| import Mathlib |
|
|
| namespace OAI |
|
|
| open MeasureTheory |
| open scoped BigOperators |
| |
| namespace Invariant233.FourierSampling |
|
|
| theorem norm_tensor_sum_le {ι κ : Type*} (s : Finset ι) (t : Finset κ) |
| (a : ι → ℂ) (b : κ → ℂ) (M : ι → κ → ℂ) (ε : ℝ) |
| (hM : ∀ i ∈ s, ∀ j ∈ t, ‖M i j‖ ≤ ε) : |
| ‖∑ i ∈ s, ∑ j ∈ t, a i * b j * M i j‖ ≤ |
| (∑ i ∈ s, ‖a i‖) * (∑ j ∈ t, ‖b j‖) * ε := by |
| calc |
| _ ≤ ∑ i ∈ s, ‖∑ j ∈ t, a i * b j * M i j‖ := norm_sum_le _ _ |
| _ ≤ ∑ i ∈ s, ∑ j ∈ t, ‖a i * b j * M i j‖ := |
| Finset.sum_le_sum fun i hi => norm_sum_le _ _ |
| _ ≤ ∑ i ∈ s, ∑ j ∈ t, ‖a i‖ * ‖b j‖ * ε := by |
| apply Finset.sum_le_sum |
| intro i hi |
| apply Finset.sum_le_sum |
| intro j hj |
| rw [norm_mul, norm_mul] |
| exact mul_le_mul_of_nonneg_left (hM i hi j hj) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) |
| _ = _ := by |
| simp_rw [← Finset.sum_mul, ← Finset.mul_sum] |
| rw [← Finset.sum_mul] |
|
|
| noncomputable def polynomial (s : Finset ℤ) (a : ℤ → ℂ) |
| (x : UnitAddCircle) : ℂ := ∑ i ∈ s, a i * fourier i x |
|
|
| theorem integrable_fourier_pair {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsFiniteMeasure μ] |
| {X Y : Ω → UnitAddCircle} (hX : Measurable X) (hY : Measurable Y) |
| (i j : ℤ) : Integrable (fun ω => fourier i (X ω) * fourier j (Y ω)) μ := by |
| apply Integrable.of_bound |
| (((fourier i).continuous.measurable.comp hX).mul |
| ((fourier j).continuous.measurable.comp hY)).aestronglyMeasurable 1 |
| filter_upwards with ω |
| change ‖fourier i (X ω) * fourier j (Y ω)‖ ≤ 1 |
| rw [norm_mul] |
| have hi : ‖fourier i (X ω)‖ ≤ 1 := by |
| simpa only [fourier_norm] using (fourier i).norm_coe_le_norm (X ω) |
| have hj : ‖fourier j (Y ω)‖ ≤ 1 := by |
| simpa only [fourier_norm] using (fourier j).norm_coe_le_norm (Y ω) |
| nlinarith [norm_nonneg (fourier i (X ω)), norm_nonneg (fourier j (Y ω))] |
|
|
| theorem integral_polynomial_mul {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsFiniteMeasure μ] |
| {X Y : Ω → UnitAddCircle} (hX : Measurable X) (hY : Measurable Y) |
| (s t : Finset ℤ) (a b : ℤ → ℂ) : |
| (∫ ω, polynomial s a (X ω) * polynomial t b (Y ω) ∂μ) = |
| ∑ i ∈ s, ∑ j ∈ t, a i * b j * |
| ∫ ω, fourier i (X ω) * fourier j (Y ω) ∂μ := by |
| have hterm : ∀ i j, Integrable |
| (fun ω => a i * b j * (fourier i (X ω) * fourier j (Y ω))) μ := |
| fun i j => (integrable_fourier_pair hX hY i j).const_mul _ |
| have hexpand : ∀ ω, |
| polynomial s a (X ω) * polynomial t b (Y ω) = |
| ∑ i ∈ s, ∑ j ∈ t, a i * b j * (fourier i (X ω) * fourier j (Y ω)) := by |
| intro ω |
| rw [polynomial, polynomial, Finset.sum_mul_sum] |
| apply Finset.sum_congr rfl |
| intro i hi |
| apply Finset.sum_congr rfl |
| intro j hj |
| ring |
| simp_rw [hexpand] |
| rw [integral_finsetSum s (fun i hi => integrable_finsetSum t (fun j hj => hterm i j))] |
| apply Finset.sum_congr rfl |
| intro i hi |
| rw [integral_finsetSum t (fun j hj => hterm i j)] |
| simp_rw [integral_const_mul] |
|
|
| theorem polynomial_error_le {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {X Y : Ω → UnitAddCircle} (hX : Measurable X) (hY : Measurable Y) |
| (s t : Finset ℤ) (hs : 0 ∈ s) (ht : 0 ∈ t) (a b : ℤ → ℂ) |
| {ε : ℝ} (hε : 0 ≤ ε) |
| (hchar : ∀ i ∈ s, ∀ j ∈ t, (i, j) ≠ (0, 0) → |
| ‖∫ ω, fourier i (X ω) * fourier j (Y ω) ∂μ‖ ≤ ε) : |
| ‖(∫ ω, polynomial s a (X ω) * polynomial t b (Y ω) ∂μ) - a 0 * b 0‖ ≤ |
| (∑ i ∈ s, ‖a i‖) * (∑ j ∈ t, ‖b j‖) * ε := by |
| classical |
| let M : ℤ → ℤ → ℂ := fun i j => |
| (∫ ω, fourier i (X ω) * fourier j (Y ω) ∂μ) - if i = 0 ∧ j = 0 then 1 else 0 |
| have hM : ∀ i ∈ s, ∀ j ∈ t, ‖M i j‖ ≤ ε := by |
| intro i hi j hj |
| by_cases h : i = 0 ∧ j = 0 |
| · rcases h with ⟨rfl, rfl⟩ |
| simpa [M] using hε |
| · simpa [M, h] using hchar i hi j hj (by simpa using h) |
| have hconstant : (∑ i ∈ s, ∑ j ∈ t, |
| a i * b j * (if i = 0 ∧ j = 0 then (1 : ℂ) else 0)) = a 0 * b 0 := by |
| rw [Finset.sum_eq_single 0] |
| · simp [ht] |
| · intro i hi hi0 |
| simp [hi0] |
| · exact fun h => (h hs).elim |
| rw [integral_polynomial_mul hX hY s t a b, ← hconstant] |
| have heq : (∑ i ∈ s, ∑ j ∈ t, a i * b j * |
| ∫ ω, fourier i (X ω) * fourier j (Y ω) ∂μ) - |
| (∑ i ∈ s, ∑ j ∈ t, a i * b j * (if i = 0 ∧ j = 0 then (1 : ℂ) else 0)) = |
| ∑ i ∈ s, ∑ j ∈ t, a i * b j * M i j := by |
| simp only [M, mul_sub, Finset.sum_sub_distrib] |
| rw [heq] |
| exact norm_tensor_sum_le s t a b M ε hM |
|
|
| theorem polynomial_error_le_card {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {X Y : Ω → UnitAddCircle} (hX : Measurable X) (hY : Measurable Y) |
| (s t : Finset ℤ) (hs : 0 ∈ s) (ht : 0 ∈ t) (a b : ℤ → ℂ) |
| (ha : ∀ i ∈ s, ‖a i‖ ≤ 1) (hb : ∀ j ∈ t, ‖b j‖ ≤ 1) |
| {ε : ℝ} (hε : 0 ≤ ε) |
| (hchar : ∀ i ∈ s, ∀ j ∈ t, (i, j) ≠ (0, 0) → |
| ‖∫ ω, fourier i (X ω) * fourier j (Y ω) ∂μ‖ ≤ ε) : |
| ‖(∫ ω, polynomial s a (X ω) * polynomial t b (Y ω) ∂μ) - a 0 * b 0‖ ≤ |
| (s.card : ℝ) * t.card * ε := by |
| apply (polynomial_error_le hX hY s t hs ht a b hε hchar).trans |
| apply mul_le_mul_of_nonneg_right _ hε |
| have has : (∑ i ∈ s, ‖a i‖) ≤ (s.card : ℝ) := by |
| simpa using Finset.sum_le_sum ha |
| have hbs : (∑ j ∈ t, ‖b j‖) ≤ (t.card : ℝ) := by |
| simpa using Finset.sum_le_sum hb |
| exact mul_le_mul has hbs (Finset.sum_nonneg fun j hj => norm_nonneg _) (by positivity) |
|
|
| theorem integral_product_replacement {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} {f g p q : Ω → ℂ} |
| (hfg : Integrable (fun ω => f ω * g ω) μ) |
| (hpq : Integrable (fun ω => p ω * q ω) μ) |
| (hfp : Integrable (fun ω => f ω - p ω) μ) |
| (hgq : Integrable (fun ω => g ω - q ω) μ) |
| (hg : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ 1) (hp : ∀ᵐ ω ∂μ, ‖p ω‖ ≤ 1) : |
| ‖(∫ ω, f ω * g ω ∂μ) - ∫ ω, p ω * q ω ∂μ‖ ≤ |
| (∫ ω, ‖f ω - p ω‖ ∂μ) + ∫ ω, ‖g ω - q ω‖ ∂μ := by |
| rw [← integral_sub hfg hpq, ← integral_add hfp.norm hgq.norm] |
| apply (norm_integral_le_integral_norm _).trans |
| apply integral_mono_ae (hfg.sub hpq).norm (hfp.norm.add hgq.norm) |
| filter_upwards [hg, hp] with ω hgω hpω |
| calc |
| ‖f ω * g ω - p ω * q ω‖ = ‖(f ω - p ω) * g ω + p ω * (g ω - q ω)‖ := by |
| congr 1 |
| ring |
| _ ≤ ‖(f ω - p ω) * g ω‖ + ‖p ω * (g ω - q ω)‖ := norm_add_le _ _ |
| _ ≤ ‖f ω - p ω‖ + ‖g ω - q ω‖ := by |
| rw [norm_mul, norm_mul] |
| have h₁ := mul_le_mul_of_nonneg_left hgω (norm_nonneg (f ω - p ω)) |
| have h₂ := mul_le_mul_of_nonneg_right hpω (norm_nonneg (g ω - q ω)) |
| simpa only [mul_one, one_mul] using add_le_add h₁ h₂ |
|
|
| theorem test_error_le {Ω : Type*} [MeasurableSpace Ω] |
| {μ : Measure Ω} [IsProbabilityMeasure μ] |
| {X Y : Ω → UnitAddCircle} (hX : Measurable X) (hY : Measurable Y) |
| (s t : Finset ℤ) (hs : 0 ∈ s) (ht : 0 ∈ t) (a b : ℤ → ℂ) |
| {f g : Ω → ℂ} |
| (hfg : Integrable (fun ω => f ω * g ω) μ) |
| (hpq : Integrable (fun ω => polynomial s a (X ω) * polynomial t b (Y ω)) μ) |
| (hfp : Integrable (fun ω => f ω - polynomial s a (X ω)) μ) |
| (hgq : Integrable (fun ω => g ω - polynomial t b (Y ω)) μ) |
| (hg : ∀ᵐ ω ∂μ, ‖g ω‖ ≤ 1) |
| (hp : ∀ᵐ ω ∂μ, ‖polynomial s a (X ω)‖ ≤ 1) |
| (ha : ∀ i ∈ s, ‖a i‖ ≤ 1) (hb : ∀ j ∈ t, ‖b j‖ ≤ 1) |
| {ε : ℝ} (hε : 0 ≤ ε) |
| (hchar : ∀ i ∈ s, ∀ j ∈ t, (i, j) ≠ (0, 0) → |
| ‖∫ ω, fourier i (X ω) * fourier j (Y ω) ∂μ‖ ≤ ε) : |
| ‖(∫ ω, f ω * g ω ∂μ) - a 0 * b 0‖ ≤ |
| (∫ ω, ‖f ω - polynomial s a (X ω)‖ ∂μ) + |
| (∫ ω, ‖g ω - polynomial t b (Y ω)‖ ∂μ) + |
| (s.card : ℝ) * t.card * ε := by |
| apply (norm_sub_le_norm_sub_add_norm_sub _ (∫ ω, polynomial s a (X ω) * polynomial t b (Y ω) ∂μ) _).trans |
| exact add_le_add |
| (integral_product_replacement hfg hpq hfp hgq hg hp) |
| (polynomial_error_le_card hX hY s t hs ht a b ha hb hε hchar) |
|
|
| theorem sampled_fourier_pair (v w n n' : ℤ) (z : ℝ) : |
| fourier v (((n : ℝ) * z : ℝ) : UnitAddCircle) * |
| fourier w (((n' : ℝ) * z : ℝ) : UnitAddCircle) = |
| Complex.exp (2 * (Real.pi : ℂ) * Complex.I * |
| ((v * n + w * n' : ℤ) : ℂ) * (z : ℂ)) := by |
| rw [fourier_coe_apply, fourier_coe_apply, ← Complex.exp_add] |
| congr 1 |
| push_cast |
| ring |
| |
| theorem sampled_fourier_pair_fract (v w n n' : ℤ) (z : ℝ) : |
| fourier v ((Int.fract ((n : ℝ) * z) : ℝ) : UnitAddCircle) * |
| fourier w ((Int.fract ((n' : ℝ) * z) : ℝ) : UnitAddCircle) = |
| Complex.exp (2 * (Real.pi : ℂ) * Complex.I * |
| ((v * n + w * n' : ℤ) : ℂ) * (z : ℂ)) := by |
| simpa only [AddCircle.coe_fract] using sampled_fourier_pair v w n n' z |
| |
| theorem marginal_integral_le {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| {μ : Measure Ω} {ν : Measure X} {F : Ω → X} (hF : Measurable F) |
| {C : ℝ} (hC : 0 ≤ C) (hmap : μ.map F ≤ ENNReal.ofReal C • ν) |
| {f : X → ℝ} (hf : Integrable f ν) (hf0 : ∀ x, 0 ≤ f x) : |
| (∫ ω, f (F ω) ∂μ) ≤ C * ∫ x, f x ∂ν := by |
| have hfi := hf.smul_measure (c := ENNReal.ofReal C) ENNReal.ofReal_ne_top |
| rw [← integral_map hF.aemeasurable (hfi.mono_measure hmap).aestronglyMeasurable] |
| calc |
| _ ≤ ∫ x, f x ∂(ENNReal.ofReal C • ν) := |
| integral_mono_measure hmap (Filter.Eventually.of_forall hf0) hfi |
| _ = _ := by rw [integral_smul_measure, ENNReal.toReal_ofReal hC, smul_eq_mul] |
| |
| theorem marginal_l1_error_le {Ω X : Type*} [MeasurableSpace Ω] [MeasurableSpace X] |
| {μ : Measure Ω} {ν : Measure X} {F : Ω → X} (hF : Measurable F) |
| {C : ℝ} (hC : 0 ≤ C) (hmap : μ.map F ≤ ENNReal.ofReal C • ν) |
| {f p : X → ℂ} (hfp : Integrable (fun x => f x - p x) ν) : |
| (∫ ω, ‖f (F ω) - p (F ω)‖ ∂μ) ≤ C * ∫ x, ‖f x - p x‖ ∂ν := |
| marginal_integral_le hF hC hmap hfp.norm (fun _ => norm_nonneg _) |
| |
| end Invariant233.FourierSampling |
| |
| end OAI |
| |
