| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Density |
|
|
| namespace OAI |
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|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory Filter |
| open scoped Topology NNReal |
| open scoped BigOperators |
| namespace MinUncut.Composition |
| open BinaryFourier OuterSmoothness |
| variable {W : Type*} [AddCommGroup W] [Module F₂ W] [Fintype W] |
| local instance tripleKernelDecidableEq : DecidableEq W := Classical.decEq W |
| attribute [local instance] BinaryFourier.dualFintype |
|
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| def tripleKernel (f : W → ℝ) (v : W × W × W) : ℝ := |
| f v.1 * f v.2.1 * f v.2.2 * f (v.1+v.2.1+v.2.2) |
|
|
| lemma tripleKernel_mean (f : W → ℝ) : |
| (𝔼 v : W × W × W, tripleKernel f v) = |
| ∑ α : Module.Dual F₂ W, coefficient f α ^ 4 := by |
| unfold tripleKernel |
| rw [expect_pair] |
| simp_rw [expect_pair] |
| exact fourth_moment f |
|
|
| omit [Module F₂ W] [Fintype W] in |
| lemma tripleKernel_abs (f : W → ℝ) {M : ℝ} (hM : 0 ≤ M) |
| (hf : ∀ w, |f w| ≤ M) (v : W × W × W) : |tripleKernel f v| ≤ M^4 := by |
| simp only [tripleKernel, abs_mul] |
| calc _ ≤ M * M * M * M := by |
| gcongr <;> exact hf _ |
| _ = _ := by ring |
|
|
| theorem adaptive_fourth_moment {B : Type*} [Fintype B] [Nonempty B] |
| (p : Density (B × (W × W × W))) (f : B → W → ℝ) {M u : ℝ} |
| (hM : 0 ≤ M) (hu : 0 < u) (hf : ∀ b w, |f b w| ≤ M) : |
| (𝔼 z, p z * tripleKernel (remainder (f z.1) u) z.2) ≤ |
| u^2 * M^2 + 2 * p.tvUniform * (M + M^2/u)^4 := by |
| let g : B × (W × W × W) → ℝ := fun z => tripleKernel (remainder (f z.1) u) z.2 |
| have hMr : 0 ≤ M + M^2/u := by positivity |
| have hg (z : B × (W × W × W)) : |g z| ≤ (M + M^2/u)^4 := |
| tripleKernel_abs _ hMr (remainder_abs_bound (f z.1) hM hu (hf z.1)) z.2 |
| have hdiff := Density.expect_difference p g (pow_nonneg hMr _) hg |
| have hmean : (𝔼 z, g z) ≤ u^2 * M^2 := by |
| change (𝔼 z : B × (W × W × W), tripleKernel (remainder (f z.1) u) z.2) ≤ _ |
| rw [expect_pair] |
| simp_rw [tripleKernel_mean] |
| calc _ ≤ 𝔼 b : B, u^2 * M^2 := Finset.expect_le_expect (fun b _ => |
| remainder_fourth_bound (f b) hM hu (hf b)) |
| _ = _ := Fintype.expect_const _ |
| have habs := le_abs_self ((𝔼 z, p z * g z) - (𝔼 z, g z)) |
| dsimp only [g] at hdiff hmean habs |
| linarith |
|
|
| end MinUncut.Composition |
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| end |
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| end OAI |
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