| import OAI.Analysis.FiniteFactor.OrbitSampling.ProductError |
| import OAI.Analysis.FiniteFactor.OrbitSampling.DiagonalEvent |
| import OAI.Analysis.FiniteFactor.Sampling.Cancellation |
|
|
| namespace OAI |
|
|
| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| noncomputable section |
|
|
| def sourceSamplingEmpiricalDensity (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (θ x : SourceCoordinate → ℝ) : ℝ := |
| (∑ n ∈ sourceSamplingPowers stage, sourceSamplingProduct stage r b E G n θ x)/ |
| ((sourceSamplingPowers stage).card : ℝ) |
|
|
| /-- The full sampling error, including every distinct-pair and mixed term. -/ |
| def sourceSamplingSquaredError (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (θ : SourceCoordinate → ℝ) : ℝ := |
| ∫ x, (sourceSamplingEmpiricalDensity stage r b E G θ x-sourceSamplingIdealDensity stage r b E G x)^2 |
| ∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) |
|
|
| theorem sourceSampling_error_joint_integrable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) : |
| Integrable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) => |
| (sourceSamplingEmpiricalDensity stage r b E G p.1 p.2-sourceSamplingIdealDensity stage r b E G p.2)^2) |
| (sourceAngleLaw.prod (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure))) := by |
| apply sampling_average_square_integrable |
| · intro n _ k _ |
| exact sourceSamplingProduct_pair_integrable stage r b E G hE hG n k |
| · intro n _ |
| exact sourceSamplingProduct_mixed_integrable stage r b E G hE hG n |
| · have hm : Measurable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) => sourceSamplingIdealDensity stage r b E G p.2) := |
| (sourceSamplingIdealDensity_measurable stage r b E G hE hG).comp measurable_snd |
| have hi := sampling_bounded_product_integrable |
| (sourceAngleLaw.prod (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure))) |
| (fun p => sourceSamplingIdealDensity stage r b E G p.2) (fun p => sourceSamplingIdealDensity stage r b E G p.2) |
| hm hm _ (fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2) |
| (fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2) |
| simpa only [pow_two] using hi |
|
|
| theorem sourceSamplingSquaredError_integrable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) : |
| Integrable (sourceSamplingSquaredError stage r b E G) sourceAngleLaw := |
| (sourceSampling_error_joint_integrable stage r b E G hE hG).integral_prod_left |
|
|
| theorem sourceSamplingSquaredError_measurable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) : |
| Measurable (sourceSamplingSquaredError stage r b E G) := by |
| have hm : Measurable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) => |
| sourceSamplingEmpiricalDensity stage r b E G p.1 p.2) := |
| (Finset.measurable_sum _ (fun n _ => sourceSamplingProduct_measurable stage r b E G hE hG n)).div_const _ |
| exact ((hm.sub ((sourceSamplingIdealDensity_measurable stage r b E G hE hG).comp measurable_snd)).pow_const 2).stronglyMeasurable.integral_prod_right.measurable |
|
|
| theorem sourceSamplingSquaredError_nonneg (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool) |
| (E G : SourceCoordinate → Set ℝ) (θ : SourceCoordinate → ℝ) : |
| 0 ≤ sourceSamplingSquaredError stage r b E G θ := integral_nonneg (fun _ => sq_nonneg _) |
|
|
| /-- Full expected squared sampling error with the source's actual sample set, |
| weights and ideal density. All distinct-pair and mixed assumptions are proved. -/ |
| theorem sourceSamplingSquaredError_expectation (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage) |
| (r : Bool) (b : SourceCoordinate → Bool) (a c a' c' : SourceCoordinate → ℝ) : |
| (∫ θ, sourceSamplingSquaredError stage r b (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ ∂sourceAngleLaw) ≤ |
| (∫ θ, sourceSamplingDiagonalContribution stage r b (fun i => Set.Icc (a i) (c i)) (fun i => Set.Icc (a' i) (c' i)) θ ∂sourceAngleLaw)+ |
| 3*sourceSamplingCorrelationBudget stage := by |
| let E := fun i => Set.Icc (a i) (c i) |
| let G := fun i => Set.Icc (a' i) (c' i) |
| have hE : ∀ i, MeasurableSet (E i) := fun _ => measurableSet_Icc |
| have hG : ∀ i, MeasurableSet (G i) := fun _ => measurableSet_Icc |
| let μ := sourceAngleLaw.prod (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure)) |
| let S := sourceSamplingPowers stage |
| let f (n : ℤ) (p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := sourceSamplingProduct stage r b E G n p.1 p.2 |
| let g (p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ)) := sourceSamplingIdealDensity stage r b E G p.2 |
| have hpair (n k : ℤ) : Integrable (fun p => f n p*f k p) μ := |
| sourceSamplingProduct_pair_integrable stage r b E G hE hG n k |
| have hmix (n : ℤ) : Integrable (fun p => f n p*g p) μ := |
| sourceSamplingProduct_mixed_integrable stage r b E G hE hG n |
| have hg : Integrable (fun p => g p^2) μ := by |
| have hm : Measurable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) => sourceSamplingIdealDensity stage r b E G p.2) := |
| (sourceSamplingIdealDensity_measurable stage r b E G hE hG).comp measurable_snd |
| simpa only [pow_two] using sampling_bounded_product_integrable μ g g hm hm _ |
| (fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2) |
| (fun p => sourceSamplingIdealDensity_bounds stage r b E G hE hG p.2) |
| have hA : (∫ p, g p^2 ∂μ) = ∫ x, sourceSamplingIdealDensity stage r b E G x^2 |
| ∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) := by |
| rw [integral_prod _ hg] |
| simp [g] |
| have hB (n k : ℤ) : (∫ p, f n p*f k p ∂μ) = |
| ∫ θ, ∫ x, sourceSamplingProduct stage r b E G n θ x*sourceSamplingProduct stage r b E G k θ x |
| ∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) ∂sourceAngleLaw := integral_prod _ (hpair n k) |
| have hM (n : ℤ) : (∫ p, f n p*g p ∂μ) = |
| ∫ θ, ∫ x, sourceSamplingProduct stage r b E G n θ x*sourceSamplingIdealDensity stage r b E G x |
| ∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure) ∂sourceAngleLaw := integral_prod _ (hmix n) |
| have hS : S.Nonempty := by |
| have hP : (0 : ℝ) < (sourceSamplingPowers stage).card := |
| (Real.rpow_pos_of_pos (sourceBlockSize_pos stage) _).trans_le (sourceSamplingPowers_spec stage hs).2.1 |
| exact Finset.card_pos.mp (by exact_mod_cast hP) |
| have hc := sampling_finite_cancellation S hS |
| (fun n k => ∫ p, f n p*f k p ∂μ) (fun n => ∫ p, f n p*g p ∂μ) |
| (∫ p, g p^2 ∂μ) (sourceSamplingCorrelationBudget stage) |
| (integral_nonneg (fun _ => sq_nonneg _)) (by unfold sourceSamplingCorrelationBudget; positivity) |
| (fun n hn k hk hne => by rw [hB,hA]; exact sourceSamplingProduct_pair_error stage hs r b a c a' c' n k hn hk hne) |
| (fun n hn => by rw [hM,hA]; exact sourceSamplingProduct_mixed_error stage hs r b E G hE hG n hn) |
| have hex := sampling_average_square_expansion μ S f g (fun n _ k _ => hpair n k) (fun n _ => hmix n) hg |
| rw [← hex] at hc |
| have hErr : (∫ p, ((∑ n ∈ S, f n p)/(S.card : ℝ)-g p)^2 ∂μ) = |
| ∫ θ, sourceSamplingSquaredError stage r b E G θ ∂sourceAngleLaw := |
| integral_prod _ (sourceSampling_error_joint_integrable stage r b E G hE hG) |
| have hDiag : (∑ n ∈ S, ∫ p, f n p*f n p ∂μ)/(S.card : ℝ)^2 = |
| ∫ θ, sourceSamplingDiagonalContribution stage r b E G θ ∂sourceAngleLaw := by |
| unfold sourceSamplingDiagonalContribution |
| rw [integral_div,integral_finsetSum _ (fun n _ => |
| (sourceSamplingProduct_square_integrable stage r b E G hE hG n).integral_prod_left)] |
| simp only [hB,pow_two,f,S] |
| rw [hErr,hDiag] at hc |
| exact hc |
| |
| end |
| end FiniteFactor |
| |
| end OAI |
| |
