| import OAI.Analysis.FiniteFactor.Sampling.SmoothedPair |
| import OAI.Analysis.FiniteFactor.Sampling.SmoothingLaw |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory |
| open scoped ENNReal |
| noncomputable section |
|
|
| theorem integrable_unit_bounded {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (f : Ω → ℝ) (hf : Measurable f) |
| (hbound : ∀ x, 0 ≤ f x ∧ f x ≤ 1) : Integrable f μ := by |
| apply (integrable_const (1 : ℝ)).mono' hf.aestronglyMeasurable |
| exact Filter.Eventually.of_forall (fun x => by simpa [Real.norm_eq_abs,abs_of_nonneg (hbound x).1] using (hbound x).2) |
|
|
| theorem product_difference_unit_bound (a b c d : ℝ) |
| (hb : 0 ≤ b ∧ b ≤ 1) (hc : 0 ≤ c ∧ c ≤ 1) : |
| |a*b-c*d| ≤ |a-c|+|b-d| := by |
| calc |
| _ = |(a-c)*b+c*(b-d)| := by congr 1; ring |
| _ ≤ |(a-c)*b|+|c*(b-d)| := abs_add_le _ _ |
| _ = |a-c| *b+c*|b-d| := by rw [abs_mul,abs_mul,abs_of_nonneg hb.1,abs_of_nonneg hc.1] |
| _ ≤ |a-c|+|b-d| := by nlinarith [abs_nonneg (a-c),abs_nonneg (b-d)] |
|
|
| /-- Uniform marginal smoothing errors control a product test under a |
| possibly dependent joint law. -/ |
| theorem integral_product_smoothing_error {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (f g a b : Ω → ℝ) |
| (hf : Measurable f) (hg : Measurable g) (ha : Measurable a) (hb : Measurable b) |
| (hfb : ∀ x, 0 ≤ f x ∧ f x ≤ 1) (hgb : ∀ x, 0 ≤ g x ∧ g x ≤ 1) |
| (hab : ∀ x, 0 ≤ a x ∧ a x ≤ 1) (hbb : ∀ x, 0 ≤ b x ∧ b x ≤ 1) : |
| |(∫ x, f x*g x ∂μ)-(∫ x, a x*b x ∂μ)| ≤ |
| (∫ x, |f x-a x| ∂μ)+(∫ x, |g x-b x| ∂μ) := by |
| have hfg : Integrable (fun x => f x*g x) μ := integrable_unit_bounded μ _ (hf.mul hg) |
| (fun x => ⟨mul_nonneg (hfb x).1 (hgb x).1, by nlinarith [(hfb x).1,(hfb x).2,(hgb x).1,(hgb x).2]⟩) |
| have habi : Integrable (fun x => a x*b x) μ := integrable_unit_bounded μ _ (ha.mul hb) |
| (fun x => ⟨mul_nonneg (hab x).1 (hbb x).1, by nlinarith [(hab x).1,(hab x).2,(hbb x).1,(hbb x).2]⟩) |
| have hfa := (integrable_unit_bounded μ f hf hfb).sub (integrable_unit_bounded μ a ha hab) |
| have hgb' := (integrable_unit_bounded μ g hg hgb).sub (integrable_unit_bounded μ b hb hbb) |
| have hfa' : Integrable (fun x => |f x-a x|) μ := by simpa [Real.norm_eq_abs] using hfa.norm |
| have hgb'' : Integrable (fun x => |g x-b x|) μ := by simpa [Real.norm_eq_abs] using hgb'.norm |
| rw [← integral_sub hfg habi] |
| calc |
| _ ≤ ∫ x, |f x*g x-a x*b x| ∂μ := by simpa [Real.norm_eq_abs] using norm_integral_le_integral_norm (fun x => f x*g x-a x*b x) |
| _ ≤ ∫ x, |f x-a x|+|g x-b x| ∂μ := by |
| apply integral_mono_ae |
| · simpa [Real.norm_eq_abs] using (hfg.sub habi).norm |
| · exact hfa'.add hgb'' |
| · exact Filter.Eventually.of_forall (fun x => product_difference_unit_bound _ _ _ _ (hgb x) (hab x)) |
| _ = _ := integral_add hfa' hgb'' |
|
|
| /-- Explicit rectangle discrepancy derived from actual marginal domination |
| and joint Fourier estimates. No two-dimensional density is required. -/ |
| theorem sampling_rectangle_discrepancy {Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) [IsProbabilityMeasure μ] (X Y : Ω → ℝ) |
| (hX : Measurable X) (hY : Measurable Y) |
| (hdomX : μ.map X ≤ (2 : ℝ≥0∞) • unitIntervalMeasure) |
| (hdomY : μ.map Y ≤ (2 : ℝ≥0∞) • unitIntervalMeasure) |
| (H : ℕ) (hH : 0 < H) (a b a' b' δ ε : ℝ) (hδ : 0 < δ) (hε : 0 ≤ ε) |
| (hFourier : ∀ v w : ℤ, |v| < (H : ℤ) → |w| < (H : ℤ) → (v ≠ 0 ∨ w ≠ 0) → |
| ‖∫ θ, samplingCharacter v (X θ)*samplingCharacter w (Y θ) ∂μ‖ ≤ ε) : |
| |(μ {θ | X θ ∈ Set.Icc a b ∧ Y θ ∈ Set.Icc a' b'}).toReal - |
| (unitIntervalMeasure (Set.Icc a b)).toReal*(unitIntervalMeasure (Set.Icc a' b')).toReal| ≤ |
| 24*δ+1/(2*(H : ℝ)*δ^2)+(H : ℝ)^2*ε := by |
| let E := Set.Icc a b |
| let E' := Set.Icc a' b' |
| let f := fun θ => E.indicator (fun _ => (1 : ℝ)) (X θ) |
| let g := fun θ => E'.indicator (fun _ => (1 : ℝ)) (Y θ) |
| let A := fun θ => samplingSmooth H E (X θ) |
| let B := fun θ => samplingSmooth H E' (Y θ) |
| have hf : Measurable f := (measurable_const.indicator measurableSet_Icc).comp hX |
| have hg : Measurable g := (measurable_const.indicator measurableSet_Icc).comp hY |
| have hA : Measurable A := (samplingSmooth_continuous H E).measurable.comp hX |
| have hB : Measurable B := (samplingSmooth_continuous H E').measurable.comp hY |
| have hfb (θ) : 0 ≤ f θ ∧ f θ ≤ 1 := by |
| dsimp [f]; by_cases he : X θ ∈ E <;> simp [he] |
| have hgb (θ) : 0 ≤ g θ ∧ g θ ≤ 1 := by |
| dsimp [g]; by_cases he : Y θ ∈ E' <;> simp [he] |
| have hprod := integral_product_smoothing_error μ f g A B hf hg hA hB hfb hgb |
| (fun θ => samplingSmooth_bounds H hH E (X θ)) (fun θ => samplingSmooth_bounds H hH E' (Y θ)) |
| have : IsProbabilityMeasure (μ.map X) := ⟨by rw [Measure.map_apply hX MeasurableSet.univ]; simp⟩ |
| have : IsProbabilityMeasure (μ.map Y) := ⟨by rw [Measure.map_apply hY MeasurableSet.univ]; simp⟩ |
| have hex := samplingSmooth_expected_error (μ.map X) hdomX H hH a b δ hδ |
| have hey := samplingSmooth_expected_error (μ.map Y) hdomY H hH a' b' δ hδ |
| have hem (a b : ℝ) : Measurable (fun t => |samplingSmooth H (Set.Icc a b) t- |
| (Set.Icc a b).indicator (fun _ => (1 : ℝ)) t|) := by |
| have hi : Measurable ((Set.Icc a b).indicator (fun _ => (1 : ℝ))) := measurable_const.indicator measurableSet_Icc |
| simpa [Real.norm_eq_abs] using ((samplingSmooth_continuous H (Set.Icc a b)).measurable.sub hi).norm |
| rw [integral_map hX.aemeasurable (hem a b).aestronglyMeasurable] at hex |
| rw [integral_map hY.aemeasurable (hem a' b').aestronglyMeasurable] at hey |
| have hfint : (∫ θ, f θ*g θ ∂μ) = (μ {θ | X θ ∈ E ∧ Y θ ∈ E'}).toReal := by |
| have he : (fun θ => f θ*g θ) = {θ | X θ ∈ E ∧ Y θ ∈ E'}.indicator (fun _ => (1 : ℝ)) := by |
| funext θ |
| dsimp [f,g] |
| by_cases hx : X θ ∈ E <;> by_cases hy : Y θ ∈ E' <;> simp [hx,hy] |
| rw [he,integral_indicator (by exact (hX measurableSet_Icc).inter (hY measurableSet_Icc))] |
| simp [measureReal_def] |
| have hsm := samplingSmooth_pair_error μ X Y hX hY H hH E E' ε hε hFourier |
| simp_rw [← Complex.ofReal_mul] at hsm |
| rw [integral_complex_ofReal, ← Complex.ofReal_sub, Complex.norm_real, Real.norm_eq_abs] at hsm |
| change |(∫ θ, A θ*B θ ∂μ)-(unitIntervalMeasure E).toReal*(unitIntervalMeasure E').toReal| ≤ _ at hsm |
| have hex' : (∫ θ, |f θ-A θ| ∂μ) ≤ 12*δ+1/(4*(H : ℝ)*δ^2) := by |
| simpa only [f,A,abs_sub_comm] using hex |
| have hey' : (∫ θ, |g θ-B θ| ∂μ) ≤ 12*δ+1/(4*(H : ℝ)*δ^2) := by |
| simpa only [g,B,abs_sub_comm] using hey |
| rw [← hfint] |
| have htri := abs_sub_le (∫ θ, f θ*g θ ∂μ) (∫ θ, A θ*B θ ∂μ) |
| ((unitIntervalMeasure E).toReal*(unitIntervalMeasure E').toReal) |
| have hc : 2*(1/(4*(H : ℝ)*δ^2)) = 1/(2*(H : ℝ)*δ^2) := by ring |
| nlinarith |
|
|
| end |
| end FiniteFactor |
|
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| end OAI |
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