File size: 9,236 Bytes
a2ee19a | 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 | import Mathlib
import OAI.Probability.Perceptron.Variational.Contact
import OAI.Probability.Perceptron.Interpolation.KernelReplicaGibbs
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Filter Set
open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction
namespace SphericalPerceptronFreeEnergy
def sourceCouplingCovariance (n k : ℕ) (p d : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ)
(j : Fin (n+1)) (x y : NormalizedSpin (n+1)×IndexedLeaf k) :=
perturbationAmplitude (n+1) (fun _ => 1) j * perturbationAmplitude (n+1) u j *
sourceJointMonomial p d j x y
lemma sourceCouplingCovariance_measurable (n k r : ℕ) (p d : Fin (n+1) → ℕ)
(u : Fin (n+1) → ℝ) (j : Fin (n+1)) (i l : Fin r) :
Measurable (fun x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k =>
sourceCouplingCovariance n k p d u j (x i) (x l)) := by
have hh := (sourceReplica_covariance_measurable (N := n+1) (k := k) p d j i l).const_mul
(perturbationAmplitude (n+1) (fun _ => 1) j * perturbationAmplitude (n+1) u j)
exact hh
lemma sourceCouplingCovariance_bound (n k : ℕ) (p d : Fin (n+1) → ℕ) (u : Fin (n+1) → ℝ)
(j : Fin (n+1)) (x y : NormalizedSpin (n+1)×IndexedLeaf k) :
|sourceCouplingCovariance n k p d u j x y| ≤
|perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j| := by
unfold sourceCouplingCovariance
rw [abs_mul]
simpa only [mul_one] using mul_le_mul_of_nonneg_left
(sourceJointMonomial_abs_le p d j x y) (abs_nonneg (perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j))
lemma sourceCoupling_conditional_ibp (n k r : ℕ) (f : ℝ →ᵇ ℝ)
(p d : Fin (n+1) → ℕ) (h : Fin (k+1) → ℝ)
(hh0 : ∀ l, 0 ≤ h l) (hh : Monotone h) (u : Fin (n+1) → ℝ)
(j : Fin (n+1)) (i : Fin r) (b : SourceBaseData n k)
{G : (Fin r → NormalizedSpin (n+1)×IndexedLeaf k) → ℝ}
(hG : Measurable G) {B : ℝ} (hB : 0 ≤ B) (hGB : ∀ x, |G x| ≤ B) :
(∫ g, gibbsReplicaMean (sourceSpinLeafKernel n k b)
(sourceCouplingHamiltonian n k f p d h u (b,g)) r
(fun x => sourceCouplingEnergy n k p d h j (b,g) (x i)*G x) ∂countableGaussianLaw) =
∫ g, gibbsReplicaMean (sourceSpinLeafKernel n k b)
(sourceCouplingHamiltonian n k f p d h u (b,g)) r
(fun x => G x*∑ l, sourceCouplingCovariance n k p d u j (x i) (x l)) -
r*gibbsReplicaMean (sourceSpinLeafKernel n k b)
(sourceCouplingHamiltonian n k f p d h u (b,g)) (r+1)
(fun x => G (fun l => x l.succ)*sourceCouplingCovariance n k p d u j (x i.succ) (x 0))
∂countableGaussianLaw := by
let c := perturbationAmplitude (n+1) (fun _ => 1) j
have hb := sourceReplica_gaussian_ibp n b.1 k r f (patternPrefix (n+1) b.1 b.2.1)
p d h hh0 hh u b.2.2 j i hG hB hGB
dsimp only at hb
have he := congrArg (fun x : ℝ => c*x) hb
simp only [sourceSpinLeafKernel_apply,sourceCouplingHamiltonian,sourceCouplingEnergy,
sourceCouplingCovariance]
have hs (x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k) :
G x*∑ l, c*perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l) =
c*(G x*∑ l, perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l)) := by
simp only [mul_assoc,Finset.mul_sum]
ring_nf
simp only [mul_assoc]
change (∫ g, gibbsReplicaMean (enrichedIndexedBaseMeasure n k b.2.2)
(enrichedIndexedHamiltonian n b.1 k f (patternPrefix (n+1) b.1 b.2.1) p d h u g) r
(fun x => c*(countableGaussianField (sourceGaussianTestRow p d h j)
(indexedGaussianRowLength (I := EnrichedIndex (n+1) (n+1) p) k) g (x i)*G x)) ∂countableGaussianLaw) = _
simp_rw [replicaMean_const_mul,integral_const_mul]
rw [hb, ← integral_const_mul]
apply integral_congr_ae
exact ae_of_all _ fun g => by
have hs₁ : (fun x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k =>
G x * ∑ l, c*(perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l))) =
(fun x => c*(G x*∑ l, perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i) (x l))) := by
funext x
rw [← Finset.mul_sum]
ring
have hs₂ : (fun x : Fin (r+1) → NormalizedSpin (n+1)×IndexedLeaf k =>
G (fun l => x l.succ)*(c*(perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i.succ) (x 0)))) =
(fun x => c*(G (fun l => x l.succ)*(perturbationAmplitude (n+1) u j*sourceJointMonomial p d j (x i.succ) (x 0)))) := by
funext x
ring
rw [hs₁, hs₂]
simp only [replicaMean_const_mul]
ring
lemma sourceCoupling_annealed_ibp (n k r : ℕ) (f : ℝ →ᵇ ℝ)
(p d : Fin (n+1) → ℕ) (h : Fin (k+1) → ℝ)
(hh0 : ∀ l, 0 ≤ h l) (hh : Monotone h) (u : Fin (n+1) → ℝ)
(j : Fin (n+1)) (i : Fin r) (z : Fin k → ℝ) (t : ℝ≥0)
{G : (Fin r → NormalizedSpin (n+1)×IndexedLeaf k) → ℝ}
(hG : Measurable G) {B : ℝ} (hB : 0 ≤ B) (hGB : ∀ x, |G x| ≤ B)
(hYi : Integrable (fun a => tiltMean (sourceSpinLeafKernel n k a.1)
(sourceCouplingHamiltonian n k f p d h u a)
(fun x => |sourceCouplingEnergy n k p d h j a x|) 1)
((sourceBaseDataLaw n k z t).prod countableGaussianLaw)) :
let H := sourceCouplingHamiltonian n k f p d h u
let Y := sourceCouplingEnergy n k p d h j
let C := sourceCouplingCovariance n k p d u j
let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw
(∫ a, gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) r (fun x => Y a (x i)*G x) ∂P) =
∫ a, gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) r (fun x => G x*∑ l, C (x i) (x l)) -
r*gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) (r+1)
(fun x => G (fun l => x l.succ)*C (x i.succ) (x 0)) ∂P := by
dsimp only
let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw
let H := sourceCouplingHamiltonian n k f p d h u
let Y := sourceCouplingEnergy n k p d h j
let C := sourceCouplingCovariance n k p d u j
let C₀ := |perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j|
have hH := sourceCouplingHamiltonian_measurable n k f p d h u
have hY := sourceCouplingEnergy_measurable n k p d h j
have he := sourceCoupling_all_exp_ae n k f p d h hh0 hh u j z t
have he0 : ∀ᵐ a ∂P, Integrable (fun x => Real.exp (H a x)) (sourceSpinLeafKernel n k a.1) := by
filter_upwards [he] with a ha
simpa only [zero_mul,add_zero] using ha 0
have he1 : ∀ᵐ a ∂P, Integrable (fun x => Real.exp (H a x)*Y a x) (sourceSpinLeafKernel n k a.1) := by
filter_upwards [he] with a ha
simpa only [zero_mul,add_zero,pow_one] using coupling_weighted_power_integrable
(sourceSpinLeafKernel n k a.1) hH.of_uncurry_left hY.of_uncurry_left ha 0 1
have hi := kernel_replica_energy_integrable (sourceFullSpinLeafKernel n k) P i hH hY hG he0 he1 hYi hB hGB
have hmG : Measurable (fun x : Fin (r+1) → NormalizedSpin (n+1)×IndexedLeaf k => G (fun l => x l.succ)) :=
hG.comp (Measurable.of_eval fun coordinate => measurable_pi_apply coordinate.succ)
have hsum : ∀ x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k, |∑ l, C (x i) (x l)| ≤ r*C₀ := by
intro x
apply (Finset.abs_sum_le_sum_abs _ _).trans
calc
_ ≤ ∑ _ : Fin r, C₀ := Finset.sum_le_sum fun l _ => sourceCouplingCovariance_bound n k p d u j _ _
_ = _ := by simp
have h1G : Measurable (fun x : Fin r → NormalizedSpin (n+1)×IndexedLeaf k => G x*∑ l, C (x i) (x l)) :=
hG.mul (Finset.measurable_sum _ fun l _ => sourceCouplingCovariance_measurable n k r p d u j i l)
have h2G : Measurable (fun x : Fin (r+1) → NormalizedSpin (n+1)×IndexedLeaf k => G (fun l => x l.succ)*C (x i.succ) (x 0)) :=
hmG.mul (sourceCouplingCovariance_measurable n k (r+1) p d u j i.succ 0)
have hi1 := kernel_replicaMean_bounded_integrable (sourceFullSpinLeafKernel n k) P
(G := fun _ x => G x*∑ l, C (x i) (x l)) hH (h1G.comp measurable_snd)
(show 0 ≤ B*(r*C₀) by positivity) (fun _ x => by
rw [abs_mul]; exact mul_le_mul (hGB x) (hsum x) (abs_nonneg _) hB)
have hi2 := kernel_replicaMean_bounded_integrable (sourceFullSpinLeafKernel n k) P
(G := fun _ x => G (fun l => x l.succ)*C (x i.succ) (x 0)) hH (h2G.comp measurable_snd)
(show 0 ≤ B*C₀ by positivity) (fun _ x => by
rw [abs_mul]; exact mul_le_mul (hGB _) (sourceCouplingCovariance_bound n k p d u j _ _) (abs_nonneg _) hB)
change (∫ a, gibbsReplicaMean (sourceFullSpinLeafKernel n k a) (H a) r (fun x => Y a (x i)*G x) ∂P) =
∫ a, gibbsReplicaMean (sourceFullSpinLeafKernel n k a) (H a) r (fun x => G x*∑ l, C (x i) (x l))-
r*gibbsReplicaMean (sourceFullSpinLeafKernel n k a) (H a) (r+1) (fun x => G (fun l => x l.succ)*C (x i.succ) (x 0)) ∂P
have hiR := hi1.sub (hi2.const_mul (r:ℝ))
dsimp only [sourceFullSpinLeafKernel,Kernel.comap_apply,P,H,Y,C,Pi.sub_apply] at hi hiR ⊢
have hprodR := integral_prod _ hiR
simp only [Pi.sub_apply] at hprodR
rw [integral_prod _ hi, hprodR]
apply integral_congr_ae
exact ae_of_all _ fun b => sourceCoupling_conditional_ibp n k r f p d h hh0 hh u j i b hG hB hGB
end SphericalPerceptronFreeEnergy
end
end OAI
|