLog eig (specialRotation p.1) c
I degree amplitude n p.2.1 p.2.2) := by
simpa only [tensorCascadeLog, noiseLeafTerminal_spinTensorTerminal, Function.comp_apply,
Kernel.comap_apply] using hi
exact hlog.const_mul (N : ℝ)⁻¹
lemma tensorDisorderPressure_conditional_mean (hpub : GaussianLipschitzVarianceInput)
{N m k : ℕ} (hN : 0 < N) (eig c : Fin N → ℝ) (U : SpecialOrthogonal N)
(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) (site : ℝ≥0) (monomial : Fin k → ℝ≥0) :
(∫ q, tensorDisorderPressure eig c I degree amplitude n (U, q)
∂tensorRootTreeLaw I degree n b v (tensorVarianceProfile I degree site monomial)) =
tensorEnrichedPressure eig (specialRotation U) c I degree amplitude n b v
(tensorVarianceProfile I degree site monomial) := by
let : IsProbabilityMeasure (tensorCascadeLaw I degree n b v) :=
(noiseCascadeLaw (SpinTensorIndex I degree → ℝ) n b
(fun i => tensorGaussianLaw I degree (v i))).prop
have h := tensorCascadeLog_root_tree_variance_le hpub hN eig (specialRotation U) c
I degree amplitude n b v hb site monomial
unfold tensorDisorderPressure tensorEnrichedPressure tensorRootTreeLaw
rw [integral_const_mul, integral_prod _ (h.1.integrable (by norm_num))]
congr 1
apply integral_congr_ae
exact ae_of_all _ fun z =>
(tensorCascadeLog_recursion hN eig (specialRotation U) c I degree amplitude n b v hb z).2.1
/-- The full pressure variance is the sum of the root/cascade conditional
contribution and the Haar variance of its conditional mean. -/
theorem tensorDisorderPressure_variance (hhaar : HaarConcentrationInput)
(hgauss : GaussianLipschitzVarianceInput) :
∃ C : ℝ, 0 < C ∧
∀ N : ℕ, 3 ≤ N →
∀ μ : Measure (SpecialOrthogonal N), IsProbabilityMeasure μ → μ.IsMulLeftInvariant →
∀ m k : ℕ, ∀ eig c : Fin N → ℝ, ∀ K : ℝ, 0 < K → (∀ i, |eig i| ≤ K) →
∀ I : Fin m → Finset (Fin N), ∀ degree : Fin k → Fin m → ℕ,
∀ amplitude : Fin k → ℝ, ∀ n : ℕ, ∀ b : ℕ → ℝ, CascadeExponents n b →
∀ site : ℕ → ℝ≥0, ∀ monomial : ℕ → Fin k → ℝ≥0,
let v := fun i => tensorVarianceProfile I degree (site i) (monomial i)
let P := tensorRootTreeLaw I degree n b (fun i => v (i + 1)) (v 0)
let F := tensorDisorderPressure eig c I degree amplitude n
let L := K + 2 * (N : ℝ)⁻¹ *
(∑ i : Fin (n + 1), ∑ j, (monomial i j : ℝ) * amplitude j ^ 2 * ∑ a, (degree j a : ℝ))
MemLp F 2 (μ.prod P) ∧ variance F (μ.prod P) ≤
(4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2
∂(rawCascadeLaw n b : Measure (RawTree n))) +
(site 0 : ℝ) * N + ∑ j, (monomial 0 j : ℝ) * amplitude j ^ 2) / (N : ℝ) ^ 2 +
C * L ^ 2 / N := by
obtain ⟨C, hC, hHaar⟩ := haar_tensorEnrichedPressure_variance hhaar
refine ⟨C, hC, ?_⟩
intro N hN μ hμ hμinv m k eig c K hK heig I degree amplitude n b hb site monomial v P F L
let : IsProbabilityMeasure μ := hμ
have hNm : 0 < N := by omega
let X := fun U : SpecialOrthogonal N => tensorEnrichedPressure eig (specialRotation U) c
I degree amplitude n b (fun i => v (i + 1)) (v 0)
have hX := hHaar N hN μ hμ hμinv m k eig c K hK heig I degree amplitude n b hb site monomial
have hsec (U : SpecialOrthogonal N) : MemLp (fun q => F (U, q)) 2 P ∧
variance (fun q => F (U, q)) P ≤
(4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2
∂(rawCascadeLaw n b : Measure (RawTree n))) +
(site 0 : ℝ) * N + ∑ j, (monomial 0 j : ℝ) * amplitude j ^ 2) / (N : ℝ) ^ 2 :=
normalized_tensorCascadeLog_root_tree_variance_le hgauss hNm eig (specialRotation U) c
I degree amplitude n b (fun i => v (i + 1)) hb (site 0) (monomial 0)
have hc (U : SpecialOrthogonal N) : (∫ q, F (U, q) ∂P) = X U :=
tensorDisorderPressure_conditional_mean hgauss hNm eig c U I degree amplitude n b
(fun i => v (i + 1)) hb (site 0) (monomial 0)
have hi (U : SpecialOrthogonal N) : Integrable (fun q => (F (U, q) - X U) ^ 2) P :=
((hsec U).1.sub (memLp_const (X U))).integrable_sq
have hbnd (U : SpecialOrthogonal N) : (∫ q, (F (U, q) - X U) ^ 2 ∂P) ≤
(4 * (∫ T, (Real.log (rawTreeTotal n T).toReal) ^ 2
∂(rawCascadeLaw n b : Measure (RawTree n))) +
(site 0 : ℝ) * N + ∑ j, (monomial 0 j : ℝ) * amplitude j ^ 2) / (N : ℝ) ^ 2 := by
have he := variance_eq_integral (hsec U).1.aemeasurable
rw [hc U] at he
rw [← he]
exact (hsec U).2
have h := variance_of_conditional_center μ P
(measurable_tensorDisorderPressure eig c I degree amplitude n) hX.1 hi hc hbnd
exact ⟨h.1, h.2.trans (add_le_add_right hX.2 _)⟩
end InvariantIsing
end
end OAI
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