| import Mathlib |
| import OAI.Probability.Perceptron.Model |
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| namespace OAI |
|
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| noncomputable section |
| open MeasureTheory ProbabilityTheory Filter Set |
| open scoped Topology NNReal ENNReal |
| namespace SphericalPerceptronFreeEnergy |
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| lemma memLp_prod_of_conditional_square_function {X Y : Type} |
| [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) (ν : Measure Y) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {F : X×Y → ℝ} {G A : X → ℝ} (hF : Measurable F) (hG : MemLp G 2 μ) |
| (hA : Integrable A μ) (hs : ∀ x, MemLp (fun y => F (x,y)) 2 ν) |
| (hb : ∀ x, (∫ y, (F (x,y)-G x)^2 ∂ν) ≤ A x) : MemLp F 2 (μ.prod ν) := by |
| apply (memLp_two_iff_integrable_sq hF.aestronglyMeasurable).mpr |
| apply (integrable_prod_iff (hF.pow_const 2).aestronglyMeasurable).mpr |
| refine ⟨Eventually.of_forall (fun x => (hs x).integrable_sq),?_⟩ |
| have hi : Integrable (fun x => 2*A x+2*(G x)^2) μ := |
| (hA.const_mul 2).add (hG.integrable_sq.const_mul 2) |
| apply hi.mono' (hF.pow_const 2).aestronglyMeasurable.norm.integral_prod_right' |
| exact ae_of_all _ fun x => by |
| simp only [Real.norm_eq_abs,abs_sq] |
| rw [abs_of_nonneg (integral_nonneg (fun y : Y => sq_nonneg (F (x,y))))] |
| have hi1 : Integrable (fun y => (F (x,y)-G x)^2) ν := |
| ((hs x).sub (memLp_const (G x))).integrable_sq |
| calc |
| _ ≤ ∫ y, 2*(F (x,y)-G x)^2+2*(G x)^2 ∂ν := by |
| apply integral_mono (hs x).integrable_sq ((hi1.const_mul 2).add (integrable_const _)) |
| intro y |
| dsimp only [Pi.add_apply] |
| nlinarith [sq_nonneg (F (x,y)-2*G x)] |
| _ = 2*(∫ y, (F (x,y)-G x)^2 ∂ν)+2*(G x)^2 := by |
| rw [integral_add (hi1.const_mul 2) (integrable_const _),integral_const_mul] |
| simp |
| _ ≤ _ := by linarith [hb x] |
|
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| lemma variance_prod_le_integral {X Y : Type} [MeasurableSpace X] [MeasurableSpace Y] |
| (μ : Measure X) (ν : Measure Y) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν] |
| {F : X × Y → ℝ} (hp : MemLp F 2 (μ.prod ν)) |
| (hs : ∀ x, MemLp (fun y => F (x,y)) 2 ν) |
| (hh : MemLp (fun x => ∫ y, F (x,y) ∂ν) 2 μ) {A : X → ℝ} {B : ℝ} |
| (hAi : Integrable A μ) (hA : ∀ x, variance (fun y => F (x,y)) ν ≤ A x) |
| (hB : variance (fun x => ∫ y, F (x,y) ∂ν) μ ≤ B) : |
| variance F (μ.prod ν) ≤ (∫ x, A x ∂μ)+B := by |
| have hi : (∫ x, ∫ y, F (x,y)^2 ∂ν ∂μ) ≤ (∫ x, A x ∂μ)+∫ x, (∫ y, F (x,y) ∂ν)^2 ∂μ := by |
| calc |
| _ ≤ ∫ x, (A x+(∫ y, F (x,y) ∂ν)^2) ∂μ := by |
| apply integral_mono hp.integrable_sq.integral_prod_left (hAi.add hh.integrable_sq) |
| intro x |
| have hv := hA x |
| rw [variance_eq_sub (hs x)] at hv |
| simpa only [Pi.pow_apply,Pi.add_apply] using sub_le_iff_le_add.mp hv |
| _ = _ := integral_add hAi hh.integrable_sq |
| rw [variance_eq_sub hh] at hB |
| rw [variance_eq_sub hp] |
| simp only [Pi.pow_apply] at hB ⊢ |
| rw [integral_prod _ hp.integrable_sq,integral_prod _ (hp.integrable (by norm_num))] |
| linarith |
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| end SphericalPerceptronFreeEnergy |
| end |
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| end OAI |
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