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a89d282 | 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 | import OAI.Probability.InvariantIsing.Gaussian.CountableGaussianVariance
namespace OAI
/-! Gaussian variance for the countable spin/leaf partition with base energy. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal
namespace InvariantIsing
lemma cylinder_cgf_memLp_two {X : Type*} [MeasurableSpace X] [Countable X]
[MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
(A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B)
(t : ℝ) :
MemLp (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g) ν t)
2 gaussianCoordinates := by
let H := fun p : (ℕ → ℝ) × X => t * cylinderField (A p.2) p.1
have hH : Measurable H := (measurable_cylinderFields A).const_mul t
have hLaw (x : X) : gaussianCoordinates.map (fun g => H (g, x)) =
gaussianReal 0 (⟨t ^ 2, sq_nonneg t⟩ * (((A x).sum (fun _ c => c ^ 2)).toNNReal)) :=
cylinder_scaled_law A x t
have hVar (x : X) :
((⟨t ^ 2, sq_nonneg t⟩ : ℝ≥0) * (((A x).sum (fun _ c => c ^ 2)).toNNReal) : ℝ) ≤ t ^ 2 * B := by
change t ^ 2 * (((A x).sum (fun _ c => c ^ 2)).toNNReal : ℝ) ≤ _
rw [Real.coe_toNNReal _ (show 0 ≤ (A x).sum (fun _ c => c ^ 2) from
Finset.sum_nonneg (fun _ _ => sq_nonneg _))]
exact mul_le_mul_of_nonneg_left (hA x) (sq_nonneg t)
exact (memLp_two_iff_integrable_sq
hH.exp.stronglyMeasurable.integral_prod_right'.measurable.log.aestronglyMeasurable).mpr
(gaussian_log_partition_second (ν := ν) hH hLaw hVar).1
/-- Splitting off a fixed integrable base energy is an almost-everywhere
identity for the actual countable Gaussian partition, including nonuniform
and cascade-leaf weights. -/
lemma cylinder_log_partition_base {X : Type*} [MeasurableSpace X] [Countable X]
[MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
(H : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
(A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ) :
(fun g : ℕ → ℝ => Real.log (∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν))
=ᵐ[gaussianCoordinates]
(fun g => Real.log (∫ x, Real.exp (H x) ∂ν) +
cgf (fun x => cylinderField (A x) g) (ν.tilted H) t) := by
have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
have hVar (x : X) : (((A x).sum (fun _ c => c ^ 2)).toNNReal : ℝ) ≤ B := by
rw [Real.coe_toNNReal _ (show 0 ≤ (A x).sum (fun _ c => c ^ 2) from
Finset.sum_nonneg (fun _ _ => sq_nonneg _))]
exact hA x
have he := (gaussian_field_exp_integrable (ν := ν.tilted H)
(measurable_cylinderFields A) (fun x => cylinderField_law (A x)) hVar t).1.prod_right_ae
filter_upwards [he] with g hg
have hz : 0 < ∫ x, Real.exp (H x) ∂ν := integral_exp_pos hH
have hz' : 0 < ∫ x, Real.exp (t * cylinderField (A x) g) ∂ν.tilted H := integral_exp_pos hg
have hi := integral_exp_tilted (μ := ν) H (fun x => t * cylinderField (A x) g)
have hid : (∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν) =
(∫ x, Real.exp (H x) ∂ν) *
(∫ x, Real.exp (t * cylinderField (A x) g) ∂ν.tilted H) := by
rw [hi]
simp only [Pi.add_apply]
field_simp
rw [hid, Real.log_mul hz.ne' hz'.ne']
rfl
lemma cylinder_log_partition_memLp_two {X : Type*} [MeasurableSpace X] [Countable X]
[MeasurableSingletonClass X] (ν : Measure X) [IsProbabilityMeasure ν]
(H : X → ℝ) (hH : Integrable (fun x => Real.exp (H x)) ν)
(A : X → ℕ →₀ ℝ) {B : ℝ} (hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ) :
MemLp (fun g : ℕ → ℝ => Real.log
(∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν)) 2 gaussianCoordinates := by
have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
have he := cylinder_log_partition_base ν H hH A hA t
exact MemLp.ae_eq (Filter.EventuallyEq.symm he)
((memLp_const (Real.log (∫ x, Real.exp (H x) ∂ν))).add
(cylinder_cgf_memLp_two (ν.tilted H) A hA t))
/-- The actual countable Gaussian partition with arbitrary integrable base
energy has the same coefficient variance bound. This is conditional only on
the specified published Gaussian Poincaré consequence. -/
theorem countable_cylinder_log_partition_variance_le (hpub : GaussianLipschitzVarianceInput)
{X : Type*} [MeasurableSpace X] [Countable X] [MeasurableSingletonClass X]
(ν : Measure X) [IsProbabilityMeasure ν] (H : X → ℝ)
(hH : Integrable (fun x => Real.exp (H x)) ν)
(A : X → ℕ →₀ ℝ) (B : ℝ) (hB : 0 ≤ B)
(hA : ∀ x, (A x).sum (fun _ c => c ^ 2) ≤ B) (t : ℝ) :
variance (fun g : ℕ → ℝ => Real.log
(∫ x, Real.exp (H x + t * cylinderField (A x) g) ∂ν)) gaussianCoordinates ≤ t ^ 2 * B := by
have : IsProbabilityMeasure (ν.tilted H) := isProbabilityMeasure_tilted hH
have he := cylinder_log_partition_base ν H hH A hA t
rw [variance_congr he, variance_const_add (cylinder_cgf_memLp_two (ν.tilted H) A hA t).aestronglyMeasurable]
exact countable_cylinder_cgf_variance_le hpub (ν.tilted H) A B hB hA t
end InvariantIsing
end
end OAI
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