| import Mathlib |
| import OAI.Probability.Perceptron.Variational.CouplingKernel |
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|
| namespace OAI |
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|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
| namespace SphericalPerceptronFreeEnergy |
|
|
| abbrev sourceFullSpinLeafKernel (n k : ℕ) : Kernel (SourceBaseData n k × (ℕ → ℝ)) |
| (NormalizedSpin (n+1) × IndexedLeaf k) := |
| (sourceSpinLeafKernel n k).comap Prod.fst measurable_fst |
|
|
| def sourceCouplingMean (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ) |
| (h : Fin (k+1) → ℝ) (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (r : ℝ) |
| (a : SourceBaseData n k × (ℕ → ℝ)) : ℝ := |
| (1/(n+1:ℕ))*tiltMean (sourceSpinLeafKernel n k a.1) |
| (fun x => sourceCouplingHamiltonian n k f p d h u a x+r*sourceCouplingEnergy n k p d h j a x) |
| (sourceCouplingEnergy n k p d h j a) 1 |
|
|
| def sourceCouplingThermal (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ) |
| (h : Fin (k+1) → ℝ) (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (r : ℝ) |
| (a : SourceBaseData n k × (ℕ → ℝ)) : ℝ := |
| let H := fun x => sourceCouplingHamiltonian n k f p d h u a x+r*sourceCouplingEnergy n k p d h j a x |
| let Y := sourceCouplingEnergy n k p d h j a |
| Real.sqrt (1/(n+1:ℕ))*tiltMean (sourceSpinLeafKernel n k a.1) H |
| (fun x => |Y x-tiltMean (sourceSpinLeafKernel n k a.1) H Y 1|) 1 |
|
|
| def sourceCouplingVariance (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ) |
| (h : Fin (k+1) → ℝ) (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (r : ℝ) |
| (a : SourceBaseData n k × (ℕ → ℝ)) : ℝ := |
| let H := fun x => sourceCouplingHamiltonian n k f p d h u a x+r*sourceCouplingEnergy n k p d h j a x |
| let Y := sourceCouplingEnergy n k p d h j a |
| (1/(n+1:ℕ))*tiltMean (sourceSpinLeafKernel n k a.1) H |
| (fun x => (Y x-tiltMean (sourceSpinLeafKernel n k a.1) H Y 1)^2) 1 |
|
|
| lemma sourceCoupling_observables_measurable (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ) |
| (h : Fin (k+1) → ℝ) (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (r : ℝ) : |
| Measurable (sourceCouplingMean n k f p d h u j r) ∧ |
| Measurable (sourceCouplingThermal n k f p d h u j r) ∧ |
| Measurable (sourceCouplingVariance n k f p d h u j r) := by |
| have hY := sourceCouplingEnergy_measurable n k p d h j |
| have hH : Measurable (Function.uncurry (fun a x => |
| sourceCouplingHamiltonian n k f p d h u a x+r*sourceCouplingEnergy n k p d h j a x)) := |
| (sourceCouplingHamiltonian_measurable n k f p d h u).add (hY.const_mul r) |
| exact ⟨(kernel_tiltMean_measurable (sourceFullSpinLeafKernel n k) hH hY).const_mul _, |
| (kernel_tiltCenteredAbs_measurable (sourceFullSpinLeafKernel n k) hH hY).const_mul _, |
| (kernel_tiltCenteredVariance_measurable (sourceFullSpinLeafKernel n k) hH hY).const_mul _⟩ |
|
|
| lemma sourceCoupling_thermal_bound_ae (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ) |
| (h : Fin (k+1) → ℝ) (hh0 : ∀ l, 0 ≤ h l) (hh : Monotone h) |
| (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (z : Fin k → ℝ) (t : ℝ≥0) (r : ℝ) : |
| ∀ᵐ a ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw, |
| 0 ≤ sourceCouplingThermal n k f p d h u j r a ∧ |
| (sourceCouplingThermal n k f p d h u j r a)^2 ≤ sourceCouplingVariance n k f p d h u j r a := by |
| filter_upwards [sourceCoupling_all_exp_ae n k f p d h hh0 hh u j z t] with a ha |
| have hH : Measurable (sourceCouplingHamiltonian n k f p d h u a) := |
| sourceCouplingHamiltonian_section n k f p d h u a |
| have hY : Measurable (sourceCouplingEnergy n k p d h j a) := |
| sourceCouplingEnergy_section n k p d h j a |
| constructor |
| · exact mul_nonneg (Real.sqrt_nonneg _) (tiltMean_nonneg _ (fun _ => abs_nonneg _) 1) |
| · unfold sourceCouplingThermal sourceCouplingVariance |
| rw [mul_pow,Real.sq_sqrt (by positivity)] |
| exact mul_le_mul_of_nonneg_left (coupling_thermal_abs_sq_le _ hH hY ha r) (by positivity) |
|
|
| lemma source_contact_thermal_disorder (n k : ℕ) (f : ℝ →ᵇ ℝ) (p d : Fin (n+1) → ℕ) |
| (h : Fin (k+1) → ℝ) (hh0 : ∀ l, 0 ≤ h l) (hh : Monotone h) |
| (u : Fin (n+1) → ℝ) (j : Fin (n+1)) (z : Fin k → ℝ) (hz : StrictMono z) |
| (hz0 : ∀ i, 0<z i) (hz1 : ∀ i, z i<1) (t : ℝ≥0) |
| {H T c a s : ℝ} (hH : 0 ≤ H) (hhH : h 0 ≤ H) (ht : (t:ℝ) ≤ T) (hs : 0<s) |
| (hu : ∀ l, u l ∈ Icc 1 2) |
| (hup : ∀ l, (u+Pi.single j s : Fin (n+1) → ℝ) l ∈ Icc 1 2) (hum : ∀ l, (u+Pi.single j (-s) : Fin (n+1) → ℝ) l ∈ Icc 1 2) |
| (hmin : IsLocalMin (fun r => c*(r-a)^2- |
| ∫ b, sourceKernelPressure n k f p d (u+Pi.single j r) h b |
| ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw) 0) |
| (hp : c*(0-a)^2-(∫ b, sourceKernelPressure n k f p d u h b |
| ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw) ≤ c*(s-a)^2- |
| ∫ b, sourceKernelPressure n k f p d (u+Pi.single j s) h b |
| ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw) |
| (hm : c*(0-a)^2-(∫ b, sourceKernelPressure n k f p d u h b |
| ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw) ≤ c*(-s-a)^2- |
| ∫ b, sourceKernelPressure n k f p d (u+Pi.single j (-s)) h b |
| ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw) : |
| let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw |
| let D := sourceCouplingMean n k f p d h u j 0 |
| let U := sourceCouplingThermal n k f p d h u j 0 |
| MemLp U 2 P ∧ Integrable D P ∧ (∫ b, U b ∂P) ≤ Real.sqrt (2*c) ∧ |
| (∫ b, |D b-∫ b', D b' ∂P| ∂P) ≤ |
| c*s+4*Real.sqrt (enrichedVarianceConstant k z f H T/(n+1:ℕ))/s := by |
| let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw |
| have hreg := sourceCoupling_regular_ae n k f p d h hh0 hh u j z t |
| have htherm := sourceCoupling_thermal_bound_ae n k f p d h hh0 hh u j z t 0 |
| have hmeas := sourceCoupling_observables_measurable n k f p d h u j |
| apply quadratic_contact_thermal_disorder_separate P |
| (F := fun r => sourceKernelPressure n k f p d (u+Pi.single j r) h) |
| (D := sourceCouplingMean n k f p d h u j) |
| (u := 0) (a := a) (c := c) (s := s) |
| (V := enrichedVarianceConstant k z f H T/(n+1:ℕ)) |
| (J := sourceCouplingVariance n k f p d h u j 0) (U := sourceCouplingThermal n k f p d h u j 0) |
| · intro r |
| have hr := sourceKernelPressure_memLp n k f p d (u+Pi.single j r) h z hz hz0 hz1 t |
| exact hr |
| · exact fun r => (hmeas r).1.aestronglyMeasurable |
| · exact (hmeas 0).2.2.aestronglyMeasurable |
| · exact (hmeas 0).2.1.aestronglyMeasurable |
| · exact htherm.mono fun _ h => h.1 |
| · exact htherm.mono fun _ h => h.2 |
| · exact hreg.mono fun _ h => ⟨h.1,h.2.1,h.2.2.1⟩ |
| · exact hs |
| · exact hreg.mono fun _ h => h.2.2.2 0 |
| · exact hmin |
| · simpa only [zero_add,Pi.single_zero,add_zero] using hp |
| · simpa only [zero_sub,Pi.single_zero,add_zero] using hm |
| · have hv := sourceKernelPressure_variance n k f p d (u+Pi.single j s) h z hz hz0 hz1 hup hH hhH t ht |
| simpa only [zero_add] using hv |
| · have hv := sourceKernelPressure_variance n k f p d u h z hz hz0 hz1 hu hH hhH t ht |
| simpa only [Pi.single_zero,add_zero] using hv |
| · have hv := sourceKernelPressure_variance n k f p d (u+Pi.single j (-s)) h z hz hz0 hz1 hum hH hhH t ht |
| simpa only [zero_sub] using hv |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
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| end OAI |
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