| import Mathlib |
| import OAI.Computability.MinUncut.Analysis.HermiteComponent |
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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 memLpMaskProjectDualFintype {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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| omit [AddCommGroup W] [Module F₂ W] in |
| lemma memLp_maskProject (S : Finset R) (u : (R → W) → (ι → ℝ) → ℝ) |
| (hu : ∀ B, MemLp (u B) 2 (γpi ι)) (B : R → W) : |
| MemLp (fun c => maskProject S (fun D => u D c) B) 2 (γpi ι) := |
| memLp_expect _ (fun D => (hu D).const_mul _) |
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| lemma expect_sqrt_le {T : Type*} [Fintype T] [Nonempty T] (f : T → ℝ) (hf : ∀ t, 0 ≤ f t) : |
| (𝔼 t, Real.sqrt (f t)) ≤ Real.sqrt (𝔼 t, f t) := by |
| let g := fun t => Real.sqrt (f t) |
| have hz := Finset.expect_nonneg (s := Finset.univ) (f := fun t => (g t - (𝔼 u, g u))^2) |
| (fun _ _ => sq_nonneg _) |
| have hg2 : ∀ t, g t ^ 2 = f t := fun t => Real.sq_sqrt (hf t) |
| have hid : (𝔼 t, (g t - (𝔼 u, g u))^2) = (𝔼 t, f t) - (𝔼 t, g t)^2 := by |
| simp only [sub_sq, hg2, |
| Finset.expect_add_distrib, Finset.expect_sub_distrib, ← Finset.expect_mul, |
| ← Finset.mul_expect, Fintype.expect_const] |
| ring |
| rw [hid] at hz |
| exact (Real.le_sqrt (Finset.expect_nonneg (fun t _ => show 0≤g t from Real.sqrt_nonneg _)) (Finset.expect_nonneg (fun t _ => hf t))).2 (by linarith) |
| end MinUncut.RowNoise |
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| end |
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| end OAI |
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