| import OAI.Analysis.FiniteFactor.OrbitSampling.ErrorEvent |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical |
| noncomputable section |
|
|
| theorem sourceSamplingEmpiricalDensity_measurable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) |
| (θ : SourceCoordinate → ℝ) : Measurable (sourceSamplingEmpiricalDensity stage r b E G θ) := |
| (Finset.measurable_sum _ (fun n _ => (sourceSamplingProduct_measurable stage r b E G hE hG n).comp |
| (measurable_const.prodMk measurable_id))).div_const _ |
|
|
| /-- Every empirical density is bounded at its finite stage, including stages |
| whose selected set is empty. No asymptotic assumption is used here. -/ |
| theorem sourceSamplingEmpiricalDensity_bounds (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (θ x : SourceCoordinate → ℝ) : |
| 0 ≤ sourceSamplingEmpiricalDensity stage r b E G θ x ∧ |
| sourceSamplingEmpiricalDensity stage r b E G θ x ≤ |
| (max 1 (1/(sourceSamplingWidth stage*sourceSamplingCutoff stage)))^(sourceSamplingPrefix stage).card := by |
| let C := (max 1 (1/(sourceSamplingWidth stage*sourceSamplingCutoff stage)))^(sourceSamplingPrefix stage).card |
| have hC : 0 ≤ C := pow_nonneg ((by norm_num : (0 : ℝ) ≤ 1).trans (le_max_left _ _)) _ |
| have hb := sourceSamplingProduct_bounds stage r b E G |
| have hn : 0 ≤ sourceSamplingEmpiricalDensity stage r b E G θ x := |
| div_nonneg (sum_nonneg (fun n _ => (hb n θ x).1)) (by positivity) |
| refine ⟨hn,?_⟩ |
| by_cases hS : (sourceSamplingPowers stage).Nonempty |
| · have hP : (0 : ℝ) < (sourceSamplingPowers stage).card := by exact_mod_cast hS.card_pos |
| apply (div_le_iff₀ hP).mpr |
| calc |
| _ ≤ ∑ _n ∈ sourceSamplingPowers stage, C := sum_le_sum (fun n _ => (hb n θ x).2) |
| _ = _ := by simp [C]; ring |
| · have he : sourceSamplingPowers stage = ∅ := Finset.not_nonempty_iff_eq_empty.mp hS |
| simp [sourceSamplingEmpiricalDensity,he] |
|
|
| /-- The empirical densities, not only the ideal products, belong to L2. -/ |
| theorem sourceSamplingEmpiricalDensity_memLp (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) |
| (θ : SourceCoordinate → ℝ) : |
| MemLp (sourceSamplingEmpiricalDensity stage r b E G θ) 2 |
| (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure)) := by |
| apply MemLp.of_bound (sourceSamplingEmpiricalDensity_measurable stage r b E G hE hG θ).aestronglyMeasurable |
| ((max 1 (1/(sourceSamplingWidth stage*sourceSamplingCutoff stage)))^(sourceSamplingPrefix stage).card) |
| apply Eventually.of_forall |
| intro x |
| rw [Real.norm_eq_abs,abs_of_nonneg (sourceSamplingEmpiricalDensity_bounds stage r b E G θ x).1] |
| exact (sourceSamplingEmpiricalDensity_bounds stage r b E G θ x).2 |
|
|
| /-- The full error and the ideal L2 bound control the actual empirical square. -/ |
| theorem sourceSamplingEmpiricalDensity_square_le (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) |
| (θ : SourceCoordinate → ℝ) : |
| (∫ x, sourceSamplingEmpiricalDensity stage r b E G θ x^2 |
| ∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure)) ≤ |
| 2*sourceSamplingSquaredError stage r b E G θ+2*Real.exp 5 := by |
| let μ := Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) |
| let z := sourceSamplingEmpiricalDensity stage r b E G θ |
| let g := sourceSamplingIdealDensity stage r b E G |
| let C := (max 1 (1/(sourceSamplingWidth stage*sourceSamplingCutoff stage)))^(sourceSamplingPrefix stage).card |
| have hz := sourceSamplingEmpiricalDensity_measurable stage r b E G hE hG θ |
| have hg := sourceSamplingIdealDensity_measurable stage r b E G hE hG |
| have hzb := sourceSamplingEmpiricalDensity_bounds stage r b E G θ |
| have hgb := sourceSamplingIdealDensity_bounds stage r b E G hE hG |
| have hzi : Integrable (fun x => z x^2) μ := sampling_square_integrable μ z C hz hzb |
| have hgi : Integrable (fun x => g x^2) μ := sampling_square_integrable μ g C hg hgb |
| have hei : Integrable (fun x => (z x-g x)^2) μ := by |
| apply sampling_integrable_bounded μ _ ((hz.sub hg).pow_const 2) (C^2) |
| intro x |
| rw [abs_of_nonneg (sq_nonneg _)] |
| have hz' : 0 ≤ z x ∧ z x ≤ C := hzb x |
| have hg' : 0 ≤ g x ∧ g x ≤ C := hgb x |
| change (z x-g x)^2 ≤ C^2 |
| nlinarith [mul_nonneg (show 0 ≤ C-(z x-g x) by linarith) (show 0 ≤ C+(z x-g x) by linarith)] |
| have hi := integral_mono_ae hzi ((hei.const_mul 2).add (hgi.const_mul 2)) |
| (Eventually.of_forall (fun x => show z x^2 ≤ 2*(z x-g x)^2+2*g x^2 by nlinarith [sq_nonneg (z x-2*g x)])) |
| simp only [Pi.add_apply] at hi |
| rw [integral_add (hei.const_mul 2) (hgi.const_mul 2),integral_const_mul,integral_const_mul] at hi |
| exact hi.trans (by |
| have hb := sourceSamplingIdealDensity_uniform_square stage r b E G hE hG |
| change (∫ x, g x^2 ∂μ) ≤ Real.exp 5 at hb |
| change _ ≤ 2*(∫ x, (z x-g x)^2 ∂μ)+2*Real.exp 5 |
| linarith) |
|
|
| end |
| end FiniteFactor |
|
|
| end OAI |
|
|
