| import Mathlib |
| import OAI.Computability.MinUncut.Estimates.FirstError |
| import OAI.Computability.MinUncut.Analysis.FreshCorrelation |
| import OAI.Computability.MinUncut.Graphs.HonestRows |
| import OAI.Computability.MinUncut.Analysis.GaussianPacking |
|
|
| namespace OAI |
|
|
| noncomputable section |
| open scoped BigOperators |
| open MeasureTheory ProbabilityTheory |
| namespace MinUncut.Inner |
| open GaussianBudget |
| attribute [local instance] Classical.propDecidable |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
| variable {m n : ℕ} |
|
|
| lemma score_query (z : Code m n) (u : Point m n → ℝ) (d : Code m n → ℝ) : |
| query u d z=decide (0 ≤ score (codeVector z) u+d z) := by |
| rw [score_codeVector] |
| rfl |
|
|
| lemma score_add_noise (v c g : Point m n → ℝ) (σ : ℝ) : |
| score v (c+σ•g)=score v c+σ*score v g := by |
| simp only [score,Pi.add_apply,Pi.smul_apply,smul_eq_mul,mul_add,Finset.sum_add_distrib] |
| rw [Finset.mul_sum] |
| congr 1 |
| apply Finset.sum_congr rfl |
| intro x _ |
| ring |
|
|
| omit [Fintype A] in |
| lemma firstFailure_evaluation (a : A) (B : FaceArray A m n) (σ η : ℝ) |
| (c g : Point m n → ℝ) (l : Code m n → ℝ) : |
| firstFailure (evaluationProof a) B σ η c g l = |
| signFailure (score (codeVector (labelCode B a)) c) |
| (score (codeVector (labelCode B a)) c+σ*score (codeVector (labelCode B a)) g+η*l (labelCode B a)) := by |
| simp only [firstFailure,evaluationProof,pullQuery,score_query,Pi.zero_apply,add_zero, |
| score_add_noise,Pi.smul_apply,smul_eq_mul,decide_eq_decide,signFailure] |
| |
| omit [Fintype A] in |
| lemma firstRejection_evaluation_le (a : A) (B : FaceArray A m n) (hn : 0<n) |
| {σ η : ℝ} (hσ : 0 ≤ σ) (hη : 0 ≤ η) : |
| (∫ c, firstRejection (evaluationProof a) B σ η c ∂gauss (Point m n)) ≤ 4*(σ+η) := by |
| simp only [firstRejection,firstFailure_evaluation,gauss] |
| exact first_triple_bound (codeVector_norm hn _) hσ hη _ |
|
|
| lemma firstError_evaluation_le (a : A) (hn : 0<n) {σ η : ℝ} (hσ : 0 ≤ σ) (hη : 0 ≤ η) : |
| firstError (m := m) (n := n) (evaluationProof a) σ η ≤ 4*(σ+η) := by |
| exact (Finset.expect_le_expect (fun B _ => firstRejection_evaluation_le a B hn hσ hη)).trans_eq |
| (Fintype.expect_const _) |
|
|
| omit [Fintype A] in |
| lemma secondFailure_evaluation (a : A) (B C : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) (p : (Point m n → ℝ) × (Code m n → ℝ)) : |
| secondFailure (evaluationProof a) B C σ η c p = |
| signFailure (score (codeVector (labelCode B a)) c+σ*score (codeVector (labelCode B a)) p.1+η*p.2 (labelCode B a)) |
| (score (codeVector (labelCode C a)) c+σ*score (codeVector (labelCode C a)) p.1+η*p.2 (labelCode C a)) := by |
| simp only [secondFailure,evaluationProof,pullQuery,score_query,score_add_noise, |
| Pi.smul_apply,smul_eq_mul,decide_eq_decide,signFailure] |
|
|
| omit [Fintype A] in |
| lemma secondRejection_evaluation_le (a : A) (B C : FaceArray A m n) (hn : 0<n) |
| {σ η : ℝ} (hσ : 0 ≤ σ) (hσ1 : σ ≤ 1) (hη : 0 ≤ η) : |
| (∫ c, secondRejection (evaluationProof a) B C σ η c ∂gauss (Point m n)) ≤ |
| 4*‖vector (codeVector (labelCode B a))-vector (codeVector (labelCode C a))‖+4*η := by |
| unfold secondRejection |
| simp only [secondFailure_evaluation,gauss] |
| have he (c : Point m n → ℝ) : |
| (∫ p : (Point m n → ℝ) × (Code m n → ℝ), |
| signFailure (score (codeVector (labelCode B a)) c+σ*score (codeVector (labelCode B a)) p.1+η*p.2 (labelCode B a)) |
| (score (codeVector (labelCode C a)) c+σ*score (codeVector (labelCode C a)) p.1+η*p.2 (labelCode C a)) |
| ∂(Measure.pi (fun _ : Point m n => gaussianReal 0 1)).prod (Measure.pi (fun _ : Code m n => gaussianReal 0 1))) = |
| ∫ g, ∫ l, |
| signFailure (score (codeVector (labelCode B a)) c+σ*score (codeVector (labelCode B a)) g+η*l (labelCode B a)) |
| (score (codeVector (labelCode C a)) c+σ*score (codeVector (labelCode C a)) g+η*l (labelCode C a)) |
| ∂Measure.pi (fun _ : Code m n => gaussianReal 0 1) ∂Measure.pi (fun _ : Point m n => gaussianReal 0 1) := by |
| apply integral_prod |
| apply signFailure_integrable < |
| simp_rw [he] |
| exact second_triple_bound (codeVector_norm hn _) (codeVector_norm hn _) hσ hσ1 hη _ _ |
|
|
| omit [Fintype A] in |
| lemma thirdFailure_evaluation (a : A) (B : FaceArray A m n) (η : ℝ) |
| (u : Point m n → ℝ) (l : Code m n → ℝ) (z : Code m n) : |
| thirdFailure (evaluationProof a) B η u l z=if z=labelCode B a then 1 else 0 := by |
| unfold thirdFailure evaluationProof |
| by_cases h : z=labelCode B a |
| · subst z |
| · simp [Function.update_of_ne (Ne.symm h),h] |
|
|
| omit [Fintype A] in |
| lemma thirdRejection_evaluation (a : A) (B : FaceArray A m n) (σ η : ℝ) |
| (c : Point m n → ℝ) : thirdRejection (evaluationProof a) B σ η c=(Fintype.card (Code m n):ℝ)⁻¹ := by |
| simp only [thirdRejection,thirdFailure_evaluation,integral_const,probReal_univ,one_smul, |
| Fintype.expect_eq_sum_div_card] |
| simp |
|
|
| lemma thirdError_evaluation (a : A) (σ η : ℝ) : |
| thirdError (m := m) (n := n) (evaluationProof a) σ η=(Fintype.card (Code m n):ℝ)⁻¹ := by |
| simp only [thirdError,thirdRejection_evaluation,integral_const,probReal_univ,one_smul,Fintype.expect_const] |
| end MinUncut.Inner |
|
|
| end |
| noncomputable section |
| open scoped BigOperators |
| namespace MinUncut.Inner |
| open BinaryFourier RowNoise GaussianBudget |
| attribute [local instance] Classical.propDecidable |
| variable {V A : Type*} [AddCommGroup V] [Module F₂ V] [AddTorsor V A] [Fintype A] |
| variable {m n : ℕ} |
|
|
| lemma codeVector_distance_sq (hn : 0<n) (z w : Code m n) : |
| ‖vector (codeVector z)-vector (codeVector w)‖^2= |
| ((n^m:ℕ):ℝ)⁻¹*∑ x, (BinaryFourier.sign (z.val x)-BinaryFourier.sign (w.val x))^2 := by |
| simp only [EuclideanSpace.real_norm_sq_eq,PiLp.sub_apply,vector,codeVector,← mul_sub,mul_pow] |
| rw [← Finset.mul_sum,inv_pow,Real.sq_sqrt (by positivity)] |
|
|
| lemma noise_vector_sq (a : A) (B : FaceArray A m n) (hn : 0<n) (ρ : ℝ) : |
| (𝔼 C : FaceArray A m n, density ρ B C*‖vector (codeVector (labelCode B a))-vector (codeVector (labelCode C a))‖^2)= |
| ((n^m:ℕ):ℝ)⁻¹*∑ x : Point m n, 2*(1-(1-ρ)^degree (labelFunctional a x)) := by |
| simp_rw [codeVector_distance_sq hn,← character_label,Finset.mul_sum,mul_left_comm (density ρ B _), |
| Finset.expect_sum_comm,← Finset.mul_expect,character_noise_distance] |
| |
| lemma noise_vector_sq_le (a : A) (B : FaceArray A m n) (hn : 0<n) |
| {ρ : ℝ} (hρ : 0 ≤ ρ) (hρ1 : ρ ≤ 1) : |
| (𝔼 C : FaceArray A m n, density ρ B C*‖vector (codeVector (labelCode B a))-vector (codeVector (labelCode C a))‖^2) ≤ 2*(m:ℝ)*ρ := by |
| have hn' : (0:ℝ)<(n^m:ℕ) := by positivity |
| rw [noise_vector_sq a B hn] |
| calc |
| _ ≤ ((n^m:ℕ):ℝ)⁻¹ * ∑ _ : Point m n, 2*(m:ℝ)*ρ := by |
| apply mul_le_mul_of_nonneg_left _ (inv_nonneg.mpr hn'.le) |
| apply Finset.sum_le_sum |
| intro x _ |
| have hb := character_noise_distance_le hρ1 (labelFunctional a x) B |
| rw [character_noise_distance] at hb |
| have hd : (degree (labelFunctional a x):ℝ) ≤ m := by exact_mod_cast labelFunctional_degree a x |
| exact hb.trans (mul_le_mul_of_nonneg_right (by linarith) hρ) |
| _ = _ := by |
| simp only [Finset.sum_const,Finset.card_univ,Fintype.card_fun,Fintype.card_fin,nsmul_eq_mul] |
| rw [← mul_assoc,inv_mul_cancel₀ hn'.ne',one_mul] |
|
|
| lemma weighted_expect_sq {T : Type*} [Fintype T] (w f : T → ℝ) |
| (hw : ∀ t, 0 ≤ w t) (hw1 : (𝔼 t, w t)=1) : |
| (𝔼 t, w t*f t)^2 ≤ 𝔼 t, w t*(f t)^2 := by |
| let M := 𝔼 t, w t*f t |
| have hz : 0 ≤ 𝔼 t, w t*(f t-M)^2 := |
| Finset.expect_nonneg (fun t _ => mul_nonneg (hw t) (sq_nonneg _)) |
| have he (t : T) : w t*(f t-M)^2=w t*(f t)^2-2*M*(w t*f t)+M^2*w t := by ring |
| simp_rw [he,Finset.expect_add_distrib,Finset.expect_sub_distrib,← Finset.mul_expect,hw1,mul_one] at hz |
| change 0 ≤ (𝔼 t, w t*(f t)^2)-2*M*M+M^2 at hz |
| change M^2 ≤ _ |
| nlinarith only [hz] |
|
|
| lemma noise_vector_le (a : A) (B : FaceArray A m n) (hn : 0<n) (hm : 0 < m) |
| {δ : ℝ} (hδ : 0 ≤ δ) (hρ1 : δ^2/(m:ℝ) ≤ 1) : |
| (𝔼 C : FaceArray A m n, density (δ^2/(m:ℝ)) B C* |
| ‖vector (codeVector (labelCode B a))-vector (codeVector (labelCode C a))‖) ≤ 2*δ := by |
| have hρ : 0 ≤ δ^2/(m:ℝ) := div_nonneg (sq_nonneg _) (Nat.cast_nonneg _) |
| have hs := weighted_expect_sq (density (δ^2/(m:ℝ)) B) |
| (fun C => ‖vector (codeVector (labelCode B a))-vector (codeVector (labelCode C a))‖) |
| (density_nonneg hρ hρ1 B) (density_mean _ B) |
| have hb := noise_vector_sq_le a B hn hρ hρ1 |
| have hmp : (0:ℝ)< m := by exact_mod_cast hm |
| have he : 2*(m:ℝ)*(δ^2/(m:ℝ))=2*δ^2 := by field_simp |
| rw [he] at hb |
| have hM : 0 ≤ 𝔼 C : FaceArray A m n, density (δ^2/(m:ℝ)) B C* |
| ‖vector (codeVector (labelCode B a))-vector (codeVector (labelCode C a))‖ := |
| Finset.expect_nonneg (fun C _ => mul_nonneg (density_nonneg hρ hρ1 B C) (norm_nonneg _)) |
| have hs' := hs.trans hb |
| exact (sq_le_sq₀ hM (by positivity)).mp (by nlinarith [sq_nonneg δ]) |
| end MinUncut.Inner |
| |
| end |
| |
| end OAI |
| |
