| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.MemLpMaskProject |
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| namespace OAI |
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| 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 maskHermiteDualFintype {U : Type*} |
| [AddCommGroup U] [Module F₂ U] [Fintype U] : |
| Fintype (Module.Dual F₂ U) := BinaryFourier.dualFintype |
| variable {R W ι : Type*} [Fintype R] [DecidableEq R] [Fintype W] [DecidableEq W] |
| [AddCommGroup W] [Module F₂ W] [Fintype ι] |
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| def maskHermite (S : Finset R) (I : ι → ℕ) (u : (R → W) → (ι → ℝ) → ℝ) |
| (B : R → W) (c : ι → ℝ) : ℝ := maskProject S (fun D => hermiteComponent I u D c) B |
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| omit [AddCommGroup W] [Module F₂ W] in |
| lemma memLp_maskHermite (S : Finset R) (I : ι → ℕ) (u : (R → W) → (ι → ℝ) → ℝ) |
| (B : R → W) : MemLp (maskHermite S I u B) 2 (γpi ι) := |
| memLp_maskProject S _ (fun _ => (memLp_psi I).mul_const _) B |
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| lemma oddAtom_maskHermite_mean (r : R) (one : W) (S : Finset R) (I : ι → ℕ) |
| (u : (R → W) → (ι → ℝ) → ℝ) (hu : ∀ B, MemLp (u B) 2 (γpi ι)) : |
| (𝔼 D, ∫ c, oddAtom r one (fun B => maskHermite S I u B c) D ∂γpi ι) ≤ |
| (2:ℝ)^Fintype.card R * Real.sqrt |
| (𝔼 D, ∫ c, (oddAtom r one (fun B => u B c) D)^2 ∂γpi ι) := by |
| have hh : ∀ B, MemLp (hermiteComponent I u B) 2 (γpi ι) := fun B => (memLp_psi I).mul_const _ |
| have hi (D : Rest (W := W) r) := (memLp_oddAtom r one (maskHermite S I u) |
| (memLp_maskHermite S I u) D).integrable (by norm_num) |
| have hj (D : Rest (W := W) r) := (memLp_oddAtom r one (hermiteComponent I u) hh D).integrable (by norm_num) |
| calc |
| _ = ∫ c, (𝔼 D, oddAtom r one (fun B => maskHermite S I u B c) D) ∂γpi ι := (integral_expect _ hi).symm |
| _ ≤ ∫ c, (2:ℝ)^Fintype.card R * (𝔼 D, oddAtom r one (fun B => hermiteComponent I u B c) D) ∂γpi ι := by |
| apply integral_mono |
| · simp only [Finset.expect_eq_sum_div_card] |
| exact (integrable_finsetSum _ (fun D _ => hi D)).div_const _ |
| · simp only [Finset.expect_eq_sum_div_card] |
| exact ((integrable_finsetSum _ (fun D _ => hj D)).div_const _).const_mul _ |
| · intro c |
| exact oddAtom_maskProject_mean r one S (fun B => hermiteComponent I u B c) |
| _ = (2:ℝ)^Fintype.card R * (𝔼 D, ∫ c, oddAtom r one (fun B => hermiteComponent I u B c) D ∂γpi ι) := by |
| rw [integral_const_mul, integral_expect _ hj] |
| _ ≤ (2:ℝ)^Fintype.card R * (𝔼 D, Real.sqrt (∫ c, (oddAtom r one (fun B => u B c) D)^2 ∂γpi ι)) := by |
| exact mul_le_mul_of_nonneg_left (Finset.expect_le_expect (fun D _ => |
| oddAtom_hermiteComponent_mean r one I u hu D)) (by positivity) |
| _ ≤ _ := mul_le_mul_of_nonneg_left (expect_sqrt_le _ (fun _ => integral_nonneg (fun _ => sq_nonneg _))) (by positivity) |
| end MinUncut.RowNoise |
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| end |
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| end OAI |
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