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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 | import OAI.Probability.InvariantIsing.Arrays.NSpinTensorRootVariance
namespace OAI
/-! Gaussian-root and cascade fluctuations for the finite tensor model. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace InvariantIsing
lemma measurable_tensorCascadeLog_joint {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) :
Measurable (fun p : (SpinTensorIndex I degree → ℝ) ×
IsingPerceptron.NoiseTree (SpinTensorIndex I degree → ℝ) n =>
tensorCascadeLog eig U c I degree amplitude n p.1 p.2) := by
have hm := ((IsingPerceptron.measurable_noiseLeafTerminal n
(fun _ => measurable_spinTensorTerminal eig U c I degree amplitude)
(fun _ => measurable_fst.add measurable_snd)).comp
(show Measurable (fun p : ((SpinTensorIndex I degree → ℝ) ×
IsingPerceptron.NoiseTree (SpinTensorIndex I degree → ℝ) n) ×
IsingPerceptron.NoiseLeaf (SpinTensorIndex I degree → ℝ) n => (p.1.1, p.2))
from by fun_prop)).exp
exact (hm.stronglyMeasurable.integral_kernel_prod_right'
(κ := (IsingPerceptron.noiseLeafKernel (SpinTensorIndex I degree → ℝ) n).comap
Prod.snd measurable_snd)).measurable.log
lemma tensorCascadeValue_root_memLp_two {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 : IsingPerceptron.CascadeExponents n b) (site : ℝ≥0) (monomial : Fin k → ℝ≥0) :
MemLp (tensorCascadeValue eig U c I degree amplitude n b v) 2
(tensorGaussianLaw I degree (tensorVarianceProfile I degree site monomial) :
Measure (SpinTensorIndex I degree → ℝ)) := by
let F := fun g : TensorGaussianSpace I degree =>
tensorCascadeValue eig U c I degree amplitude n b v
(tensorRootMark I degree (tensorVarianceProfile I degree site monomial) g)
have hLip := tensorCascadeValue_root_lipschitz hN eig U c I degree amplitude n b v hb site monomial 0
simp only [zero_add] at hLip
let B : ℝ≥0 := ⟨(site : ℝ) * N + ∑ j, (monomial j : ℝ) * amplitude j ^ 2,
add_nonneg (mul_nonneg (NNReal.coe_nonneg _) (Nat.cast_nonneg _))
(Finset.sum_nonneg fun _j _ => mul_nonneg (NNReal.coe_nonneg _) (sq_nonneg _))⟩
have hLip0 : LipschitzWith (NNReal.sqrt B) (fun g => F g - F 0) := by
apply LipschitzWith.of_dist_le_mul
intro g h
simpa only [dist_sub_right] using hLip.dist_le_mul g h
have hL := hLip0.comp_memLp (by simp)
(IsGaussian.memLp_two_id : MemLp id 2 (stdGaussian (TensorGaussianSpace I degree)))
have hF : MemLp F 2 (stdGaussian (TensorGaussianSpace I degree)) := by
convert hL.add (memLp_const (F 0)) using 1
ext g
simp only [Pi.add_apply, Function.comp_apply, id_eq, sub_add_cancel]
have hm : Measurable (tensorCascadeValue eig U c I degree amplitude n b v) :=
IsingPerceptron.measurable_cascadeRecursion n b (fun i => tensorGaussianLaw I degree (v i))
(fun _ => measurable_fst.add measurable_snd) (measurable_spinTensorTerminal eig U c I degree amplitude)
have hmp := tensorRootMark_measurePreserving I degree (tensorVarianceProfile I degree site monomial)
rw [← hmp.map_eq]
exact (memLp_map_measure_iff hm.aestronglyMeasurable hmp.aemeasurable).mpr hF
/-- The actual log partition on the product of root marks and marked
cascade trees has variance bounded by the sum of the two contributions. -/
theorem tensorCascadeLog_root_tree_variance_le (hpub : GaussianLipschitzVarianceInput)
{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 : IsingPerceptron.CascadeExponents n b) (site : ℝ≥0) (monomial : Fin k → ℝ≥0) :
let root := tensorGaussianLaw I degree (tensorVarianceProfile I degree site monomial)
let P := (root : Measure (SpinTensorIndex I degree → ℝ)).prod (tensorCascadeLaw I degree n b v)
let F := fun p : (SpinTensorIndex I degree → ℝ) ×
IsingPerceptron.NoiseTree (SpinTensorIndex I degree → ℝ) n =>
tensorCascadeLog eig U c I degree amplitude n p.1 p.2
MemLp F 2 P ∧ variance F P ≤
4 * (∫ T, (Real.log (IsingPerceptron.rawTreeTotal n T).toReal) ^ 2
∂(IsingPerceptron.rawCascadeLaw n b : Measure (IsingPerceptron.RawTree n))) +
(site : ℝ) * N + ∑ j, (monomial j : ℝ) * amplitude j ^ 2 := by
intro root P F
let : IsProbabilityMeasure (tensorCascadeLaw I degree n b v) :=
(IsingPerceptron.noiseCascadeLaw (SpinTensorIndex I degree → ℝ) n b
(fun i => tensorGaussianLaw I degree (v i))).prop
have h := IsingPerceptron.variance_of_conditional_center
(root : Measure (SpinTensorIndex I degree → ℝ)) (tensorCascadeLaw I degree n b v)
(measurable_tensorCascadeLog_joint eig U c I degree amplitude n)
(tensorCascadeValue_root_memLp_two hN eig U c I degree amplitude n b v hb site monomial)
(fun z => (tensorCascadeLog_recursion hN eig U c I degree amplitude n b v hb z).2.2.1)
(fun z => (tensorCascadeLog_recursion hN eig U c I degree amplitude n b v hb z).2.1)
(fun z => (tensorCascadeLog_recursion hN eig U c I degree amplitude n b v hb z).2.2.2)
refine ⟨h.1, ?_⟩
have hv := tensorCascadeValue_root_variance_le hpub hN eig U c I degree amplitude n b v hb site monomial
exact h.2.trans (by linarith)
/-- For pressure, the Gaussian-root and cascade contributions scale by
the square of the volume. The rotation remains fixed in this statement. -/
theorem normalized_tensorCascadeLog_root_tree_variance_le (hpub : GaussianLipschitzVarianceInput)
{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 : IsingPerceptron.CascadeExponents n b) (site : ℝ≥0) (monomial : Fin k → ℝ≥0) :
let root := tensorGaussianLaw I degree (tensorVarianceProfile I degree site monomial)
let P := (root : Measure (SpinTensorIndex I degree → ℝ)).prod (tensorCascadeLaw I degree n b v)
let F := fun p : (SpinTensorIndex I degree → ℝ) ×
IsingPerceptron.NoiseTree (SpinTensorIndex I degree → ℝ) n =>
tensorCascadeLog eig U c I degree amplitude n p.1 p.2
MemLp (fun p => (N : ℝ)⁻¹ * F p) 2 P ∧
variance (fun p => (N : ℝ)⁻¹ * F p) P ≤
(4 * (∫ T, (Real.log (IsingPerceptron.rawTreeTotal n T).toReal) ^ 2
∂(IsingPerceptron.rawCascadeLaw n b : Measure (IsingPerceptron.RawTree n))) +
(site : ℝ) * N + ∑ j, (monomial j : ℝ) * amplitude j ^ 2) / (N : ℝ) ^ 2 := by
intro root P F
have h := tensorCascadeLog_root_tree_variance_le hpub hN eig U c I degree amplitude n b v hb site monomial
refine ⟨h.1.const_mul (N : ℝ)⁻¹, ?_⟩
rw [variance_const_mul]
have hbnd := mul_le_mul_of_nonneg_left h.2 (sq_nonneg (N : ℝ)⁻¹)
convert hbnd using 1
ring
end InvariantIsing
end
end OAI
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