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e320e82 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 | import OAI.Combinatorics.GotsmanLinial.CoordinateMultipliers
import OAI.Combinatorics.GotsmanLinial.HilbertSchmidt
import Mathlib.InformationTheory.Hamming
import Mathlib.Tactic
namespace OAI
/-!
# Coordinate commutators and Hamming-distance energy
The coordinate sign matrices are the actual diagonal operators of the Boolean
cube. The identities here turn their Hilbert--Schmidt and entry estimates into
the energy estimates used by the edge-recovery argument.
-/
open scoped BigOperators
namespace LeanBlast.GotsmanLinial
variable {n : ℕ}
/-- In the point basis, a coordinate commutator records the difference of the
two endpoint signs. -/
theorem coordinate_commutator_apply
(M : Matrix (Cube n) (Cube n) ℂ) (i : Fin n) (x y : Cube n) :
(M * coordinateSignMatrix i - coordinateSignMatrix i * M) x y =
(((cubeCoord y i - cubeCoord x i : ℝ) : ℂ)) * M x y := by
simp only [coordinateSignMatrix, signMatrix, Matrix.sub_apply,
Matrix.mul_diagonal, Matrix.diagonal_mul, Complex.ofReal_sub]
ring
/-- Squaring a coordinate difference gives four exactly on the coordinates
where the Boolean endpoints differ. -/
theorem coordinate_commutator_sq_norm
(M : Matrix (Cube n) (Cube n) ℂ) (i : Fin n) (x y : Cube n) :
‖(M * coordinateSignMatrix i - coordinateSignMatrix i * M) x y‖ ^ 2 =
(if x i ≠ y i then (4 : ℝ) else 0) * ‖M x y‖ ^ 2 := by
rw [coordinate_commutator_apply, norm_mul, mul_pow]
congr 1
cases hx : x i <;> cases hy : y i <;> norm_num [cubeCoord, hx, hy]
/-- Summing the squared entries over coordinates counts Hamming distance. -/
theorem sum_coordinate_commutator_sq_norm
(M : Matrix (Cube n) (Cube n) ℂ) (x y : Cube n) :
(∑ i : Fin n,
‖(M * coordinateSignMatrix i - coordinateSignMatrix i * M) x y‖ ^ 2) =
4 * (hammingDist x y : ℝ) * ‖M x y‖ ^ 2 := by
classical
simp_rw [coordinate_commutator_sq_norm, ite_mul, zero_mul]
calc
(∑ i : Fin n, if x i ≠ y i then 4 * ‖M x y‖ ^ 2 else 0) =
∑ i ∈ Finset.univ.filter (fun i => x i ≠ y i), 4 * ‖M x y‖ ^ 2 := by
rw [Finset.sum_filter]
_ = (hammingDist x y : ℝ) * (4 * ‖M x y‖ ^ 2) := by
simp [hammingDist, nsmul_eq_mul]
_ = 4 * (hammingDist x y : ℝ) * ‖M x y‖ ^ 2 := by ring
/-- The total coordinate-commutator energy is four times the distance-weighted
entry energy. -/
theorem sum_hsNormSq_coordinate_commutator
(M : Matrix (Cube n) (Cube n) ℂ) :
(∑ i : Fin n, hsNormSq (M * coordinateSignMatrix i - coordinateSignMatrix i * M)) =
4 * ∑ x : Cube n, ∑ y : Cube n, (hammingDist x y : ℝ) * ‖M x y‖ ^ 2 := by
classical
simp only [hsNormSq]
calc
(∑ i : Fin n, ∑ x : Cube n, ∑ y : Cube n,
‖(M * coordinateSignMatrix i - coordinateSignMatrix i * M) x y‖ ^ 2) =
∑ x : Cube n, ∑ y : Cube n, ∑ i : Fin n,
‖(M * coordinateSignMatrix i - coordinateSignMatrix i * M) x y‖ ^ 2 := by
rw [Finset.sum_comm]
exact Finset.sum_congr rfl fun _ _ => Finset.sum_comm
_ = ∑ x : Cube n, ∑ y : Cube n,
4 * (hammingDist x y : ℝ) * ‖M x y‖ ^ 2 := by
simp_rw [sum_coordinate_commutator_sq_norm]
_ = 4 * ∑ x : Cube n, ∑ y : Cube n,
(hammingDist x y : ℝ) * ‖M x y‖ ^ 2 := by
simp only [Finset.mul_sum, mul_assoc]
/-- A uniform coordinate-commutator Hilbert--Schmidt bound gives the distance
energy inequality for the actual cube matrix. -/
theorem distance_energy_le_of_coordinate_commutator_bound
(M : Matrix (Cube n) (Cube n) ℂ)
(hbound : ∀ i : Fin n,
hsNormSq (M * coordinateSignMatrix i - coordinateSignMatrix i * M) ≤ (2 : ℝ) ^ n) :
(∑ x : Cube n, ∑ y : Cube n, (hammingDist x y : ℝ) * ‖M x y‖ ^ 2) ≤
(n : ℝ) * (2 : ℝ) ^ n / 4 := by
have h := Finset.sum_le_sum (fun i (_ : i ∈ (Finset.univ : Finset (Fin n))) => hbound i)
rw [sum_hsNormSq_coordinate_commutator] at h
simp only [Finset.sum_const, Finset.card_univ, Fintype.card_fin, nsmul_eq_mul] at h
linarith
/-- A coordinate at which distinct endpoints differ turns an entrywise
commutator estimate into the edge-entry cap `1/4`. -/
theorem off_diagonal_sq_norm_le_quarter_of_commutator_bound
(M : Matrix (Cube n) (Cube n) ℂ)
(hbound : ∀ (i : Fin n) (x y : Cube n),
‖(M * coordinateSignMatrix i - coordinateSignMatrix i * M) x y‖ ^ 2 ≤ 1)
(x y : Cube n) (hxy : x ≠ y) : ‖M x y‖ ^ 2 ≤ (1 : ℝ) / 4 := by
classical
obtain ⟨i, hi⟩ : ∃ i : Fin n, x i ≠ y i := by
by_contra h
apply hxy
funext i
exact not_not.mp (not_exists.mp h i)
have h := hbound i x y
rw [coordinate_commutator_sq_norm, ite_eq_left hi] at h
linarith
end LeanBlast.GotsmanLinial
end OAI
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