| import Mathlib |
| import OAI.Probability.Perceptron.Variational.UnboundedContact |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal BigOperators |
| namespace SphericalPerceptronFreeEnergy |
| variable {A S : Type*} [MeasurableSpace A] [MeasurableSpace S] |
| variable (κ : Kernel A S) [IsMarkovKernel κ] (P : Measure A) [IsProbabilityMeasure P] |
| variable {H Y : A → S → ℝ} |
|
|
| lemma kernel_coupling_energy_concentration |
| (hH : Measurable (Function.uncurry H)) (hY : Measurable (Function.uncurry Y)) |
| (he : ∀ᵐ a ∂P, ∀ r : ℝ, Integrable (fun x => Real.exp (H a x+r*Y a x)) (κ a)) |
| (hm : Integrable (fun a => tiltMean (κ a) (H a) (Y a) 1) P) |
| (ht : Integrable (fun a => tiltMean (κ a) (H a) |
| (fun x => |Y a x-tiltMean (κ a) (H a) (Y a) 1|) 1) P) : |
| let EY := ∫ a, tiltMean (κ a) (H a) (Y a) 1 ∂P |
| Integrable (fun a => tiltMean (κ a) (H a) (fun x => |Y a x|) 1) P ∧ |
| Integrable (fun a => tiltMean (κ a) (H a) (fun x => |Y a x-EY|) 1) P ∧ |
| (∫ a, tiltMean (κ a) (H a) (fun x => |Y a x-EY|) 1 ∂P) ≤ |
| (∫ a, tiltMean (κ a) (H a) (fun x => |Y a x-tiltMean (κ a) (H a) (Y a) 1|) 1 ∂P)+ |
| ∫ a, |tiltMean (κ a) (H a) (Y a) 1-EY| ∂P := by |
| dsimp only |
| let m := fun a => tiltMean (κ a) (H a) (Y a) 1 |
| let U := fun a => tiltMean (κ a) (H a) (fun x => |Y a x-m a|) 1 |
| have ht' : Integrable U P := ht |
| have hbound (c : ℝ) : ∀ᵐ a ∂P, |
| tiltMean (κ a) (H a) (fun x => |Y a x-c|) 1 ≤ U a+|m a-c| := by |
| filter_upwards [he] with a he |
| simpa only [zero_mul,add_zero] using |
| coupling_tilt_center_triangle (κ a) hH.of_uncurry_left hY.of_uncurry_left he 0 (m a) c |
| have hib (c : ℝ) : Integrable (fun a => tiltMean (κ a) (H a) (fun x => |Y a x-c|) 1) P := by |
| apply (ht'.add ((hm.sub (integrable_const c)).abs)).mono' |
| (kernel_tiltMean_measurable κ (Y := fun a x => |Y a x-c|) hH (hY.sub measurable_const |>.abs)).aestronglyMeasurable |
| filter_upwards [hbound c] with a ha |
| simpa only [Pi.add_apply,Pi.sub_apply,Real.norm_of_nonneg (tiltMean_nonneg (κ a) (fun _ => abs_nonneg _) 1)] using ha |
| refine ⟨by simpa only [sub_zero] using hib 0,hib (∫ a, m a ∂P),?_⟩ |
| calc |
| _ ≤ ∫ a, U a+|m a-∫ b, m b ∂P| ∂P := integral_mono_ae (hib _) (ht'.add ((hm.sub (integrable_const _)).abs)) (hbound _) |
| _ = _ := integral_add ht' ((hm.sub (integrable_const _)).abs) |
|
|
| lemma kernel_scaled_energy_concentration |
| (hH : Measurable (Function.uncurry H)) (hY : Measurable (Function.uncurry Y)) |
| (he : ∀ᵐ a ∂P, ∀ r : ℝ, Integrable (fun x => Real.exp (H a x+r*Y a x)) (κ a)) |
| {v C D : ℝ} (hv : 0<v) |
| (hm : Integrable (fun a => v*tiltMean (κ a) (H a) (Y a) 1) P) |
| (ht : Integrable (fun a => Real.sqrt v*tiltMean (κ a) (H a) |
| (fun x => |Y a x-tiltMean (κ a) (H a) (Y a) 1|) 1) P) |
| (hC : (∫ a, Real.sqrt v*tiltMean (κ a) (H a) |
| (fun x => |Y a x-tiltMean (κ a) (H a) (Y a) 1|) 1 ∂P) ≤ C) |
| (hD : (∫ a, |v*tiltMean (κ a) (H a) (Y a) 1- |
| ∫ b, v*tiltMean (κ b) (H b) (Y b) 1 ∂P| ∂P) ≤ D) : |
| let EY := ∫ a, tiltMean (κ a) (H a) (Y a) 1 ∂P |
| Integrable (fun a => tiltMean (κ a) (H a) (fun x => |Y a x|) 1) P ∧ |
| Integrable (fun a => tiltMean (κ a) (H a) (fun x => |Y a x-EY|) 1) P ∧ |
| (∫ a, tiltMean (κ a) (H a) (fun x => |Y a x-EY|) 1 ∂P) ≤ C/Real.sqrt v+D/v := by |
| have hm' : Integrable (fun a => tiltMean (κ a) (H a) (Y a) 1) P := by |
| convert hm.div_const v using 1 |
| funext a |
| exact (mul_div_cancel_left₀ _ hv.ne').symm |
| have ht' : Integrable (fun a => tiltMean (κ a) (H a) |
| (fun x => |Y a x-tiltMean (κ a) (H a) (Y a) 1|) 1) P := by |
| convert ht.div_const (Real.sqrt v) using 1 |
| funext a |
| exact (mul_div_cancel_left₀ _ (Real.sqrt_pos.mpr hv).ne').symm |
| have hres := kernel_coupling_energy_concentration κ P hH hY he hm' ht' |
| refine ⟨hres.1,hres.2.1,hres.2.2.trans (add_le_add ?_ ?_)⟩ |
| · rw [integral_const_mul] at hC |
| exact (le_div_iff₀ (Real.sqrt_pos.mpr hv)).mpr (by simpa only [mul_comm] using hC) |
| · simp_rw [integral_const_mul,← mul_sub,abs_mul,abs_of_pos hv] at hD |
| rw [integral_const_mul] at hD |
| exact (le_div_iff₀ hv).mpr (by simpa only [mul_comm] using hD) |
|
|
| end SphericalPerceptronFreeEnergy |
| end |
|
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| end OAI |
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