File size: 3,152 Bytes
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 | import OAI.Probability.InvariantIsing.Fields.CascadeMarking
namespace OAI
/-! The actual finite one-site functional as a Gaussian cascade expectation. -/
noncomputable section
open MeasureTheory ProbabilityTheory
open scoped BigOperators NNReal
namespace InvariantIsing
def fieldCascadeVariance (h : FieldStep) (i : ℕ) : ℝ≥0 :=
NNReal.mk ((IsingPerceptron.pathAmplitude (heightSequence h) (i + 1)) ^ 2)
(sq_nonneg _)
def fieldCascadeLaw (h : FieldStep) : Measure (IsingPerceptron.NoiseTree ℝ h.depth) :=
gaussianCascadeLaw h.depth (IsingPerceptron.chainExponent h.cut) (fieldCascadeVariance h)
lemma fieldCascadeFold_eq_magneticRecursion (h : FieldStep) :
(List.ofFn (fun i : Fin h.depth =>
(IsingPerceptron.chainExponent h.cut i, (fieldCascadeVariance h i : ℝ)))).foldr
(fun av U => gaussianOperator av.1 av.2 U) (fun x => Real.log (Real.cosh x)) =
IsingPerceptron.magneticRecursion h.depth
(fun i => IsingPerceptron.pathAmplitude (heightSequence h) (i + 1))
(IsingPerceptron.chainExponent h.cut) := by
simp_rw [gaussianOperator_eq_transform]
rw [← List.foldr_map (f := fun av : ℝ × ℝ => (av.2, av.1))
(g := fun sd U => IsingPerceptron.gaussianTransform sd.1 sd.2 U), List.map_ofFn]
simp only [Function.comp_def, fieldCascadeVariance, NNReal.coe_mk]
exact (IsingPerceptron.magneticRecursion_eq_fold h.depth
(fun i => IsingPerceptron.pathAmplitude (heightSequence h) (i + 1))
(IsingPerceptron.chainExponent h.cut)).symm
theorem fieldCascadeLog_recursion (h : FieldStep) (x : ℝ) :
Integrable (gaussianCascadeLog h.depth x) (fieldCascadeLaw h) ∧
(∫ T, gaussianCascadeLog h.depth x T ∂fieldCascadeLaw h) =
IsingPerceptron.magneticRecursion h.depth
(fun i => IsingPerceptron.pathAmplitude (heightSequence h) (i + 1))
(IsingPerceptron.chainExponent h.cut) x := by
have hr := gaussianCascadeLog_recursion h.depth (IsingPerceptron.chainExponent h.cut)
(fieldCascadeVariance h)
(IsingPerceptron.chainExponent_admissible h.ordered_cut h.first h.last) x
rw [fieldCascadeFold_eq_magneticRecursion] at hr
exact hr
/-- The finite one-site entropy value equals the root Gaussian average of
the logarithm of the Gaussian-marked cascade integral, with its prescribed
diagonal covariance subtraction. -/
theorem fieldValue_eq_cascade_expectation (h : FieldStep) :
fieldValue h 0 =
(∫ g, (∫ T, gaussianCascadeLog h.depth (Real.sqrt (h.height 0) * g) T
∂fieldCascadeLaw h) ∂gaussianReal 0 1) - h.height (Fin.last h.depth) / 2 := by
rw [fieldValue_eq_magneticFieldValue]
unfold IsingPerceptron.magneticFieldValue
have hlast : heightSequence h h.depth = h.height (Fin.last h.depth) :=
heightSequence_eq h _ le_rfl
rw [hlast]
congr 1
apply integral_congr_ae
filter_upwards [] with g
rw [(fieldCascadeLog_recursion h (Real.sqrt (h.height 0) * g)).2]
have hroot : IsingPerceptron.pathAmplitude (heightSequence h) 0 = Real.sqrt (h.height 0) := by
simp [IsingPerceptron.pathAmplitude, IsingPerceptron.pathIncrement, heightSequence]
rw [hroot]
end InvariantIsing
end
end OAI
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