| import OAI.Analysis.FiniteFactor.Sampling.IntegratedCorrelation |
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| namespace OAI |
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| namespace FiniteFactor |
| open MeasureTheory Filter Finset |
| open scoped Classical |
| noncomputable section |
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| /-- Exact finite-average expansion, before bounding or discarding any term. -/ |
| theorem sampling_average_square_expansion {ι Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) (S : Finset ι) (f : ι → Ω → ℝ) (g : Ω → ℝ) |
| (hpair : ∀ n ∈ S, ∀ k ∈ S, Integrable (fun x => f n x*f k x) μ) |
| (hmix : ∀ n ∈ S, Integrable (fun x => f n x*g x) μ) |
| (hg : Integrable (fun x => g x^2) μ) : |
| (∫ x, ((∑ n ∈ S, f n x)/(S.card : ℝ)-g x)^2 ∂μ) = |
| (∑ n ∈ S, ∑ k ∈ S, ∫ x, f n x*f k x ∂μ)/(S.card : ℝ)^2 - |
| 2*(∑ n ∈ S, ∫ x, f n x*g x ∂μ)/(S.card : ℝ)+(∫ x, g x^2 ∂μ) := by |
| have he (x : Ω) : ((∑ n ∈ S, f n x)/(S.card : ℝ)-g x)^2 = |
| (∑ n ∈ S, ∑ k ∈ S, f n x*f k x)/(S.card : ℝ)^2 - |
| 2*(∑ n ∈ S, f n x*g x)/(S.card : ℝ)+g x^2 := by |
| have hsq : (∑ n ∈ S, ∑ k ∈ S, f n x*f k x) = (∑ n ∈ S, f n x)^2 := by |
| rw [pow_two]; simp_rw [Finset.sum_mul,Finset.mul_sum] |
| rw [hsq,← Finset.sum_mul] |
| ring |
| have hpi := integrable_finsetSum S (fun n hn => integrable_finsetSum S (fun k hk => hpair n hn k hk)) |
| have hmi := integrable_finsetSum S (fun n hn => hmix n hn) |
| have hiP : Integrable (fun x => (∑ n ∈ S, ∑ k ∈ S, f n x*f k x)/(S.card : ℝ)^2) μ := hpi.div_const _ |
| have hiM : Integrable (fun x => 2*(∑ n ∈ S, f n x*g x)/(S.card : ℝ)) μ := (hmi.const_mul 2).div_const _ |
| have hiSub : Integrable (fun x => (∑ n ∈ S, ∑ k ∈ S, f n x*f k x)/(S.card : ℝ)^2 - |
| 2*(∑ n ∈ S, f n x*g x)/(S.card : ℝ)) μ := hiP.sub hiM |
| have hpis : (∫ x, ∑ n ∈ S, ∑ k ∈ S, f n x*f k x ∂μ) = |
| ∑ n ∈ S, ∑ k ∈ S, ∫ x, f n x*f k x ∂μ := by |
| rw [integral_finsetSum S (fun n hn => integrable_finsetSum S (fun k hk => hpair n hn k hk))] |
| apply sum_congr rfl |
| intro n hn |
| exact integral_finsetSum S (fun k hk => hpair n hn k hk) |
| simp_rw [he] |
| rw [integral_add hiSub hg,integral_sub hiP hiM,integral_div,integral_div,integral_const_mul, |
| hpis,integral_finsetSum S (fun n hn => hmix n hn)] |
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|
| /-- Integrability of the full error follows from the same finite expansion. -/ |
| theorem sampling_average_square_integrable {ι Ω : Type*} [MeasurableSpace Ω] |
| (μ : Measure Ω) (S : Finset ι) (f : ι → Ω → ℝ) (g : Ω → ℝ) |
| (hpair : ∀ n ∈ S, ∀ k ∈ S, Integrable (fun x => f n x*f k x) μ) |
| (hmix : ∀ n ∈ S, Integrable (fun x => f n x*g x) μ) |
| (hg : Integrable (fun x => g x^2) μ) : |
| Integrable (fun x => ((∑ n ∈ S, f n x)/(S.card : ℝ)-g x)^2) μ := by |
| have hpi := integrable_finsetSum S (fun n hn => integrable_finsetSum S (fun k hk => hpair n hn k hk)) |
| have hmi := integrable_finsetSum S (fun n hn => hmix n hn) |
| have h : Integrable (fun x => (∑ n ∈ S, ∑ k ∈ S, f n x*f k x)/(S.card : ℝ)^2 - |
| 2*(∑ n ∈ S, f n x*g x)/(S.card : ℝ)+g x^2) μ := |
| ((hpi.div_const _).sub ((hmi.const_mul 2).div_const _)).add hg |
| apply h.congr |
| apply Eventually.of_forall |
| intro x |
| have hsq : (∑ n ∈ S, ∑ k ∈ S, f n x*f k x) = (∑ n ∈ S, f n x)^2 := by |
| rw [pow_two]; simp_rw [Finset.sum_mul,Finset.mul_sum] |
| dsimp only |
| rw [hsq,← Finset.sum_mul] |
| ring |
|
|
| /-- The distinct-pair and mixed main terms cancel to -A/P. Since A is |
| nonnegative, the surviving upper bound is the actual diagonal term plus 3ε. -/ |
| theorem sampling_finite_cancellation {ι : Type*} (S : Finset ι) (hS : S.Nonempty) |
| (B : ι → ι → ℝ) (M : ι → ℝ) (A ε : ℝ) (hA : 0 ≤ A) (hε : 0 ≤ ε) |
| (hpair : ∀ n ∈ S, ∀ k ∈ S, n ≠ k → |B n k-A| ≤ ε) |
| (hmix : ∀ n ∈ S, |M n-A| ≤ ε) : |
| (∑ n ∈ S, ∑ k ∈ S, B n k)/(S.card : ℝ)^2 - |
| 2*(∑ n ∈ S, M n)/(S.card : ℝ)+A ≤ |
| (∑ n ∈ S, B n n)/(S.card : ℝ)^2+3*ε := by |
| let N : ℝ := S.card |
| have hN : 0 < N := by dsimp [N]; exact_mod_cast hS.card_pos |
| have hN1 : 1 ≤ N := by dsimp [N]; exact_mod_cast (show 1 ≤ S.card from hS.card_pos) |
| have hp (n : ι) (hn : n ∈ S) : |
| (∑ k ∈ S, B n k) ≤ B n n+(N-1)*(A+ε) := by |
| have hoff : (∑ k ∈ S.erase n, B n k) ≤ ((S.erase n).card : ℝ)*(A+ε) := by |
| calc |
| _ ≤ ∑ _k ∈ S.erase n, (A+ε) := sum_le_sum (fun k hk => by |
| have hmem := mem_erase.mp hk |
| have hb := (abs_le.mp (hpair n hn k hmem.2 (Ne.symm hmem.1))).2 |
| linarith) |
| _ = _ := by simp; ring |
| have hc : ((S.erase n).card : ℝ) = N-1 := by |
| rw [card_erase_of_mem hn,Nat.cast_sub (by exact hS.card_pos),Nat.cast_one] |
| rw [hc] at hoff |
| have he := sum_erase_add S (fun k => B n k) hn |
| linarith |
| have hpall : (∑ n ∈ S, ∑ k ∈ S, B n k) ≤ |
| (∑ n ∈ S, B n n)+N*(N-1)*(A+ε) := by |
| calc |
| _ ≤ ∑ n ∈ S, (B n n+(N-1)*(A+ε)) := sum_le_sum hp |
| _ = _ := by rw [sum_add_distrib]; simp [N]; ring |
| have hmall : N*(A-ε) ≤ ∑ n ∈ S, M n := by |
| calc |
| _ = ∑ _n ∈ S, (A-ε) := by simp [N]; ring |
| _ ≤ _ := sum_le_sum (fun n hn => by have hh := (abs_le.mp (hmix n hn)).1; linarith) |
| change _/N^2-2*_/N+A ≤ _/N^2+3*ε |
| have hpdiv := div_le_div_of_nonneg_right hpall (sq_nonneg N) |
| have hmdiv := div_le_div_of_nonneg_right hmall hN.le |
| have hcancel : ((∑ n ∈ S, B n n)+N*(N-1)*(A+ε))/N^2- |
| 2*(N*(A-ε))/N+A = |
| (∑ n ∈ S, B n n)/N^2+3*ε-(A+ε)/N := by field_simp; ring |
| have hnneg : 0 ≤ (A+ε)/N := div_nonneg (add_nonneg hA hε) hN.le |
| simp only [mul_div_assoc] at hmdiv hcancel ⊢ |
| nlinarith |
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|
| end |
| end FiniteFactor |
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| end OAI |
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