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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 | import OAI.Probability.InvariantIsing.Fields.FieldSmoothFamily
namespace OAI
/-! The actual finite scalar log-cosh recursion with affine covariance
increments is a smooth field family on its strict covariance interval. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal
namespace InvariantIsing
def fieldLogCoshFamily (I : Set ℝ) : FieldSmoothFamily I where
U := fun p => Real.log (Real.cosh p.2)
X := fun p => Real.tanh p.2
XX := fun p => 1 / (Real.cosh p.2) ^ 2
T := fun _ => 0
KX := 1
KXX := 1
KT := 0
mU := by fun_prop
mX := by
have ht : Measurable (fun x : ℝ => Real.tanh x) := by
simp only [Real.tanh_eq]
fun_prop
exact ht.comp measurable_snd
mXX := by fun_prop
mT := measurable_const
bX := fun p => field_abs_tanh_le_one p.2
bXX := fun p => field_tanh_second_bound p.2
bT := fun _ _ => by simp
derivative := fun p _ => by
have h : HasDerivAt (fun x => Real.log (Real.cosh x)) (Real.tanh p.2) p.2 := by
simpa only [Real.tanh_eq_sinh_div_cosh] using
(Real.hasDerivAt_cosh p.2).log (Real.cosh_pos p.2).ne'
have hs : HasFDerivAt (fun q : ℝ × ℝ => q.2) (ContinuousLinearMap.snd ℝ ℝ ℝ) p :=
hasFDerivAt_snd
convert h.hasFDerivAt.comp p hs using 1
· rfl
· apply ContinuousLinearMap.ext
intro q
simp [pairLinear, mul_comm]
spatial := fun _ y => by
simpa only [Real.tanh_eq_sinh_div_cosh] using
(Real.hasDerivAt_cosh y).log (Real.cosh_pos y).ne'
second := fun _ y => field_hasDerivAt_tanh y
structure FieldAffineStep where
base : ℝ
slope : ℝ
exponent : ℝ
def fieldAffineValue : List FieldAffineStep → ℝ × ℝ → ℝ
| [], p => Real.log (Real.cosh p.2)
| av :: L, p => gaussianTransform (av.base + av.slope * p.1) av.exponent
(fun y => fieldAffineValue L (p.1, y)) p.2
def fieldAffineFamily (I : Set ℝ) (hI : IsOpen I) :
(L : List FieldAffineStep) →
(∀ av ∈ L, ∀ t ∈ I, 0 < av.base + av.slope * t) → FieldSmoothFamily I
| [], _ => fieldLogCoshFamily I
| av :: L, hL =>
(fieldAffineFamily I hI L (fun bv hb => hL bv (List.mem_cons_of_mem av hb))).transform
hI av.base av.slope av.exponent (hL av List.mem_cons_self)
theorem fieldAffineFamily_value (I : Set ℝ) (hI : IsOpen I) (L : List FieldAffineStep)
(hL : ∀ av ∈ L, ∀ t ∈ I, 0 < av.base + av.slope * t) :
(fieldAffineFamily I hI L hL).U = fieldAffineValue L := by
induction L with
| nil => rfl
| cons av L ih =>
funext p
simp only [fieldAffineFamily, FieldSmoothFamily.transform, FieldSmoothFamily.value,
fieldAffineValue]
rw [ih]
theorem fieldAffineValue_hasFDerivAt (I : Set ℝ) (hI : IsOpen I)
(L : List FieldAffineStep)
(hL : ∀ av ∈ L, ∀ t ∈ I, 0 < av.base + av.slope * t)
(p : ℝ × ℝ) (hp : p.1 ∈ I) :
HasFDerivAt (fieldAffineValue L)
(pairLinear ((fieldAffineFamily I hI L hL).T p)
((fieldAffineFamily I hI L hL).X p)) p := by
rw [← fieldAffineFamily_value I hI L hL]
exact (fieldAffineFamily I hI L hL).derivative p hp
end InvariantIsing
end
end OAI
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