| import OAI.Probability.InvariantIsing.Fields.FieldPublishedPairInput |
| import OAI.Probability.IsingPerceptron.TiltedLabelLaw |
|
|
| namespace OAI |
|
|
| /-! The actual scalar-field forest as a finite Ising energy-mark forest. -/ |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory IsingPerceptron |
| open scoped BigOperators NNReal |
|
|
| namespace InvariantIsing |
|
|
| lemma measurable_fieldEnergy_map (N : ℕ) : Measurable (fieldEnergy (N := N)) := by |
| unfold fieldEnergy |
| fun_prop |
|
|
| lemma fieldEnergy_map_norm_le (N : ℕ) (z : Fin N → ℝ) : |
| ‖fieldEnergy z‖ ≤ (N : ℝ) * ‖z‖ := by |
| apply (pi_norm_le_iff_of_nonneg (mul_nonneg (Nat.cast_nonneg _) (norm_nonneg z))).mpr |
| intro σ |
| change |∑ i, z i * spinValue (σ i)| ≤ _ |
| calc |
| _ ≤ ∑ i, |z i * spinValue (σ i)| := Finset.abs_sum_le_sum_abs _ _ |
| _ = ∑ i, |z i| := by simp only [abs_mul, abs_spinValue, mul_one] |
| _ ≤ ∑ _ : Fin N, ‖z‖ := Finset.sum_le_sum fun i _ => by |
| simpa only [Real.norm_eq_abs] using norm_le_pi_norm z i |
| _ = _ := by simp |
|
|
| def fieldEnergyMarkLaw (N : ℕ) (v : ℝ≥0) : ProbabilityMeasure (Spin N → ℝ) := |
| (vectorGaussianLaw N v).map (fieldEnergy (N := N)) |
|
|
| lemma fieldEnergyMarkLaw_moments (N : ℕ) (hN : 0 < N) (v : ℝ≥0) : |
| ExponentialNormMoments (fieldEnergyMarkLaw N v : Measure (Spin N → ℝ)) := by |
| intro a |
| apply (integrable_map_measure (by fun_prop) (measurable_fieldEnergy_map N).aemeasurable).mpr |
| have hi := vectorGaussianLaw_moments hN v (|a| * N) |
| apply hi.mono' (((measurable_fieldEnergy_map N).norm.const_mul a).exp.aestronglyMeasurable) |
| apply ae_of_all |
| intro z |
| rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)] |
| apply Real.exp_le_exp.mpr |
| calc |
| a * ‖fieldEnergy z‖ ≤ |a| * ‖fieldEnergy z‖ := |
| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _) |
| _ ≤ |a| * ((N : ℝ) * ‖z‖) := |
| mul_le_mul_of_nonneg_left (fieldEnergy_map_norm_le N z) (abs_nonneg _) |
| _ = _ := by ring |
|
|
| def fieldEnergyCoordinates (N : ℕ) (h : FieldStep) |
| (p : LabeledTree h.depth × (ForestVertex h.depth → Fin N → ℝ)) : |
| LabeledTree h.depth × (ForestVertex h.depth → Spin N → ℝ) := |
| (p.1, fun a => fieldEnergy (p.2 a)) |
|
|
| lemma measurable_fieldEnergyCoordinates (N : ℕ) (h : FieldStep) : |
| Measurable (fieldEnergyCoordinates N h) := by |
| exact measurable_fst.prodMk (Measurable.of_eval fun a => |
| (measurable_fieldEnergy_map N).comp ((measurable_pi_apply a).comp measurable_snd)) |
|
|
| theorem fieldEnergyCoordinates_law (N : ℕ) (h : FieldStep) : |
| MeasurePreserving (fieldEnergyCoordinates N h) (fieldVectorCoordinateLaw N h) |
| (cascadeCoordinateLaw h.depth (chainExponent h.cut) |
| (fun i => fieldEnergyMarkLaw N (fieldStepVariance h i))) := by |
| have hm : MeasurePreserving |
| (fun g : ForestVertex h.depth → Fin N → ℝ => fun a => fieldEnergy (g a)) |
| (Measure.infinitePi (fun a : ForestVertex h.depth => |
| (vectorGaussianLaw N (fieldStepVariance h (forestVertexDepth h.depth a)) : |
| Measure (Fin N → ℝ)))) |
| (Measure.infinitePi (fun a : ForestVertex h.depth => |
| (fieldEnergyMarkLaw N (fieldStepVariance h (forestVertexDepth h.depth a)) : |
| Measure (Spin N → ℝ)))) := by |
| refine ⟨Measurable.of_eval (fun a => |
| (measurable_fieldEnergy_map N).comp (measurable_pi_apply a)), ?_⟩ |
| rw [Measure.infinitePi_map_pi _ (fun _ => measurable_fieldEnergy_map N)] |
| rfl |
| exact (MeasurePreserving.id (labeledCascadeLaw h.depth (chainExponent h.cut) : |
| Measure (LabeledTree h.depth))).prod hm |
|
|
| lemma fieldEnergyCoordinates_energy (N : ℕ) (h : FieldStep) |
| (p : LabeledTree h.depth × (ForestVertex h.depth → Fin N → ℝ)) |
| (z : Fin N → ℝ) (s : Spin N × LabeledLeaf h.depth) : |
| cascadeCoordinateEnergy h.depth (fieldEnergy z) (fieldEnergyCoordinates N h p) s = |
| fieldEnergy (fieldVectorEndpoint N h p z s.2) s.1 := by |
| change fieldEnergy z s.1 + ∑ i : Fin h.depth, fieldEnergy (p.2 (edgeAt h.depth s.2 i)) s.1 = _ |
| have he : fieldVectorEndpoint N h p z s.2 = |
| z + ∑ i : Fin h.depth, p.2 (edgeAt h.depth s.2 i) := by |
| ext j |
| simpa only [fieldVectorEndpoint, Pi.add_apply, Finset.sum_apply] using |
| labeledEnergy_edge_sum h.depth p.2 s.2 z j |
| rw [he] |
| simp only [fieldEnergy, Pi.add_apply, Finset.sum_apply, add_mul, Finset.sum_add_distrib, |
| Finset.sum_mul] |
| rw [Finset.sum_comm] |
|
|
| theorem field_vector_spin_exp_integrable (N : ℕ) (hN : 0 < N) (h : FieldStep) |
| (z : Fin N → ℝ) : |
| ∀ᵐ p ∂fieldVectorCoordinateLaw N h, |
| Integrable (fun s : Spin N × LabeledLeaf h.depth => |
| Real.exp (fieldEnergy (fieldVectorEndpoint N h p z s.2) s.1)) |
| ((uniformSpinPrior N : Measure (Spin N)).prod (labeledLeafLaw h.depth p.1)) := by |
| have he := (fieldEnergyCoordinates_law N h).quasiMeasurePreserving.ae |
| (cascadeCoordinate_exp_integrable h.depth (chainExponent h.cut) |
| (chainExponent_admissible h.ordered_cut h.first h.last) |
| (fun i => fieldEnergyMarkLaw N (fieldStepVariance h i)) |
| (uniformSpinPrior N : Measure (Spin N)) |
| (fun i _ => fieldEnergyMarkLaw_moments N hN (fieldStepVariance h i)) (fieldEnergy z)) |
| filter_upwards [he] with p hp |
| change Integrable (fun s => Real.exp (cascadeCoordinateEnergy h.depth (fieldEnergy z) |
| (fieldEnergyCoordinates N h p) s)) |
| ((uniformSpinPrior N : Measure (Spin N)).prod (labeledLeafLaw h.depth p.1)) at hp |
| simpa only [fieldEnergyCoordinates_energy] using hp |
|
|
| end InvariantIsing |
|
|
| end |
|
|
| end OAI |
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|
