File size: 4,400 Bytes
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 | import OAI.Analysis.FiniteFactor.OrbitSampling.SelfSummability
namespace OAI
namespace FiniteFactor
open MeasureTheory Filter Finset
noncomputable section
/-- The self-pair part of the empirical squared norm, with its actual P^-2 weight. -/
def sourceSamplingDiagonalContribution (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (θ : SourceCoordinate → ℝ) : ℝ :=
(∑ n ∈ sourceSamplingPowers stage,
∫ x, sourceSamplingProduct stage r b E G n θ x ^ 2
∂Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure)) / ((sourceSamplingPowers stage).card : ℝ)^2
theorem sourceSamplingDiagonalContribution_nonneg (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (θ : SourceCoordinate → ℝ) :
0 ≤ sourceSamplingDiagonalContribution stage r b E G θ :=
div_nonneg (sum_nonneg (fun _ _ => integral_nonneg (fun _ => sq_nonneg _))) (sq_nonneg _)
theorem sourceSamplingProduct_square_integrable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) (n : ℤ) :
Integrable (fun p : (SourceCoordinate → ℝ) × (SourceCoordinate → ℝ) =>
sourceSamplingProduct stage r b E G n p.1 p.2 ^ 2)
(sourceAngleLaw.prod (Measure.infinitePi (fun _ : SourceCoordinate => unitIntervalMeasure))) :=
sampling_square_integrable _ _
((max 1 (1/(sourceSamplingWidth stage*sourceSamplingCutoff stage)))^(sourceSamplingPrefix stage).card)
(sourceSamplingProduct_measurable stage r b E G hE hG n)
(fun p => sourceSamplingProduct_bounds stage r b E G n p.1 p.2)
theorem sourceSamplingDiagonalContribution_integrable (stage : ℕ) (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
Integrable (sourceSamplingDiagonalContribution stage r b E G) sourceAngleLaw := by
apply Integrable.div_const
exact integrable_finsetSum _ (fun n _ =>
(sourceSamplingProduct_square_integrable stage r b E G hE hG n).integral_prod_left)
/-- The diagonal contribution's expectation is bounded by the proved
summable budget; the finite set is the actual deterministic sample set. -/
theorem sourceSamplingDiagonalContribution_expectation (stage : ℕ) (hs : 200000 ≤ sourceBlockSize stage)
(r : Bool) (b : SourceCoordinate → Bool) (E G : SourceCoordinate → Set ℝ)
(hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
(∫ θ, sourceSamplingDiagonalContribution stage r b E G θ ∂sourceAngleLaw) ≤ sourceSamplingSelfBudget stage := by
have hP : (0 : ℝ) < (sourceSamplingPowers stage).card :=
(Real.rpow_pos_of_pos (sourceBlockSize_pos stage) _).trans_le (sourceSamplingPowers_spec stage hs).2.1
unfold sourceSamplingDiagonalContribution
rw [integral_div,integral_finsetSum _ (fun n _ =>
(sourceSamplingProduct_square_integrable stage r b E G hE hG n).integral_prod_left)]
have hsum := Finset.sum_le_sum (fun n hn => sourceSamplingProduct_self_moment stage hs r b E G hE hG n hn)
simp only [Finset.sum_const,nsmul_eq_mul] at hsum
calc
_ ≤ ((sourceSamplingPowers stage).card : ℝ)*
(2+8*(sourceBlockSize stage : ℝ)^(3/5 : ℝ)*Real.log (sourceBlockSize stage : ℝ))^(sourceSamplingPrefix stage).card /
((sourceSamplingPowers stage).card : ℝ)^2 := div_le_div_of_nonneg_right hsum (sq_nonneg _)
_ = _ := by unfold sourceSamplingSelfBudget; field_simp
/-- Actual self-pair expectations are summable across the constructed stages. -/
theorem sourceSamplingDiagonalContribution_summable_integral (r : Bool) (b : SourceCoordinate → Bool)
(E G : SourceCoordinate → Set ℝ) (hE : ∀ i, MeasurableSet (E i)) (hG : ∀ i, MeasurableSet (G i)) :
Summable (fun stage => ∫ θ, sourceSamplingDiagonalContribution stage r b E G θ ∂sourceAngleLaw) := by
apply sourceSamplingSelfBudget_summable.of_norm_bounded_eventually_nat
filter_upwards [sourceBlockSize_tendsto.eventually (eventually_ge_atTop 200000)] with stage hs
rw [Real.norm_eq_abs,abs_of_nonneg (integral_nonneg (sourceSamplingDiagonalContribution_nonneg stage r b E G))]
exact sourceSamplingDiagonalContribution_expectation stage hs r b E G hE hG
end
end FiniteFactor
end OAI
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