| import Mathlib |
| import OAI.Combinatorics.IndependentSets.Reduction.RowBitsEquiv |
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| namespace OAI |
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| namespace LargeIndependentSets.BooleanJunta |
| open MeasureTheory Set Filter |
| open scoped BigOperators Classical NNReal ENNReal Topology |
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| lemma exists_dyadic_precision (n : ℕ) (L : ℝ≥0) {u : ℝ} (hu : 0 < u) : |
| ∃ m : ℕ, (L*(1/2:ℝ)^m)^2 < u/16 ∧ (n:ℝ)*m*(L*(1/2:ℝ)^m) < 1 := by |
| have hp : Tendsto (fun m : ℕ => (1/2:ℝ)^m) atTop (𝓝 0) := |
| tendsto_pow_atTop_nhds_zero_of_lt_one (by norm_num) (by norm_num) |
| have he : Tendsto (fun m : ℕ => (L*(1/2:ℝ)^m)^2) atTop (𝓝 0) := by |
| simpa using (hp.const_mul (L:ℝ)).pow 2 |
| have hn : Tendsto (fun m : ℕ => (n:ℝ)*m*(L*(1/2:ℝ)^m)) atTop (𝓝 0) := by |
| have h := (tendsto_self_mul_const_pow_of_lt_one (r:=(1/2:ℝ)) (by norm_num) (by norm_num)).const_mul ((n:ℝ)*L) |
| convert h using 1 <;> (try funext m) <;> ring_nf |
| exact ((he.eventually_lt_const (by positivity : 0 < u/16)).and |
| (hn.eventually_lt_const (by norm_num : (0:ℝ)<1))).exists |
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| lemma cubeSampler_mix {n m : ℕ} (S : Finset (Fin n)) |
| (p q : Fin n → Cube m × ℝ) : |
| cubeSampler (ProductAveraging.mix S (p,q)) = |
| ProductAveraging.mix S (cubeSampler p,cubeSampler q) := by |
| funext i |
| by_cases hi : i ∈ S <;> simp [cubeSampler, ProductAveraging.mix, hi] |
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| lemma seed_average_eq {n m : ℕ} (S : Finset (Fin n)) |
| {f : (Fin n → ℝ) → ℝ} (hf : Measurable f) (p : Fin n → Cube m × ℝ) : |
| ProductAveraging.average (seedLaw m) S (f ∘ cubeSampler) p = |
| ProductAveraging.average unitLaw S f (cubeSampler p) := by |
| unfold ProductAveraging.average |
| simp only [Function.comp_def, cubeSampler_mix] |
| have hm : Measurable (fun y : Fin n → ℝ => f (ProductAveraging.mix S (cubeSampler p,y))) := |
| hf.comp ((ProductAveraging.mix_measurable S).comp (measurable_const.prodMk measurable_id)) |
| exact ProductAveraging.integral_preserving (cubeSampler_preserving n m) hm.aestronglyMeasurable |
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|
| theorem real_cube_junta_uniform (L : ℝ≥0) {u : ℝ} (hu : 0 < u) : |
| ∃ J : ℕ, 1 ≤ J ∧ ∀ n : ℕ, ∀ f : (Fin n → ℝ) → ℝ, |
| LipschitzWith L f → (∀ x, |f x| ≤ 1) → |
| ∃ S : Finset (Fin n), S.card ≤ J ∧ |
| Integrable (fun x => (f x-ProductAveraging.average unitLaw S f x)^2) (unitCubeLaw n) ∧ |
| (∫ x, (f x-ProductAveraging.average unitLaw S f x)^2 ∂unitCubeLaw n) < u := by |
| obtain ⟨J,hJ,H⟩ := rows_junta_uniform (L:=(L:ℝ)+1) (u:=u/16) (by positivity) (by positivity) |
| refine ⟨J,hJ,?_⟩ |
| intro n f hf hb |
| obtain ⟨m,hmerr,hminf⟩ := exists_dyadic_precision n L hu |
| have hI : (∑ i, ∑ k, rowInfluence (f ∘ cellPoint (n:=n) (m:=m)) i k) ≤ (L:ℝ)+1 := |
| (total_rowInfluence_cell_bound hf hb).trans (by linarith) |
| obtain ⟨S,hS,g,hg,hdep,he⟩ := H n m (f ∘ cellPoint) (fun x => hb _) hI |
| let F : (Fin n → Cube m × ℝ) → ℝ := f ∘ cubeSampler |
| let G : (Fin n → Cube m × ℝ) → ℝ := fun p => g (fun i => (p i).1) |
| let C : (Fin n → Cube m × ℝ) → ℝ := fun p => f (cellPoint (fun i => (p i).1)) |
| have hFm : Measurable F := hf.continuous.measurable.comp (cubeSampler_preserving n m).measurable |
| have hGm : Measurable G := (measurable_of_countable g).comp (cells_preserving n m).measurable |
| have hCm : Measurable C := (measurable_of_countable (f ∘ cellPoint)).comp (cells_preserving n m).measurable |
| have hFb : ∀ p, |F p| ≤ 1 := fun p => hb _ |
| have hGb : ∀ p, |G p| ≤ 1 := fun p => hg _ |
| have hCb : ∀ p, |C p| ≤ 1 := fun p => hb _ |
| have herr (a b : ℝ) (ha : |a| ≤ 1) (hb : |b| ≤ 1) : |a-b| ≤ 2 := |
| (abs_sub a b).trans (by linarith) |
| have hij : Integrable (fun p => (F p-G p)^2) (seedCubeLaw n m) := |
| ProductAveraging.bounded_sq_integrable (hFm.sub hGm) (fun p => herr _ _ (hFb p) (hGb p)) |
| have hic : Integrable (fun p => (C p-G p)^2) (seedCubeLaw n m) := |
| ProductAveraging.bounded_sq_integrable (hCm.sub hGm) (fun p => herr _ _ (hCb p) (hGb p)) |
| have hice : (∫ p, (C p-G p)^2 ∂seedCubeLaw n m) = 𝔼 q, (f (cellPoint q)-g q)^2 := by |
| calc |
| _ = ∫ q, (f (cellPoint q)-g q)^2 ∂cellLaw n m := |
| ProductAveraging.integral_preserving (cells_preserving n m) Integrable.of_finite.aestronglyMeasurable |
| _ = _ := integral_uniformLaw _ |
| have herrInt : (∫ p, (F p-G p)^2 ∂seedCubeLaw n m) < u/4 := by |
| have hle : (∫ p, (F p-G p)^2 ∂seedCubeLaw n m) ≤ |
| 2*(L*(1/2:ℝ)^m)^2 + 2*(𝔼 q, (f (cellPoint q)-g q)^2) := by |
| have hdom : Integrable (fun p => 2*(L*(1/2:ℝ)^m)^2 + 2*(C p-G p)^2) (seedCubeLaw n m) := |
| (integrable_const _).add (hic.const_mul 2) |
| have hpt : ∀ᵐ p ∂seedCubeLaw n m, (F p-G p)^2 ≤ |
| 2*(L*(1/2:ℝ)^m)^2+2*(C p-G p)^2 := by |
| filter_upwards [sample_cell_error hf] with p hp |
| have hp2 : (F p-C p)^2 ≤ (L*(1/2:ℝ)^m)^2 := by |
| have hn : 0 ≤ L*(1/2:ℝ)^m := by positivity |
| simpa only [sq_abs, F, C, Function.comp_def] using (sq_le_sq₀ (abs_nonneg _) hn).mpr hp |
| nlinarith [sq_nonneg ((F p-C p)-(C p-G p))] |
| have hn := integral_mono_ae hij hdom hpt |
| have him : Integrable (fun p => 2*(C p-G p)^2) (seedCubeLaw n m) := hic.const_mul 2 |
| rw [integral_add (integrable_const _) him] at hn |
| simp only [integral_const_mul, integral_const, measureReal_def, measure_univ, |
| ENNReal.toReal_one, one_smul, hice] at hn |
| simpa using hn |
| dsimp only [Function.comp_def] at he |
| linarith |
| have hGdep : ∀ p q, (∀ i ∈ S, p i = q i) → G p = G q := by |
| intro p q hpq |
| exact hdep _ _ (fun i hi => congrArg Prod.fst (hpq i hi)) |
| have hproj := ProductAveraging.junta_average_error (seedLaw m) S hFm hGm hFb hGb hGdep |
| have hAm := ProductAveraging.average_measurable unitLaw S hf.continuous.measurable |
| have hAb := ProductAveraging.average_bound unitLaw S hb |
| have hfinali : Integrable (fun x => (f x-ProductAveraging.average unitLaw S f x)^2) (unitCubeLaw n) := |
| ProductAveraging.bounded_sq_integrable (hf.continuous.measurable.sub hAm) |
| (fun x => herr _ _ (hb x) (hAb x)) |
| have hfinale : (∫ p, (F p-ProductAveraging.average (seedLaw m) S F p)^2 ∂seedCubeLaw n m) = |
| ∫ x, (f x-ProductAveraging.average unitLaw S f x)^2 ∂unitCubeLaw n := by |
| simp only [F, seed_average_eq S hf.continuous.measurable] |
| exact ProductAveraging.integral_preserving (cubeSampler_preserving n m) hfinali.aestronglyMeasurable |
| refine ⟨S,hS,hfinali,?_⟩ |
| change (∫ p, (F p-ProductAveraging.average (seedLaw m) S F p)^2 ∂seedCubeLaw n m) ≤ |
| 4*(∫ p, (F p-G p)^2 ∂seedCubeLaw n m) at hproj |
| rw [hfinale] at hproj |
| linarith |
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| end LargeIndependentSets.BooleanJunta |
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| end OAI |
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