| import Mathlib |
| import OAI.Computability.MinUncut.Analysis.OddAtom |
| import OAI.Computability.MinUncut.Estimates.IntegralExpect |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Polynomial Filter |
| open scoped BigOperators Topology |
| open MeasureTheory ProbabilityTheory WithLp |
| open scoped BigOperators RealInnerProductSpace |
| namespace MinUncut.RowNoise |
| open MeasureTheory ProbabilityTheory BinaryFourier GaussianHermite |
| open scoped BigOperators |
| local instance hermiteComponentDualFintype {U : Type*} |
| [AddCommGroup U] [Module F₂ U] [Fintype U] : |
| Fintype (Module.Dual F₂ U) := BinaryFourier.dualFintype |
| variable {R W ι : Type*} [DecidableEq R] [Fintype W] |
| [AddCommGroup W] [Module F₂ W] [Fintype ι] |
|
|
| lemma memLp_expect {T Ω : Type*} [Fintype T] [MeasurableSpace Ω] {μ : Measure Ω} |
| (f : T → Ω → ℝ) (hf : ∀ t, MemLp (f t) 2 μ) : |
| MemLp (fun x => 𝔼 t, f t x) 2 μ := by |
| simp only [Finset.expect_eq_sum_div_card, div_eq_mul_inv] |
| exact (memLp_finsetSum _ (fun t _ => hf t)).mul_const _ |
|
|
| lemma memLp_rowCoefficient (r : R) (u : (R → W) → (ι → ℝ) → ℝ) |
| (hu : ∀ B, MemLp (u B) 2 (γpi ι)) (α : Module.Dual F₂ W) (D : Rest (W := W) r) : |
| MemLp (fun c => rowCoefficient r (fun B => u B c) α D) 2 (γpi ι) := by |
| exact memLp_expect _ (fun b => (hu (joinRow r b D)).mul_const _) |
|
|
| lemma memLp_oddAtom (r : R) (one : W) (u : (R → W) → (ι → ℝ) → ℝ) |
| (hu : ∀ B, MemLp (u B) 2 (γpi ι)) (D : Rest (W := W) r) : |
| MemLp (fun c => oddAtom r one (fun B => u B c) D) 2 (γpi ι) := by |
| let fs : Module.Dual F₂ W → (ι → ℝ) → ℝ := fun α c => |
| if α one=1 then |rowCoefficient r (fun B => u B c) α D| else 0 |
| have hfs (α : Module.Dual F₂ W) : MemLp (fs α) 2 (γpi ι) := by |
| by_cases h : α one=1 |
| · simpa [fs, h, ← Real.norm_eq_abs] using (memLp_rowCoefficient r u hu α D).norm |
| · simp [fs, h] |
| have hh : MemLp (Finset.univ.sup' Finset.univ_nonempty fs) 2 (γpi ι) := |
| Finset.sup'_induction (p := fun f => MemLp f 2 (γpi ι)) Finset.univ_nonempty fs (fun _ hf _ hg => hf.sup hg) (fun α _ => hfs α) |
| convert hh using 1 |
| funext c |
| simp only [Finset.sup'_apply, fs, oddAtom] |
| |
| def hermiteComponent (I : ι → ℕ) (u : (R → W) → (ι → ℝ) → ℝ) |
| (B : R → W) (c : ι → ℝ) : ℝ := psi I c * GaussianHermite.coeff (u B) I |
| |
| lemma rowCoefficient_hermiteComponent (r : R) (I : ι → ℕ) |
| (u : (R → W) → (ι → ℝ) → ℝ) (hu : ∀ B, MemLp (u B) 2 (γpi ι)) |
| (α : Module.Dual F₂ W) (D : Rest (W := W) r) (c : ι → ℝ) : |
| rowCoefficient r (fun B => hermiteComponent I u B c) α D = |
| psi I c * GaussianHermite.coeff (fun c' => rowCoefficient r (fun B => u B c') α D) I := by |
| simp only [rowCoefficient, hermiteComponent, GaussianHermite.coeff, mul_assoc, ← Finset.mul_expect] |
| congr 1 |
| simp_rw [← integral_mul_const] |
| rw [← integral_expect] |
| · congr 1 |
| funext c' |
| simp only [mul_assoc, ← Finset.mul_expect] |
| · intro b |
| exact ((memLp_psi I).integrable_mul (hu (joinRow r b D))).mul_const _ |
| |
| lemma oddAtom_hermiteComponent (r : R) (one : W) (I : ι → ℕ) |
| (u : (R → W) → (ι → ℝ) → ℝ) (hu : ∀ B, MemLp (u B) 2 (γpi ι)) |
| (D : Rest (W := W) r) (c : ι → ℝ) : |
| oddAtom r one (fun B => hermiteComponent I u B c) D ≤ |
| |psi I c| * ∫ c', |psi I c'| * oddAtom r one (fun B => u B c') D ∂γpi ι := by |
| apply oddAtom_le |
| · exact mul_nonneg (abs_nonneg _) (integral_nonneg (fun _ => mul_nonneg (abs_nonneg _) (oddAtom_nonneg _ _ _ _))) |
| intro α hα |
| rw [rowCoefficient_hermiteComponent r I u hu, abs_mul] |
| apply mul_le_mul_of_nonneg_left _ (abs_nonneg _) |
| calc |
| _ ≤ ∫ c', |psi I c' * rowCoefficient r (fun B => u B c') α D| ∂γpi ι := abs_integral_le_integral_abs |
| _ ≤ _ := by |
| apply integral_mono |
| · exact ((memLp_psi I).integrable_mul (memLp_rowCoefficient r u hu α D)).abs |
| · exact (memLp_psi I).norm.integrable_mul (memLp_oddAtom r one u hu D) |
| · intro c' |
| dsimp only |
| rw [abs_mul] |
| exact mul_le_mul_of_nonneg_left (rowCoefficient_le_oddAtom _ _ _ α hα _) (abs_nonneg _) |
| |
| lemma oddAtom_hermiteComponent_mean (r : R) (one : W) (I : ι → ℕ) |
| (u : (R → W) → (ι → ℝ) → ℝ) (hu : ∀ B, MemLp (u B) 2 (γpi ι)) |
| (D : Rest (W := W) r) : |
| (∫ c, oddAtom r one (fun B => hermiteComponent I u B c) D ∂γpi ι) ≤ |
| Real.sqrt (∫ c, (oddAtom r one (fun B => u B c) D)^2 ∂γpi ι) := by |
| have ha := memLp_oddAtom r one u hu D |
| have hh : ∀ B, MemLp (hermiteComponent I u B) 2 (γpi ι) := fun B => |
| (memLp_psi I).mul_const _ |
| have hc : 0 ≤ ∫ c', |psi I c'| * oddAtom r one (fun B => u B c') D ∂γpi ι := |
| integral_nonneg (fun c' => mul_nonneg (abs_nonneg _) (oddAtom_nonneg _ _ _ _)) |
| calc |
| _ ≤ ∫ c, |psi I c| * ∫ c', |psi I c'| * oddAtom r one (fun B => u B c') D ∂γpi ι ∂γpi ι := by |
| exact integral_mono ((memLp_oddAtom r one (hermiteComponent I u) hh D).integrable (by norm_num)) |
| (((memLp_psi I).integrable (by norm_num)).norm.mul_const _) |
| (oddAtom_hermiteComponent r one I u hu D) |
| _ = (∫ c, |psi I c| ∂γpi ι) * ∫ c', |psi I c'| * oddAtom r one (fun B => u B c') D ∂γpi ι := integral_mul_const _ _ |
| _ ≤ ∫ c', |psi I c'| * oddAtom r one (fun B => u B c') D ∂γpi ι := by |
| simpa only [one_mul] using mul_le_mul_of_nonneg_right (psi_l1 I) hc |
| _ ≤ _ := by |
| have h := integral_norm_mul_le_sqrt (memLp_psi I) ha |
| have hn : (∫ c, psi I c^2 ∂γpi ι)=1 := by simpa [sq] using psi_product I I |
| simpa only [abs_of_nonneg (oddAtom_nonneg _ _ _ _), hn, Real.sqrt_one, one_mul] using h |
| end MinUncut.RowNoise |
|
|
| end |
|
|
| end OAI |
|
|
