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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 | import OAI.Probability.InvariantIsing.Fields.FieldTailMgf
import OAI.Probability.InvariantIsing.Fields.FieldFiniteDomain
import OAI.Probability.InvariantIsing.Magnetic.MagneticFieldInverseConvex
namespace OAI
/-! The canonical terminal field has a depth-independent second moment,
controlled by the initial bias and the final covariance height. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal BigOperators
namespace InvariantIsing
lemma fieldTailVariance_total (h : FieldStep) :
h.height 0 + fieldTailVariance (scalarFieldIncrements h) =
h.height (Fin.last h.depth) := by
have hs := fieldIncrement_sum h
rw [Fin.sum_univ_succ] at hs
have htail : fieldTailVariance (scalarFieldIncrements h) =
∑ i : Fin h.depth, (fieldIncrement h i.succ).2 := by
simp only [fieldTailVariance, scalarFieldIncrements, List.map_ofFn,
Function.comp_apply, NNReal.coe_mk, List.sum_ofFn]
rw [htail]
simpa [fieldIncrement] using hs
theorem fieldCanonicalEndpointLaw_mgf (h : FieldStep) (b t : ℝ) :
Integrable (fun y => Real.exp (t * y)) (fieldCanonicalEndpointLaw h b) ∧
(∫ y, Real.exp (t * y) ∂fieldCanonicalEndpointLaw h b) ≤
Real.exp (t * b + (t ^ 2 / 2 + |t|) * h.height (Fin.last h.depth)) := by
let L := scalarFieldIncrements h
have hL := scalarFieldIncrements_positive h
have hB := scalarFieldIncrements_exponent_le_one h
have hreg := fieldScalarValue_regular L hL measurable_logCosh logCosh_linearGrowth
let root : ℝ≥0 := ⟨h.height 0, h.nonneg 0⟩
let κ := fieldTransitionKernel 0 root
(fieldScalarValue L (fun y => Real.log (Real.cosh y))) hreg.1
have hκ (z : ℝ) := And.intro
(fieldTransitionKernel_integrable_exp root z 0 t _ hreg.1 hreg.2)
(fieldScalarTransition_mgf L hL root z 0 t le_rfl zero_le_one)
have he := kernel_exp_bound_comp κ (fieldTailEndpointKernel L hL) t
((t ^ 2 / 2 + |t|) * root) ((t ^ 2 / 2 + |t|) * fieldTailVariance L)
hκ (fieldTailEndpointKernel_mgf L hL hB t) b
have hlaw : (fieldTailEndpointKernel L hL ∘ₖ κ) b = fieldCanonicalEndpointLaw h b := by
rw [Kernel.comp_apply]
dsimp only [κ]
rw [fieldTransitionKernel_zero _ _ hreg.1 hreg.2]
rfl
rw [hlaw] at he
refine ⟨he.1, he.2.trans_eq ?_⟩
congr 1
change t * b + (t ^ 2 / 2 + |t|) * h.height 0 +
(t ^ 2 / 2 + |t|) * fieldTailVariance (scalarFieldIncrements h) = _
calc
_ = t * b + (t ^ 2 / 2 + |t|) *
(h.height 0 + fieldTailVariance (scalarFieldIncrements h)) := by ring
_ = _ := by rw [fieldTailVariance_total]
lemma sq_le_exp_two_add (x : ℝ) : x ^ 2 ≤ Real.exp (2 * x) + Real.exp (-2 * x) := by
have ha : |x| ≤ Real.exp |x| := by linarith [Real.add_one_le_exp |x|]
have hs := (sq_le_sq₀ (abs_nonneg x) (Real.exp_pos _).le).mpr ha
rw [sq_abs] at hs
rw [show (Real.exp |x|) ^ 2 = Real.exp (|x| + |x|) by
rw [pow_two, Real.exp_add]] at hs
by_cases hx : 0 ≤ x
· rw [abs_of_nonneg hx, show x + x = 2 * x by ring] at hs
linarith [Real.exp_pos (-2 * x)]
· rw [abs_of_nonpos (le_of_not_ge hx), show -x + -x = -2 * x by ring] at hs
linarith [Real.exp_pos (2 * x)]
theorem fieldCanonicalEndpointLaw_second_moment (h : FieldStep) (b : ℝ) :
Integrable (fun y : ℝ => y ^ 2) (fieldCanonicalEndpointLaw h b) ∧
(∫ y, y ^ 2 ∂fieldCanonicalEndpointLaw h b) ≤
Real.exp (2 * b + 4 * h.height (Fin.last h.depth)) +
Real.exp (-2 * b + 4 * h.height (Fin.last h.depth)) := by
have hp := fieldCanonicalEndpointLaw_mgf h b 2
have hn := fieldCanonicalEndpointLaw_mgf h b (-2)
have hi : Integrable (fun y : ℝ => y ^ 2) (fieldCanonicalEndpointLaw h b) := by
apply (hp.1.add hn.1).mono' (by fun_prop)
exact ae_of_all _ (fun y => by
rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
exact sq_le_exp_two_add y)
refine ⟨hi, ?_⟩
have hle := integral_mono hi (hp.1.add hn.1) sq_le_exp_two_add
simp only [Pi.add_apply] at hle
rw [integral_add hp.1 hn.1] at hle
norm_num at hp hn
have hn' : (∫ y, Real.exp (-2 * y) ∂fieldCanonicalEndpointLaw h b) ≤
Real.exp (-2 * b + 4 * h.height (Fin.last h.depth)) := by
simpa only [neg_mul] using hn.2
exact hle.trans (add_le_add hp.2 hn')
end InvariantIsing
end
end OAI
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