| import Mathlib | |
| namespace OAI | |
| universe u v | |
| noncomputable section | |
| open scoped BigOperators Classical Topology | |
| open Set | |
| namespace DiamondDistortion | |
| namespace FiniteAverage | |
| variable {Ω : Type*} [Fintype Ω] | |
| lemma jensen_sq (w f : Ω → ℝ) (hw : ∀ ω, 0 ≤ w ω) (hs : ∑ ω, w ω = 1) : | |
| (∑ ω, w ω * f ω) ^ 2 ≤ ∑ ω, w ω * f ω ^ 2 := by | |
| let m := ∑ ω, w ω * f ω | |
| have hpos : 0 ≤ ∑ ω, w ω * (f ω - m) ^ 2 := by | |
| exact Finset.sum_nonneg (fun ω _ => mul_nonneg (hw ω) (sq_nonneg _)) | |
| have he : (∑ ω, w ω * (f ω - m) ^ 2) = | |
| (∑ ω, w ω * f ω ^ 2) - 2*m*(∑ ω, w ω*f ω) + m^2*(∑ ω,w ω) := by | |
| calc | |
| _ = ∑ ω, (w ω * f ω ^ 2 - (2*m)*(w ω*f ω) + m^2*w ω) := by | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| ring | |
| _ = _ := by | |
| rw [Finset.sum_add_distrib, Finset.sum_sub_distrib, | |
| ← Finset.mul_sum, ← Finset.mul_sum] | |
| rw [he, hs] at hpos | |
| dsimp [m] at hpos | |
| nlinarith | |
| lemma sum_mul_sq_le (w : Ω → ℝ) (hw : ∀ ω, 0 ≤ w ω) | |
| (hs : ∑ ω,w ω = 1) (f : Ω → ℝ) {V : ℝ} (hV : 0 ≤ V) | |
| (hf : (∑ ω, w ω * f ω ^ 2) ≤ V) : | |
| |∑ ω, w ω * f ω| ≤ Real.sqrt V := by | |
| have h := (jensen_sq w f hw hs).trans hf | |
| exact (sq_le_sq₀ (abs_nonneg _) (Real.sqrt_nonneg _)).mp (by | |
| rw [sq_abs, Real.sq_sqrt hV] | |
| exact h) | |
| lemma double_product (w f g : Ω → ℝ) : | |
| (∑ ω, w ω * f ω) * (∑ ω, w ω * g ω) = | |
| ∑ ω, ∑ ν, w ω * w ν * (f ω * g ν) := by | |
| rw [Finset.sum_mul] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| rw [Finset.mul_sum] | |
| apply Finset.sum_congr rfl | |
| intro ν _ | |
| ring | |
| lemma norm_mean_le_sqrt {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] | |
| (w : Ω → ℝ) (hw : ∀ ω, 0 ≤ w ω) (hs : ∑ ω,w ω = 1) | |
| (z : Ω → E) {V : ℝ} (hV : 0 ≤ V) | |
| (hz : (∑ ω, w ω * ‖z ω‖ ^ 2) ≤ V) : | |
| ‖∑ ω, w ω • z ω‖ ≤ Real.sqrt V := by | |
| calc | |
| _ ≤ ∑ ω, ‖w ω • z ω‖ := norm_sum_le _ _ | |
| _ = ∑ ω, w ω * ‖z ω‖ := by | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| rw [norm_smul, Real.norm_eq_abs, abs_of_nonneg (hw ω)] | |
| _ ≤ |∑ ω, w ω * ‖z ω‖| := le_abs_self _ | |
| _ ≤ _ := sum_mul_sq_le w hw hs (fun ω => ‖z ω‖) hV hz | |
| /-- Averaged opposite-sign cancellation, including the error's gate energy. -/ | |
| lemma pooled_mass {κ : Type*} [Fintype κ] | |
| (w : Ω → ℝ) (hw : ∀ ω, 0 ≤ w ω) (hs : ∑ ω,w ω = 1) | |
| (b W : Ω → κ → ℝ) (V L : Ω → ℝ) | |
| (hmean : ∀ g, (∑ ω,w ω*b ω g) = -(∑ ω,w ω*W ω g)) | |
| (hW : ∀ ω, (∑ g,W ω g ^ 2) ≤ V ω) | |
| (hopp : ∀ ω ν, (∑ g, (|b ω g| * |b ν g| - b ω g*b ν g)) ≤ L ω + L ν) : | |
| (∑ g, (∑ ω,w ω* |b ω g|)^2) ≤ (∑ ω,w ω*V ω) + 2*(∑ ω,w ω*L ω) := by | |
| have hsign : (∑ g, (∑ ω,w ω*b ω g)^2) ≤ ∑ ω,w ω*V ω := by | |
| calc | |
| _ ≤ ∑ g, ∑ ω,w ω*(W ω g)^2 := by | |
| apply Finset.sum_le_sum | |
| intro g _ | |
| rw [hmean g, neg_sq] | |
| exact jensen_sq w (fun ω => W ω g) hw hs | |
| _ = ∑ ω,w ω*(∑ g,W ω g^2) := by | |
| rw [Finset.sum_comm] | |
| simp only [Finset.mul_sum] | |
| _ ≤ _ := Finset.sum_le_sum fun ω _ => mul_le_mul_of_nonneg_left (hW ω) (hw ω) | |
| have hleft : (∑ g, ((∑ ω,w ω* |b ω g|)^2 - (∑ ω,w ω*b ω g)^2)) = | |
| ∑ ω,∑ ν,w ω*w ν*(∑ g,(|b ω g| * |b ν g| - b ω g*b ν g)) := by | |
| calc | |
| _ = ∑ g, ∑ ω,∑ ν,w ω*w ν*(|b ω g| * |b ν g| - b ω g*b ν g) := by | |
| apply Finset.sum_congr rfl | |
| intro g _ | |
| rw [sq, sq, double_product, double_product, ← Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| rw [← Finset.sum_sub_distrib] | |
| apply Finset.sum_congr rfl | |
| intro ν _ | |
| ring | |
| _ = _ := by | |
| rw [Finset.sum_comm] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| rw [Finset.sum_comm] | |
| simp only [Finset.mul_sum] | |
| have hright : (∑ ω,∑ ν,w ω*w ν*(L ω+L ν)) = 2*(∑ ω,w ω*L ω) := by | |
| calc | |
| _ = (∑ ω,w ω*L ω)*(∑ ν,w ν) + (∑ ω,w ω)*(∑ ν,w ν*L ν) := by | |
| simp only [Finset.sum_mul, Finset.mul_sum, ← Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| apply Finset.sum_congr rfl | |
| intro ν _ | |
| ring | |
| _ = _ := by rw [hs]; ring | |
| have hcross : (∑ ω,∑ ν,w ω*w ν*(∑ g,(|b ω g| * |b ν g| - b ω g*b ν g))) ≤ | |
| ∑ ω,∑ ν,w ω*w ν*(L ω+L ν) := by | |
| apply Finset.sum_le_sum | |
| intro ω _ | |
| apply Finset.sum_le_sum | |
| intro ν _ | |
| exact mul_le_mul_of_nonneg_left (hopp ω ν) (mul_nonneg (hw ω) (hw ν)) | |
| rw [← hleft, hright, Finset.sum_sub_distrib] at hcross | |
| linarith | |
| lemma joint_budget (w : Ω → ℝ) (hw : ∀ ω,0 ≤ w ω) (hs : ∑ ω,w ω=1) | |
| (I : Ω → Type*) [∀ ω,Fintype (I ω)] (a b : ∀ ω,I ω→ℝ) {r R : ℝ} | |
| (he : ∀ ω,(∑ i,a ω i^2) ≤ R^2) | |
| (hc : (∑ ω,w ω*(∑ i,a ω i*b ω i)) = r^2) : | |
| (∑ ω,w ω*(∑ i,(a ω i-b ω i)^2)) + | |
| (∑ ω,w ω*(r^2-∑ i,b ω i^2)) ≤ R^2-r^2 := by | |
| have hrow (ω : Ω) : (∑ i,(a ω i-b ω i)^2) + (r^2-∑ i,b ω i^2) = | |
| (∑ i,a ω i^2) - 2*(∑ i,a ω i*b ω i) + r^2 := by | |
| have hex : (∑ i,(a ω i-b ω i)^2) = | |
| (∑ i,a ω i^2) - 2*(∑ i,a ω i*b ω i) + (∑ i,b ω i^2) := by | |
| calc | |
| _ = ∑ i,(a ω i^2 - 2*(a ω i*b ω i) + b ω i^2) := by | |
| apply Finset.sum_congr rfl | |
| intro i _ | |
| ring | |
| _ = _ := by rw [Finset.sum_add_distrib, Finset.sum_sub_distrib, ← Finset.mul_sum] | |
| rw [hex] | |
| ring | |
| have hid : (∑ ω,w ω*(∑ i,(a ω i-b ω i)^2)) + | |
| (∑ ω,w ω*(r^2-∑ i,b ω i^2)) = (∑ ω,w ω*(∑ i,a ω i^2)) - r^2 := by | |
| rw [← Finset.sum_add_distrib] | |
| simp_rw [← mul_add, hrow] | |
| calc | |
| _ = (∑ ω,w ω*(∑ i,a ω i^2)) - 2*(∑ ω,w ω*(∑ i,a ω i*b ω i)) + | |
| r^2*(∑ ω,w ω) := by | |
| rw [Finset.mul_sum, Finset.mul_sum, ← Finset.sum_sub_distrib, ← Finset.sum_add_distrib] | |
| apply Finset.sum_congr rfl | |
| intro ω _ | |
| ring | |
| _ = _ := by rw [hc,hs]; ring | |
| rw [hid] | |
| apply sub_le_sub_right | |
| calc | |
| _ ≤ ∑ ω,w ω*R^2 := Finset.sum_le_sum fun ω _ => mul_le_mul_of_nonneg_left (he ω) (hw ω) | |
| _ = R^2 := by rw [← Finset.sum_mul,hs,one_mul] | |
| lemma sqrt_pooled {η θ : ℝ} (hη : 0 ≤ η) (hθ : 0 ≤ θ) : | |
| Real.sqrt η + Real.sqrt (η+2*θ) ≤ 2*Real.sqrt (η+θ) := by | |
| have h2 : 0 ≤ η+2*θ := by positivity | |
| have h1 : 0 ≤ η+θ := add_nonneg hη hθ | |
| have he0 := Real.sq_sqrt hη | |
| have he1 := Real.sq_sqrt h1 | |
| have he2 := Real.sq_sqrt h2 | |
| have hp : Real.sqrt η * Real.sqrt (η+2*θ) ≤ η+θ := by | |
| have he : (Real.sqrt η * Real.sqrt (η+2*θ))^2 = η*(η+2*θ) := by | |
| rw [mul_pow, he0, he2] | |
| have hn := mul_nonneg (Real.sqrt_nonneg η) (Real.sqrt_nonneg (η+2*θ)) | |
| nlinarith [sq_nonneg θ] | |
| apply (sq_le_sq₀ (add_nonneg (Real.sqrt_nonneg _) (Real.sqrt_nonneg _)) | |
| (mul_nonneg (by norm_num) (Real.sqrt_nonneg _))).mp | |
| nlinarith | |
| end FiniteAverage | |
| end DiamondDistortion | |
| end | |
| end OAI | |