| import Mathlib |
| import OAI.Probability.Perceptron.Pressure.KernelEnergyConcentration |
| import OAI.Probability.Perceptron.Variational.AnnealedIBP |
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| namespace OAI |
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| noncomputable section |
| namespace SphericalPerceptronFreeEnergy |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators BoundedContinuousFunction |
|
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| lemma gibbsReplicaMean_const_of_integrable {S : Type*} [MeasurableSpace S] |
| (μ : Measure S) [IsProbabilityMeasure μ] {H : S → ℝ} (hH : Measurable H) |
| (he : Integrable (fun x => Real.exp (H x)) μ) (r : ℕ) (c : ℝ) : |
| gibbsReplicaMean μ H r (fun _ => c) = c := by |
| let := tilt_law_probability_of_integrable μ (by simpa only [one_mul] using he : |
| Integrable (fun x => Real.exp (1*H x)) μ) |
| rw [gibbsReplicaMean_integral_of_integrable μ hH he] |
| simp |
|
|
| lemma sourceCoupling_exp_integrable_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) : |
| ∀ᵐ a ∂(sourceBaseDataLaw n k z t).prod countableGaussianLaw, |
| Integrable (fun x => Real.exp (sourceCouplingHamiltonian n k f p d h u a x)) (sourceSpinLeafKernel n k a.1) ∧ |
| Integrable (fun x => Real.exp (sourceCouplingHamiltonian n k f p d h u a x)* |
| sourceCouplingEnergy n k p d h j a x) (sourceSpinLeafKernel n k a.1) := by |
| have hH := sourceCouplingHamiltonian_measurable n k f p d h u |
| have hY := sourceCouplingEnergy_measurable n k p d h j |
| filter_upwards [sourceCoupling_all_exp_ae n k f p d h hh0 hh u j z t] with a ha |
| constructor |
| · simpa only [zero_mul,add_zero] using ha 0 |
| · simpa only [zero_mul,add_zero,pow_one] using coupling_weighted_power_integrable |
| (sourceSpinLeafKernel n k a.1) hH.of_uncurry_left hY.of_uncurry_left ha 0 1 |
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| lemma sourceCoupling_energy_mean (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) |
| (hYi : Integrable (fun a => tiltMean (sourceSpinLeafKernel n k a.1) |
| (sourceCouplingHamiltonian n k f p d h u a) |
| (fun x => |sourceCouplingEnergy n k p d h j a x|) 1) |
| ((sourceBaseDataLaw n k z t).prod countableGaussianLaw)) : |
| let H := sourceCouplingHamiltonian n k f p d h u |
| let Y := sourceCouplingEnergy n k p d h j |
| let C := sourceCouplingCovariance n k p d u j |
| let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw |
| (∫ a, tiltMean (sourceSpinLeafKernel n k a.1) (H a) (Y a) 1 ∂P) = |
| perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j* |
| ((k:ℝ)/(k+1:ℕ))^(d j)- |
| ∫ a, gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) 2 (fun x => C (x 1) (x 0)) ∂P := by |
| dsimp only |
| let P := (sourceBaseDataLaw n k z t).prod countableGaussianLaw |
| let H := sourceCouplingHamiltonian n k f p d h u |
| let C := sourceCouplingCovariance n k p d u j |
| let D := perturbationAmplitude (n+1) (fun _ => 1) j*perturbationAmplitude (n+1) u j*((k:ℝ)/(k+1:ℕ))^(d j) |
| have he := sourceCoupling_annealed_ibp n k 1 f p d h hh0 hh u j 0 z t |
| (G := fun _ => 1) measurable_const (by norm_num : (0:ℝ) ≤ 1) (fun _ => by norm_num) hYi |
| dsimp only at he |
| simp only [mul_one,one_mul,Fin.sum_univ_one,Nat.cast_one,Fin.succ_zero_eq_one] at he |
| have hAe := sourceCoupling_exp_integrable_ae n k f p d h hh0 hh u j z t |
| have hleft : (fun a => gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) 1 |
| (fun x => sourceCouplingEnergy n k p d h j a (x 0))) =ᵐ[P] |
| (fun a => tiltMean (sourceSpinLeafKernel n k a.1) (H a) (sourceCouplingEnergy n k p d h j a) 1) := by |
| filter_upwards [hAe] with a ha |
| exact gibbsReplicaMean_coordinate_of_integrable _ (sourceCouplingHamiltonian_section n k f p d h u a) |
| (sourceCouplingEnergy_section n k p d h j a) ha.1 1 0 |
| have hdiag : ∀ x : NormalizedSpin (n+1)×IndexedLeaf k, C x x = D := by |
| intro x |
| dsimp [C,sourceCouplingCovariance,D] |
| rw [sourceJointMonomial_diagonal] |
| have hright : (fun a => gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) 1 |
| (fun x => C (x 0) (x 0))) =ᵐ[P] (fun _ => D) := by |
| filter_upwards [hAe] with a ha |
| simp only [hdiag] |
| exact gibbsReplicaMean_const_of_integrable _ (sourceCouplingHamiltonian_section n k f p d h u a) ha.1 1 D |
| have hI := kernel_replicaMean_bounded_integrable (sourceFullSpinLeafKernel n k) P |
| (H := H) (G := fun _ (x : Fin 2 → NormalizedSpin (n+1)×IndexedLeaf k) => C (x 1) (x 0)) |
| (sourceCouplingHamiltonian_measurable n k f p d h u) |
| ((sourceCouplingCovariance_measurable n k 2 p d u j 1 0).comp measurable_snd) |
| (abs_nonneg _) (fun _ x => sourceCouplingCovariance_bound n k p d u j _ _) |
| dsimp only [sourceFullSpinLeafKernel,Kernel.comap_apply] at hI |
| rw [integral_congr_ae hleft] at he |
| have hr : (∫ a, gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) 1 |
| (fun x => C (x 0) (x 0))-gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) 2 |
| (fun x => C (x 1) (x 0)) ∂P) = |
| D-∫ a, gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) 2 (fun x => C (x 1) (x 0)) ∂P := by |
| calc |
| _ = ∫ a, D-gibbsReplicaMean (sourceSpinLeafKernel n k a.1) (H a) 2 (fun x => C (x 1) (x 0)) ∂P := by |
| apply integral_congr_ae |
| filter_upwards [hright] with a ha |
| rw [ha] |
| _ = _ := by rw [integral_sub (integrable_const _) hI]; simp |
| exact he.trans hr |
|
|
| lemma source_contact_energy_concentration (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 H₀ := sourceCouplingHamiltonian n k f p d h u |
| let Y := sourceCouplingEnergy n k p d h j |
| let EY := ∫ b, tiltMean (sourceSpinLeafKernel n k b.1) (H₀ b) (Y b) 1 ∂P |
| Integrable (fun b => tiltMean (sourceSpinLeafKernel n k b.1) (H₀ b) (fun x => |Y b x|) 1) P ∧ |
| Integrable (fun b => tiltMean (sourceSpinLeafKernel n k b.1) (H₀ b) (fun x => |Y b x-EY|) 1) P ∧ |
| (∫ b, tiltMean (sourceSpinLeafKernel n k b.1) (H₀ b) (fun x => |Y b x-EY|) 1 ∂P) ≤ |
| Real.sqrt (2*c)/Real.sqrt (1/(n+1:ℕ))+ |
| (c*s+4*Real.sqrt (enrichedVarianceConstant k z f H T/(n+1:ℕ))/s)/(1/(n+1:ℕ)) := by |
| have hc := source_contact_thermal_disorder n k f p d h hh0 hh u j z hz hz0 hz1 t |
| hH hhH ht hs hu hup hum hmin hp hm |
| dsimp only at hc ⊢ |
| have hmean : Integrable (fun b => (1/(n+1:ℕ))*tiltMean (sourceSpinLeafKernel n k b.1) |
| (sourceCouplingHamiltonian n k f p d h u b) (sourceCouplingEnergy n k p d h j b) 1) |
| ((sourceBaseDataLaw n k z t).prod countableGaussianLaw) := by |
| have hm := hc.2.1 |
| change Integrable (fun b => (1/(n+1:ℕ))*tiltMean (sourceSpinLeafKernel n k b.1) |
| (fun x => sourceCouplingHamiltonian n k f p d h u b x+0*sourceCouplingEnergy n k p d h j b x) |
| (sourceCouplingEnergy n k p d h j b) 1) _ at hm |
| simpa only [zero_mul,add_zero] using hm |
| have htherm := hc.1.integrable (by norm_num : (1:ENNReal) ≤ 2) |
| change Integrable (fun b => Real.sqrt (1/(n+1:ℕ))*tiltMean (sourceSpinLeafKernel n k b.1) |
| (fun x => sourceCouplingHamiltonian n k f p d h u b x+0*sourceCouplingEnergy n k p d h j b x) |
| (fun x => |sourceCouplingEnergy n k p d h j b x-tiltMean (sourceSpinLeafKernel n k b.1) |
| (fun x => sourceCouplingHamiltonian n k f p d h u b x+0*sourceCouplingEnergy n k p d h j b x) |
| (sourceCouplingEnergy n k p d h j b) 1|) 1) _ at htherm |
| simp only [zero_mul,add_zero] at htherm |
| have hC := hc.2.2.1 |
| simp only [sourceCouplingThermal,zero_mul,add_zero] at hC |
| have hD := hc.2.2.2 |
| simp only [sourceCouplingMean,zero_mul,add_zero] at hD |
| have hr := kernel_scaled_energy_concentration (sourceFullSpinLeafKernel n k) |
| ((sourceBaseDataLaw n k z t).prod countableGaussianLaw) |
| (sourceCouplingHamiltonian_measurable n k f p d h u) (sourceCouplingEnergy_measurable n k p d h j) |
| (sourceCoupling_all_exp_ae n k f p d h hh0 hh u j z t) |
| (show 0<(1/(n+1:ℕ):ℝ) by positivity) hmean htherm hC hD |
| exact hr |
|
|
| end SphericalPerceptronFreeEnergy |
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| end |
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| end OAI |
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