| import OAI.Analysis.FiniteFactor.Sampling.LayerDiscrepancy |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter |
| open scoped Classical ENNReal |
| noncomputable section |
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| def samplingClosedIndicator (a b t : ℝ) : ℝ := |
| (Set.Icc a b).indicator (fun _ => 1) t |
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| theorem samplingClosedIndicator_measurable (a b : ℝ) : |
| Measurable (samplingClosedIndicator a b) := measurable_const.indicator measurableSet_Icc |
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| theorem samplingClosedIndicator_bounds (a b t : ℝ) : |
| 0 ≤ samplingClosedIndicator a b t ∧ samplingClosedIndicator a b t ≤ 1 := by |
| simp only [samplingClosedIndicator,Set.indicator]; split_ifs <;> norm_num |
|
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| theorem samplingClosedIndicator_integral (a b : ℝ) : |
| (∫ t, samplingClosedIndicator a b t ∂unitIntervalMeasure) = |
| (unitIntervalMeasure (Set.Icc a b)).toReal := by |
| unfold samplingClosedIndicator |
| rw [integral_indicator measurableSet_Icc] |
| simp [measureReal_def] |
|
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| theorem samplingClosedIndicator_pair {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X Y : Ω → ℝ) |
| (hX : Measurable X) (hY : Measurable Y) (a b c d : ℝ) : |
| (∫ x, samplingClosedIndicator a b (X x)*samplingClosedIndicator c d (Y x) ∂μ) = |
| (μ {x | X x ∈ Set.Icc a b ∧ Y x ∈ Set.Icc c d}).toReal := by |
| have hs : MeasurableSet {x | X x ∈ Set.Icc a b ∧ Y x ∈ Set.Icc c d} := |
| (measurableSet_Icc.preimage hX).inter (measurableSet_Icc.preimage hY) |
| have he : (fun x => samplingClosedIndicator a b (X x)*samplingClosedIndicator c d (Y x)) = |
| {x | X x ∈ Set.Icc a b ∧ Y x ∈ Set.Icc c d}.indicator (fun _ => (1 : ℝ)) := by |
| funext x |
| simp only [samplingClosedIndicator,Set.indicator,Set.mem_ofPred_eq] |
| split_ifs <;> simp_all |
| rw [he,integral_indicator hs] |
| simp [measureReal_def] |
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| / |
| theorem sampling_two_intervals_pair {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X Y : Ω → ℝ) |
| (hX : Measurable X) (hY : Measurable Y) (δ : ℝ) |
| (hrect : ∀ a b c d : ℝ, |
| |(μ {x | X x ∈ Set.Icc a b ∧ Y x ∈ Set.Icc c d}).toReal - |
| (unitIntervalMeasure (Set.Icc a b)).toReal*(unitIntervalMeasure (Set.Icc c d)).toReal| ≤ δ) |
| (a b c d : Fin 2 → ℝ) : |
| |(∫ x, (∑ j, samplingClosedIndicator (a j) (b j) (X x))* |
| (∑ k, samplingClosedIndicator (c k) (d k) (Y x)) ∂μ) - |
| (∫ t, ∑ j, samplingClosedIndicator (a j) (b j) t ∂unitIntervalMeasure)* |
| (∫ t, ∑ k, samplingClosedIndicator (c k) (d k) t ∂unitIntervalMeasure)| ≤ 4*δ := by |
| let f (j : Fin 2) := samplingClosedIndicator (a j) (b j) |
| let g (k : Fin 2) := samplingClosedIndicator (c k) (d k) |
| have hi (j k : Fin 2) : Integrable (fun x => f j (X x)*g k (Y x)) μ := by |
| apply sampling_integrable_bounded μ _ |
| (((samplingClosedIndicator_measurable _ _).comp hX).mul |
| ((samplingClosedIndicator_measurable _ _).comp hY)) 1 |
| intro x |
| have h1 := samplingClosedIndicator_bounds (a j) (b j) (X x) |
| have h2 := samplingClosedIndicator_bounds (c k) (d k) (Y x) |
| simp only [Pi.mul_apply,Function.comp_apply] |
| rw [abs_of_nonneg (mul_nonneg h1.1 h2.1)] |
| nlinarith |
| have hif (j : Fin 2) : Integrable (f j) unitIntervalMeasure := |
| integrable_unit_bounded _ _ (samplingClosedIndicator_measurable _ _) (samplingClosedIndicator_bounds _ _) |
| have hig (k : Fin 2) : Integrable (g k) unitIntervalMeasure := |
| integrable_unit_bounded _ _ (samplingClosedIndicator_measurable _ _) (samplingClosedIndicator_bounds _ _) |
| change |(∫ x, (∑ j, f j (X x))*(∑ k, g k (Y x)) ∂μ) - |
| (∫ t, ∑ j, f j t ∂unitIntervalMeasure)*(∫ t, ∑ k, g k t ∂unitIntervalMeasure)| ≤ _ |
| simp_rw [Finset.sum_mul,Finset.mul_sum] |
| rw [integral_finsetSum _ (fun j _ => integrable_finsetSum _ (fun k _ => hi j k))] |
| simp_rw [integral_finsetSum _ (fun k _ => hi _ k)] |
| rw [integral_finsetSum _ (fun j _ => hif j),integral_finsetSum _ (fun k _ => hig k)] |
| simp_rw [Finset.sum_mul,Finset.mul_sum] |
| rw [← Finset.sum_sub_distrib] |
| simp_rw [← Finset.sum_sub_distrib] |
| calc |
| _ ≤ ∑ j : Fin 2, ∑ k : Fin 2, |
| |(∫ x, f j (X x)*g k (Y x) ∂μ)- |
| (∫ t, f j t ∂unitIntervalMeasure)*(∫ t, g k t ∂unitIntervalMeasure)| := |
| (Finset.abs_sum_le_sum_abs _ _).trans (Finset.sum_le_sum (fun j _ => Finset.abs_sum_le_sum_abs _ _)) |
| _ ≤ ∑ _j : Fin 2, ∑ _k : Fin 2, δ := by |
| apply Finset.sum_le_sum |
| intro j _ |
| apply Finset.sum_le_sum |
| intro k _ |
| simpa only [f,g,samplingClosedIndicator_pair μ X Y hX hY, |
| samplingClosedIndicator_integral] using hrect (a j) (b j) (c k) (d k) |
| _ = _ := by simp; ring |
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|
| end |
| end FiniteFactor |
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| end OAI |
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