| import OAI.Probability.DilutedSpin.ClippedCavityProxy |
| import OAI.Probability.DilutedSpin.PhysicalCavityPoisson |
| import OAI.Probability.DilutedSpin.PhysicalCavityRate |
|
|
| namespace OAI |
|
|
| section |
| namespace DilutedSpinGlass.UniversalDictionary |
| open _root_.MeasureTheory _root_.OAI.MeasureTheory ProbabilityTheory HeterogeneousMarks PhysicalRoot PrescribedTree ConcreteReservoir KernelTower SizeCoupling |
| open scoped NNReal BigOperators |
| variable {N L q : ℕ} [NeZero N] |
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| lemma clippedSiteInsertion_rate_bound (M : Model (q+1)) {C H : ℝ} (hC : 0≤C) (hH : 0≤H) |
| (u : Spec L×ℕ → ℝ) (r t : ℝ≥0) : |
| |clippedSiteInsertion M C H N L u r-clippedSiteInsertion M C H N L u t|≤ |
| C*|(r:ℝ)-t|+4*(scoreRate N:ℝ)/(N+1) := by |
| have hθm σ : Measurable (fun z : InteractionSample (q+1) => (clipSample C z).1 σ) := |
| (measurable_clipReal C).comp ((measurable_pi_apply σ).comp measurable_fst) |
| have hθ z : ‖(clipSample (p := q+1) C z).1‖≤C := |
| (pi_norm_le_iff_of_nonneg hC).mpr (fun σ => by simpa only [Real.norm_eq_abs] using clipSample_bound hC z σ) |
| have hρ := reservoirEnergyLaw_integrable (N := N) M hC |
| have hF := reservoirEnergyRoot_lipschitz (N := N) M hH u |
| have hFi := integrable_lipschitz_of_id (reservoirEnergyLaw M C N) hρ hF |
| have hFb (E : (Fin N → Spin) → ℝ) : |reservoirEnergyRoot M H N L u E|≤‖E‖+(H*N+1*scoreRate N) := by |
| simpa only [reservoirEnergyRoot,add_assoc] using averagedEnergyRoot_bound M.field.toMeasure |
| ((weights L).prod (finiteUniform (Fin N))) (scoreRate N) |
| (fun i : (Spec L×ℕ)×Fin N => prior i.1.1) (gridExponents L) (gridExponents_pos L) |
| (clipReal H) (locatedFactor (spinFactor direction anchor u)) (fun y => clipReal_bound hH y) |
| (fun i y a => spinFactor_log_bound direction anchor direction_bound anchor_bound u i.1 _ a) E |
| have hCi a := integrable_cavityPoissonValue M.disorder.toMeasure M.field.toMeasure |
| (reservoirEnergyLaw M C N) hρ (clipSample C) (clipReal H) hθm (measurable_clipReal H) |
| hC hH hθ (fun y => clipReal_bound hH y) hF.continuous hFb a |
| have hh := averagedEnergyRoot_cavity_rate_bound M.disorder.toMeasure M.field.toMeasure (weights L) |
| (reservoirEnergyLaw M C N) hρ r t (scoreRate N) (clipSample C) (clipReal H) hθm (measurable_clipReal H) |
| (fun i : Spec L×ℕ => prior i.1) (gridExponents L) (gridExponents_pos L) (gridExponents_last L) |
| (spinFactor direction anchor u) hH hC hθ (fun y => clipReal_bound hH y) |
| (fun i σ a => spinFactor_log_bound direction anchor direction_bound anchor_bound u i σ a) |
| unfold clippedSiteInsertion |
| rw [insertionPoisson_site_energy M hC hH,insertionPoisson_site_energy M hC hH, |
| integral_sub (hCi r) hFi,integral_sub (hCi t) hFi,sub_sub_sub_cancel_right] |
| simpa only [mul_one,reservoirEnergyRoot] using hh |
|
|
| lemma clippedBondInsertion_rate_bound {p : ℕ} (M : Model p) {C H : ℝ} (hC : 0≤C) (hH : 0≤H) |
| (u : Spec L×ℕ → ℝ) (r t : ℝ≥0) : |
| |clippedBondInsertion M C H N L u r-clippedBondInsertion M C H N L u t|≤C*|(r:ℝ)-t| := by |
| let μ := Measure.map (indexedPotential (N := N) (clipSample (p := p) C)) |
| (M.disorder.toMeasure.prod (finiteUniform (Fin p → Fin N))) |
| have hm := measurable_indexedPotential (N := N) (clipSample (p := p) C) |
| (fun σ => (measurable_clipReal C).comp ((measurable_pi_apply σ).comp measurable_fst)) |
| have hb (z : InteractionSample p×(Fin p → Fin N)) : ‖indexedPotential (clipSample C) z‖≤C := |
| (pi_norm_le_iff_of_nonneg hC).mpr (fun σ => by simpa only [Real.norm_eq_abs,indexedPotential,clipSample] using clipReal_bound hC (z.1.1 (fun i => σ (z.2 i)))) |
| have hi : Integrable id μ := |
| (integrable_map_measure aestronglyMeasurable_id hm.aemeasurable).mpr |
| (Integrable.of_bound hm.aestronglyMeasurable C (ae_of_all _ hb)) |
| have hnorm : (∫ v,‖v‖ ∂μ)≤C := by |
| rw [show (∫ v,‖v‖ ∂μ)=∫ z,‖indexedPotential (clipSample C) z‖ |
| ∂(M.disorder.toMeasure.prod (finiteUniform (Fin p → Fin N))) from |
| integral_map hm.aemeasurable continuous_norm.aestronglyMeasurable] |
| exact (le_abs_self _).trans (abs_integral_le_bound (fun z => by simpa only [abs_of_nonneg (norm_nonneg _)] using hb z)) |
| have hF := reservoirEnergyRoot_lipschitz (N := N) M hH u |
| unfold clippedBondInsertion |
| rw [insertionPoisson_bond_energy M hC hH,insertionPoisson_bond_energy M hC hH, |
| ← integral_energyInsertionFunctional (reservoirEnergyLaw M C N) (compoundPoisson r μ) (compoundPoisson_integrable_id r μ hi) hF, |
| ← integral_energyInsertionFunctional (reservoirEnergyLaw M C N) (compoundPoisson t μ) (compoundPoisson_integrable_id t μ hi) hF] |
| exact (compoundPoisson_rate_bound r t μ hi (energyInsertionFunctional_lipschitz (reservoirEnergyLaw M C N) hF)).trans |
| (by simpa only [NNReal.coe_one,one_mul,mul_one,mul_comm] using mul_le_mul_of_nonneg_left hnorm (abs_nonneg ((r:ℝ)-t))) |
|
|
| end DilutedSpinGlass.UniversalDictionary |
|
|
| end |
|
|
| end OAI |
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|
