| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.Density |
| import OAI.Computability.MinUncut.Estimates.EuclideanInvariantScore |
|
|
| 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.Inner |
| open scoped BigOperators |
|
|
| variable {m n : ℕ} |
|
|
| def faceCorrelation (v : Point m n → ℝ) : ℝ := |
| Finset.univ.sup' Finset.univ_nonempty |
| (fun z : Code m n => |𝔼 x : Point m n, v x * BinaryFourier.sign (z.val x)|) |
| |
| lemma code_correlation_le (v : Point m n → ℝ) (z : Code m n) : |
| |𝔼 x : Point m n, v x * BinaryFourier.sign (z.val x)| ≤ faceCorrelation v := by |
| exact Finset.le_sup' (fun w : Code m n => |𝔼 x : Point m n, v x * BinaryFourier.sign (w.val x)|) (Finset.mem_univ z) |
|
|
| lemma faceCorrelation_nonneg (v : Point m n → ℝ) : 0 ≤ faceCorrelation v := |
| (abs_nonneg _).trans (code_correlation_le v 0) |
|
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| lemma faceCorrelation_le {v : Point m n → ℝ} {a : ℝ} |
| (ha : ∀ z : Code m n, |𝔼 x : Point m n, v x * BinaryFourier.sign (z.val x)| ≤ a) : |
| faceCorrelation v ≤ a := Finset.sup'_le _ _ (fun z _ => ha z) |
|
|
| lemma faceCorrelation_le_l1 (v : Point m n → ℝ) : |
| faceCorrelation v ≤ 𝔼 x : Point m n, |v x| := by |
| apply faceCorrelation_le |
| intro z |
| calc |
| _ ≤ 𝔼 x : Point m n, |v x * BinaryFourier.sign (z.val x)| := |
| Finset.abs_expect_le _ _ |
| _ = _ := by simp only [abs_mul, BinaryFourier.sign_abs, mul_one] |
|
|
| lemma faceCorrelation_add_le (v w : Point m n → ℝ) : |
| faceCorrelation (v+w) ≤ faceCorrelation v + faceCorrelation w := by |
| apply faceCorrelation_le |
| intro z |
| simp only [Pi.add_apply, add_mul, Finset.expect_add_distrib] |
| exact (abs_add_le _ _).trans (add_le_add (code_correlation_le v z) (code_correlation_le w z)) |
|
|
| lemma faceCorrelation_neg (v : Point m n → ℝ) : faceCorrelation (-v) = faceCorrelation v := by |
| unfold faceCorrelation |
| simp only [Pi.neg_apply, neg_mul, Finset.expect_neg_distrib, abs_neg] |
|
|
| lemma faceCorrelation_lipschitz (v w : Point m n → ℝ) : |
| |faceCorrelation v-faceCorrelation w| ≤ 𝔼 x : Point m n, |v x-w x| := by |
| apply abs_le.mpr |
| have h₁ := faceCorrelation_add_le (v-w) w |
| have h₂ := faceCorrelation_add_le (w-v) v |
| have h₃ := faceCorrelation_le_l1 (v-w) |
| have h₄ := faceCorrelation_le_l1 (w-v) |
| simp only [sub_add_cancel] at h₁ h₂ |
| simp only [Pi.sub_apply] at h₃ h₄ |
| simp_rw [abs_sub_comm (w _)] at h₄ |
| constructor <;> linarith |
|
|
| lemma representation_correlation (v : Point m n → ℝ) (μ : Code m n → ℝ) : |
| |𝔼 x : Point m n, v x * (∑ z : Code m n, μ z * BinaryFourier.sign (z.val x))| ≤ |
| (∑ z : Code m n, |μ z|)*faceCorrelation v := by |
| simp_rw [Finset.mul_sum] |
| rw [Finset.expect_sum_comm] |
| calc |
| _ ≤ ∑ z : Code m n, |𝔼 x : Point m n, v x * (μ z * BinaryFourier.sign (z.val x))| := |
| Finset.abs_sum_le_sum_abs _ _ |
| _ = ∑ z : Code m n, |μ z| * |𝔼 x : Point m n, v x * BinaryFourier.sign (z.val x)| := by |
| apply Finset.sum_congr rfl |
| intro z _ |
| simp_rw [mul_left_comm (v _), ← Finset.mul_expect] |
| rw [abs_mul] |
| _ ≤ ∑ z : Code m n, |μ z| * faceCorrelation v := |
| Finset.sum_le_sum (fun z _ => mul_le_mul_of_nonneg_left (code_correlation_le v z) (abs_nonneg _)) |
| _ = _ := (Finset.sum_mul _ _ _).symm |
|
|
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
|
|
| theorem gradient_convex_correlation (f : FoldedProof A) (B : FaceArray A m n) |
| (hn : 0 < n) {σ η : ℝ} (hσ : σ ≠ 0) (hη : η ≠ 0) |
| (c : Point m n → ℝ) (v : Point m n → ℝ) : |
| |𝔼 x : Point m n, v x * gradient f B σ η c x| ≤ tableMass f B σ η c * faceCorrelation v := by |
| simp_rw [gradient_table_representation f B hn hσ hη] |
| exact representation_correlation v (tableCoefficient f B σ η c) |
| end MinUncut.Inner |
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|
| end |
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| end OAI |
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