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67aee1d | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 | import OAI.Analysis.FiniteFactor.Sampling.LayerDiscrepancy
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter
open scoped Classical ENNReal
noncomputable section
def samplingClosedIndicator (a b t : ℝ) : ℝ :=
(Set.Icc a b).indicator (fun _ => 1) t
theorem samplingClosedIndicator_measurable (a b : ℝ) :
Measurable (samplingClosedIndicator a b) := measurable_const.indicator measurableSet_Icc
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
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]
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]
/-- Two intervals in each variable cost four rectangle discrepancies. -/
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
end
end FiniteFactor
end OAI
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