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c4cdbbd | 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 | import OAI.Probability.InvariantIsing.Haar.PublishedInputs
import OAI.Probability.IsingPerceptron.EnrichedFieldCap
import OAI.Probability.IsingPerceptron.NormalizedRestrictionPi
namespace OAI
/-! Finite Gaussian cutoff partition functions in the Euclidean norm. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal InnerProductSpace
namespace InvariantIsing
/-- The finite-cutoff Gaussian log partition, with arbitrary probability
weights and deterministic base energy. In the application the finite state
space is a positive-mass restriction of the spin/cascade-leaf prior. -/
def finiteGaussianLogPartition {X : Type*} [MeasurableSpace X]
(ν : Measure X) (H : X → ℝ) {d : ℕ}
(A : X → EuclideanSpace ℝ (Fin d)) (g : EuclideanSpace ℝ (Fin d)) : ℝ :=
Real.log (∫ x, Real.exp (H x + ⟪A x, g⟫_ℝ) ∂ν)
theorem finiteGaussianLogPartition_lipschitz {X : Type*} [Fintype X]
[MeasurableSpace X] [MeasurableSingletonClass X]
(ν : Measure X) [IsProbabilityMeasure ν] (H : X → ℝ) {d : ℕ}
(A : X → EuclideanSpace ℝ (Fin d)) (L : ℝ≥0) (hA : ∀ x, ‖A x‖ ≤ L) :
LipschitzWith L (finiteGaussianLogPartition ν H A) := by
apply LipschitzWith.of_dist_le_mul
intro g h
have hb (x : X) : |(H x + ⟪A x, g⟫_ℝ) - (H x + ⟪A x, h⟫_ℝ)| ≤
(L : ℝ) * ‖g - h‖ := by
rw [add_sub_add_left_eq_sub, ← inner_sub_right]
exact (abs_real_inner_le_norm _ _).trans
(mul_le_mul_of_nonneg_right (hA x) (norm_nonneg _))
have hcomp := logMean_abs_sub_le ν (by norm_num : (0 : ℝ) < 1)
(show Integrable (fun x => Real.exp (1 * (H x + ⟪A x, g⟫_ℝ))) ν from Integrable.of_finite)
(show Integrable (fun x => Real.exp (1 * (H x + ⟪A x, h⟫_ℝ))) ν from Integrable.of_finite) hb
simpa only [Real.dist_eq, dist_eq_norm, Real.norm_eq_abs, logMean, one_mul, div_one,
finiteGaussianLogPartition] using hcomp
/-- The Gaussian Poincaré inequality bounds the finite-cutoff
log partition variance by the coefficient squared norm,
uniformly in the cutoff and base energy. -/
theorem finiteGaussianLogPartition_variance_le (hpub : GaussianLipschitzVarianceInput)
{X : Type*} [Fintype X] [MeasurableSpace X] [MeasurableSingletonClass X]
(ν : Measure X) [IsProbabilityMeasure ν] (H : X → ℝ) {d : ℕ}
(A : X → EuclideanSpace ℝ (Fin d)) (B : ℝ≥0)
(hA : ∀ x, ‖A x‖ ^ 2 ≤ B) :
variance (finiteGaussianLogPartition ν H A)
(stdGaussian (EuclideanSpace ℝ (Fin d))) ≤ B := by
have hs : ((NNReal.sqrt B : ℝ≥0) : ℝ) ^ 2 = B := by exact_mod_cast NNReal.sq_sqrt B
have hnorm (x : X) : ‖A x‖ ≤ ((NNReal.sqrt B : ℝ≥0) : ℝ) := by
have hn := NNReal.coe_nonneg (NNReal.sqrt B)
nlinarith [hA x, norm_nonneg (A x)]
exact (hpub d (NNReal.sqrt B) _
(finiteGaussianLogPartition_lipschitz ν H A (NNReal.sqrt B) hnorm)).trans_eq hs
/-- Finite-state countable-cylinder restriction. A finite family of finitely supported coefficients uses a common
finite prefix, whose Gaussian law is the standard Euclidean Gaussian. -/
theorem finite_cylinder_cgf_variance_le (hpub : GaussianLipschitzVarianceInput)
{X : Type*} [Fintype X] [MeasurableSpace X] [MeasurableSingletonClass X]
(ν : Measure X) [IsProbabilityMeasure 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 => cgf (fun x => cylinderField (A x) g) ν t)
gaussianCoordinates ≤ t ^ 2 * B := by
classical
obtain ⟨n, hn⟩ := finite_family_support_bound A
have hs (x : X) : ∀ j ∈ (A x).support, j < n + 1 :=
fun j hj => hn x j (Finsupp.mem_support_iff.mp hj)
let V : X → EuclideanSpace ℝ (Fin (n + 1)) :=
fun x => WithLp.toLp 2 (fun j : Fin (n + 1) => t * A x j)
let B' : ℝ≥0 := ⟨t ^ 2 * B, mul_nonneg (sq_nonneg t) hB⟩
have hv (x : X) : ‖V x‖ ^ 2 ≤ B' := by
rw [EuclideanSpace.real_norm_sq_eq]
simp only [V, mul_pow, ← Finset.mul_sum,
← cylinder_variance_prefix (A x) (hs x)]
exact mul_le_mul_of_nonneg_left (hA x) (sq_nonneg t)
let F := finiteGaussianLogPartition ν (fun _ => 0) V
have hF : Measurable F := by
have hsqrt : (((NNReal.sqrt B' : ℝ≥0) : ℝ)) ^ 2 = B' :=
by exact_mod_cast NNReal.sq_sqrt B'
have hnorm (x : X) : ‖V x‖ ≤ ((NNReal.sqrt B' : ℝ≥0) : ℝ) := by
have hn' := NNReal.coe_nonneg (NNReal.sqrt B')
nlinarith [hv x, norm_nonneg (V x)]
exact (finiteGaussianLogPartition_lipschitz ν (fun _ => 0) V (NNReal.sqrt B') hnorm).continuous.measurable
have hto : MeasurePreserving
(WithLp.toLp 2 : (Fin (n + 1) → ℝ) → EuclideanSpace ℝ (Fin (n + 1)))
(Measure.pi (fun _ : Fin (n + 1) => gaussianReal 0 1))
(stdGaussian (EuclideanSpace ℝ (Fin (n + 1)))) :=
⟨by fun_prop, map_pi_eq_stdGaussian⟩
have hmp := hto.comp (gaussian_prefix_measurePreserving (n + 1))
have he : (fun g : ℕ → ℝ => cgf (fun x => cylinderField (A x) g) ν t) =
F ∘ ((WithLp.toLp 2) ∘ (fun g : ℕ → ℝ => fun j : Fin (n + 1) => g j)) := by
funext g
have hi (x : X) : ⟪V x, WithLp.toLp 2 (fun j : Fin (n + 1) => g j)⟫_ℝ =
t * cylinderField (A x) g := by
rw [cylinderField_eq_prefix (A x) (hs x)]
simp only [V, PiLp.inner_apply, RCLike.inner_apply', conj_trivial]
rw [Finset.mul_sum]
apply Finset.sum_congr rfl
intro j _
ring
simp only [Function.comp_apply, F, finiteGaussianLogPartition, cgf, mgf, zero_add, hi]
rw [he]
change variance (fun g => F ((WithLp.toLp 2 ∘
(fun g : ℕ → ℝ => fun j : Fin (n + 1) => g j)) g)) gaussianCoordinates ≤ _
rw [hmp.variance_fun_comp hF.aemeasurable]
exact finiteGaussianLogPartition_variance_le hpub ν (fun _ => 0) V B' hv
end InvariantIsing
end
end OAI
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