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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 | import OAI.Probability.InvariantIsing.Fields.FieldAffineRecursion
import OAI.Probability.InvariantIsing.Fields.FieldSpinLinear
namespace OAI
/-! Compensation of the covariance derivative by the spatial generator.
The identity telescopes adjacent covariance increments; it uses the actual
Gaussian transition and does not require exchanging mixed derivatives. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped NNReal
namespace InvariantIsing
lemma field_gaussianTiltAverage_const (s ζ : ℝ) {F : ℝ → ℝ}
(hF : Measurable F) (hg : HasLinearGrowth F) (c z : ℝ) :
gaussianTiltAverage s ζ F (fun _ => c) z = c := by
rw [field_gaussianTiltAverage_eq_transition s ζ hF measurable_const]
exact fieldSpinTransition_const ζ (Real.toNNReal s) hF hg c z
lemma field_gaussianTiltAverage_const_mul (s ζ : ℝ) {F a : ℝ → ℝ}
(hF : Measurable F) (ha : Measurable a) (c z : ℝ) :
gaussianTiltAverage s ζ F (fun y => c * a y) z =
c * gaussianTiltAverage s ζ F a z := by
rw [field_gaussianTiltAverage_eq_transition s ζ hF (ha.const_mul c),
field_gaussianTiltAverage_eq_transition s ζ hF ha]
exact fieldSpinTransition_const_mul ζ (Real.toNNReal s) F a c z
lemma field_gaussianTiltAverage_add (s ζ : ℝ) {F a b : ℝ → ℝ}
(hF : Measurable F) (hg : HasLinearGrowth F) (ha : Measurable a) (hb : Measurable b)
{A B : ℝ} (haB : ∀ y, |a y| ≤ A) (hbB : ∀ y, |b y| ≤ B) (z : ℝ) :
gaussianTiltAverage s ζ F (fun y => a y + b y) z =
gaussianTiltAverage s ζ F a z + gaussianTiltAverage s ζ F b z := by
have hm : Measurable (fun y => a y + b y) := ha.add hb
rw [field_gaussianTiltAverage_eq_transition s ζ hF hm,
field_gaussianTiltAverage_eq_transition s ζ hF ha,
field_gaussianTiltAverage_eq_transition s ζ hF hb]
exact fieldSpinTransition_add ζ (Real.toNNReal s) hF hg ha hb haB hbB z
namespace FieldSmoothFamily
theorem compensated_tangent {I : Set ℝ} (F : FieldSmoothFamily I)
(a d dprev ζ η : ℝ) {p : ℝ × ℝ} (hp : p.1 ∈ I) :
F.tangent a (d - dprev) ζ p + dprev / 2 *
(F.curvature a (d - dprev) ζ p + ζ * (F.mean a (d - dprev) ζ p) ^ 2) =
gaussianTiltAverage (a + (d - dprev) * p.1) ζ (fun y => F.U (p.1, y))
(fun y => F.T (p.1, y) + d / 2 * (F.XX (p.1, y) + η * (F.X (p.1, y)) ^ 2)) p.2 -
d / 2 * (η - ζ) *
gaussianTiltAverage (a + (d - dprev) * p.1) ζ (fun y => F.U (p.1, y))
(fun y => (F.X (p.1, y)) ^ 2) p.2 := by
let s := a + (d - dprev) * p.1
have hU : Measurable (fun y => F.U (p.1, y)) := F.mU.comp (by fun_prop)
have hT : Measurable (fun y => F.T (p.1, y)) := F.mT.comp (by fun_prop)
have hXX : Measurable (fun y => F.XX (p.1, y)) := F.mXX.comp (by fun_prop)
have hXsq : Measurable (fun y => (F.X (p.1, y)) ^ 2) :=
(F.mX.comp (by fun_prop)).pow_const 2
have hsq : ∀ y, |(F.X (p.1, y)) ^ 2| ≤ F.KX ^ 2 := fun y => by
rw [abs_pow]
exact pow_le_pow_left₀ (abs_nonneg _) (F.bX (p.1, y)) 2
have hm (r : ℝ) : Measurable (fun y => F.XX (p.1, y) + r * (F.X (p.1, y)) ^ 2) :=
hXX.add (hXsq.const_mul r)
have hb (r : ℝ) : ∀ y, |F.XX (p.1, y) + r * (F.X (p.1, y)) ^ 2| ≤
F.KXX + |r| * F.KX ^ 2 :=
bounded_generator (fun y => F.bX (p.1, y)) (fun y => F.bXX (p.1, y)) r
have hexpand (r : ℝ) :
gaussianTiltAverage s ζ (fun y => F.U (p.1, y))
(fun y => F.XX (p.1, y) + r * (F.X (p.1, y)) ^ 2) p.2 =
gaussianTiltAverage s ζ (fun y => F.U (p.1, y)) (fun y => F.XX (p.1, y)) p.2 +
r * gaussianTiltAverage s ζ (fun y => F.U (p.1, y))
(fun y => (F.X (p.1, y)) ^ 2) p.2 := by
rw [field_gaussianTiltAverage_add s ζ hU (F.growth p.1) hXX (hXsq.const_mul r)
(fun y => F.bXX (p.1, y)) (fun y => by
rw [abs_mul]
exact mul_le_mul_of_nonneg_left (hsq y) (abs_nonneg r)),
field_gaussianTiltAverage_const_mul s ζ hU hXsq]
have hfull :
gaussianTiltAverage s ζ (fun y => F.U (p.1, y))
(fun y => F.T (p.1, y) + d / 2 * (F.XX (p.1, y) + η * (F.X (p.1, y)) ^ 2)) p.2 =
gaussianTiltAverage s ζ (fun y => F.U (p.1, y)) (fun y => F.T (p.1, y)) p.2 +
d / 2 * gaussianTiltAverage s ζ (fun y => F.U (p.1, y))
(fun y => F.XX (p.1, y) + η * (F.X (p.1, y)) ^ 2) p.2 := by
rw [field_gaussianTiltAverage_add s ζ hU (F.growth p.1) hT ((hm η).const_mul (d / 2))
(fun y => F.bT (p.1, y) hp) (fun y => by
rw [abs_mul]
exact mul_le_mul_of_nonneg_left (hb η y) (abs_nonneg (d / 2))),
field_gaussianTiltAverage_const_mul s ζ hU (hm η)]
change F.tangent a (d - dprev) ζ p + dprev / 2 *
(F.curvature a (d - dprev) ζ p + ζ * (F.mean a (d - dprev) ζ p) ^ 2) = _
rw [hfull, hexpand η]
unfold tangent curvature
rw [show a + (d - dprev) * p.1 = s from rfl, hexpand ζ]
ring
end FieldSmoothFamily
end InvariantIsing
end
end OAI
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