File size: 5,442 Bytes
a89d282 | 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 | import OAI.Probability.InvariantIsing.Magnetic.RestrictedFieldRootLoss
import OAI.Probability.InvariantIsing.Magnetic.MagneticBlockTerminalLoss
namespace OAI
/-! Exact bias cancellation on a constrained block and identification of
the unrestricted block with the average scalar field functional. -/
noncomputable section
open MeasureTheory ProbabilityTheory IsingPerceptron
open scoped BigOperators NNReal
namespace InvariantIsing
def biasedConstrainedBlockValue {N : ℕ} (S : Finset (Spin N)) (h : FieldStep)
(b : Fin N → ℝ) : ℝ :=
(N : ℝ)⁻¹ * (∫ z, restrictedFieldRecursion S h.depth (chainExponent h.cut)
(fieldStepVariance h) z
∂Measure.pi (fun i => gaussianReal (b i) (NNReal.mk (h.height 0) (h.nonneg 0)))) -
h.height (Fin.last h.depth) / 2
lemma fieldValue_gaussian_integral (h : FieldStep) (b : ℝ) :
(∫ z, fieldScalarValue (scalarFieldIncrements h) (fun y => Real.log (Real.cosh y)) z
∂gaussianReal b (NNReal.mk (h.height 0) (h.nonneg 0))) =
fieldValue h b + h.height (Fin.last h.depth) / 2 := by
have he := gaussianOperator_eq_gaussian_integral (NNReal.mk (h.height 0) (h.nonneg 0))
(fieldScalarValue_regular _ (scalarFieldIncrements_positive h)
measurable_logCosh logCosh_linearGrowth).1 b
change gaussianOperator 0 (h.height 0)
(fieldScalarValue (scalarFieldIncrements h) (fun y => Real.log (Real.cosh y))) b = _ at he
rw [fieldValue_root]
linarith [he]
theorem biasedConstrainedBlockValue_univ {N : ℕ} (hN : 0 < N) (h : FieldStep) (b : Fin N → ℝ) :
biasedConstrainedBlockValue Finset.univ h b = (N : ℝ)⁻¹ * ∑ i, fieldValue h (b i) := by
have hb := chainExponent_admissible h.ordered_cut h.first h.last
have hrec : restrictedFieldRecursion (Finset.univ : Finset (Spin N)) h.depth
(chainExponent h.cut) (fieldStepVariance h) = fun z => ∑ i,
fieldScalarValue (scalarFieldIncrements h) (fun y => Real.log (Real.cosh y)) (z i) := by
funext z
simpa only [fieldStepIncrements] using restrictedFieldRecursion_univ h.depth
(chainExponent h.cut) (fieldStepVariance h) (fun i hi => (hb.1 i hi).1) z
have hreg := fieldScalarValue_regular _ (scalarFieldIncrements_positive h)
measurable_logCosh logCosh_linearGrowth
have hi (i : Fin N) : Integrable (fun z : Fin N → ℝ =>
fieldScalarValue (scalarFieldIncrements h) (fun y => Real.log (Real.cosh y)) (z i))
(Measure.pi (fun j => gaussianReal (b j) (NNReal.mk (h.height 0) (h.nonneg 0)))) :=
(measurePreserving_eval _ i).integrable_comp_of_integrable
(field_gaussian_linear_integrable _ (b i) hreg.1 hreg.2)
rw [biasedConstrainedBlockValue, hrec, integral_finsetSum _ (fun i _ => hi i)]
have he (i : Fin N) : (∫ z : Fin N → ℝ,
fieldScalarValue (scalarFieldIncrements h) (fun y => Real.log (Real.cosh y)) (z i)
∂Measure.pi (fun j => gaussianReal (b j) (NNReal.mk (h.height 0) (h.nonneg 0)))) =
fieldValue h (b i) + h.height (Fin.last h.depth) / 2 := by
rw [integral_comp_eval (μ := fun j : Fin N =>
gaussianReal (b j) (NNReal.mk (h.height 0) (h.nonneg 0))) (i := i) hreg.1.aestronglyMeasurable]
exact fieldValue_gaussian_integral h (b i)
simp_rw [he]
rw [Finset.sum_add_distrib]
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul]
have hNr : (N : ℝ) ≠ 0 := by exact_mod_cast hN.ne'
field_simp
ring
theorem biasedConstrainedBlockValue_group {N : ℕ} (hN : 0 < N)
{A : Type*} [Fintype A] [DecidableEq A] (group : Fin N → A) (k : A → ℕ)
(hk : ∀ a, k a ≤ spinGroupSize group a) (bias : A → ℝ) (h : FieldStep) :
biasedConstrainedBlockValue (spinGroupSlice group k) h (fun i => bias (group i)) =
constrainedBlockValue (spinGroupSlice group k) h +
(N : ℝ)⁻¹ * spinGroupFieldConstant group k bias := by
let S := spinGroupSlice group k
let F := restrictedFieldRecursion S h.depth (chainExponent h.cut) (fieldStepVariance h)
have hb := chainExponent_admissible h.ordered_cut h.first h.last
have hreg := restrictedFieldRecursion_regular hN S (spinGroupSlice_nonempty group k hk)
h.depth (chainExponent h.cut) (fieldStepVariance h) (fun i hi => (hb.1 i hi).1)
have he : (∫ z, F z ∂Measure.pi (fun i => gaussianReal (bias (group i))
(NNReal.mk (h.height 0) (h.nonneg 0)))) =
∫ z, F z + spinGroupFieldConstant group k bias
∂(vectorGaussianLaw N (NNReal.mk (h.height 0) (h.nonneg 0)) : Measure (Fin N → ℝ)) := by
rw [← (vectorGaussian_shift _ (fun i => bias (group i))).map_eq,
integral_map (show AEMeasurable (fun z : Fin N → ℝ => (fun i => bias (group i)) + z)
(vectorGaussianLaw N (NNReal.mk (h.height 0) (h.nonneg 0)) : Measure (Fin N → ℝ)) from
(measurable_const.add measurable_id).aemeasurable) hreg.1.aestronglyMeasurable]
apply integral_congr_ae
apply ae_of_all
intro z
change F ((fun i => bias (group i)) + z) = _
rw [add_comm]
exact restrictedFieldRecursion_group_bias hN group k hk bias _ _ _
(fun i hi => (hb.1 i hi).1) z
change (N : ℝ)⁻¹ * (∫ z, F z ∂_) - _ = (N : ℝ)⁻¹ * (∫ z, F z ∂_) - _ + _
rw [he, integral_add (constrainedBlockValue_root_integrable hN S
(spinGroupSlice_nonempty group k hk) h) (integrable_const _)]
simp only [integral_const, measureReal_def, measure_univ, ENNReal.toReal_one, one_smul]
ring
end InvariantIsing
end
end OAI
|