| 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 |
| |
