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f18e7de | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.PermutationIndicatorMoments
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
noncomputable section
namespace MatrixMultiplication.PermutationMatching
open scoped BigOperators
variable {P : Type*} [Fintype P] [DecidableEq P]
def matchingCount (A B : Finset P) (p : Equiv.Perm P) : ℝ :=
∑ i ∈ A, membership B p i
theorem average_matchingCount [Nonempty P] (A B : Finset P) :
average (matchingCount A B) =
(A.card : ℝ) * (B.card : ℝ) / (Fintype.card P : ℝ) := by
unfold matchingCount
rw [average_finset_sum]
simp only [average_membership, Finset.sum_const, nsmul_eq_mul]
ring
theorem average_matchingCount_sq [Nonempty P] (A B : Finset P) :
average (fun p => matchingCount A B p ^ 2) =
(A.card : ℝ) * (B.card : ℝ) / (Fintype.card P : ℝ) +
(A.card : ℝ) * ((A.card : ℝ) - 1) * (B.card : ℝ) *
((B.card : ℝ) - 1) /
((Fintype.card P : ℝ) * ((Fintype.card P : ℝ) - 1)) := by
let u : ℝ := (B.card : ℝ) / (Fintype.card P : ℝ)
let v : ℝ := (B.card : ℝ) * ((B.card : ℝ) - 1) /
((Fintype.card P : ℝ) * ((Fintype.card P : ℝ) - 1))
have hpair (i j : P) :
average (fun p => membership B p i * membership B p j) =
v + if i = j then u - v else 0 := by
by_cases hij : i = j
· subst j
simp only [membership_sq]
rw [average_membership]
dsimp [u]
ring
· rw [average_membership_mul_of_ne B hij]
simp [hij, v]
have hsq (p : Equiv.Perm P) : matchingCount A B p ^ 2 =
∑ i ∈ A, ∑ j ∈ A, membership B p i * membership B p j := by
unfold matchingCount
rw [pow_two, Finset.sum_mul_sum]
simp_rw [hsq]
rw [average_finset_sum]
simp_rw [average_finset_sum, hpair]
have hinner (i : P) (hi : i ∈ A) :
(∑ j ∈ A, (v + (if i = j then u - v else 0))) =
(A.card : ℝ) * v + (u - v) := by
rw [Finset.sum_add_distrib]
simp [hi]
rw [Finset.sum_congr rfl hinner]
simp only [Finset.sum_const, nsmul_eq_mul]
dsimp [u, v]
ring
theorem matchingCount_variance [Nonempty P] (A B : Finset P)
(hn : 2 ≤ Fintype.card P) :
average (fun p => (matchingCount A B p - average (matchingCount A B)) ^ 2) =
(A.card : ℝ) * ((Fintype.card P : ℝ) - (A.card : ℝ)) *
(B.card : ℝ) * ((Fintype.card P : ℝ) - (B.card : ℝ)) /
((Fintype.card P : ℝ) ^ 2 * ((Fintype.card P : ℝ) - 1)) := by
rw [average_centered_sq, average_matchingCount_sq, average_matchingCount]
have hn' : (2 : ℝ) ≤ Fintype.card P := by exact_mod_cast hn
have hn0 : (Fintype.card P : ℝ) ≠ 0 := by linarith
have hn1 : (Fintype.card P : ℝ) - 1 ≠ 0 := by linarith
field_simp [hn0, hn1]
ring
theorem matchingCount_variance_le (A B : Finset P)
(hn : 2 ≤ Fintype.card P) :
average (fun p => (matchingCount A B p - average (matchingCount A B)) ^ 2) ≤
(Fintype.card P : ℝ) / 8 := by
have : Nonempty P := Fintype.card_pos_iff.mp (lt_of_lt_of_le (by decide : 0 < 2) hn)
rw [matchingCount_variance A B hn]
have hn' : (2 : ℝ) ≤ Fintype.card P := by exact_mod_cast hn
have ha : (A.card : ℝ) ≤ (Fintype.card P : ℝ) := by
exact_mod_cast Finset.card_le_univ A
have hb : (B.card : ℝ) ≤ (Fintype.card P : ℝ) := by
exact_mod_cast Finset.card_le_univ B
have ha0 : (0 : ℝ) ≤ A.card := Nat.cast_nonneg _
have hb0 : (0 : ℝ) ≤ B.card := Nat.cast_nonneg _
have hna : 0 ≤ (A.card : ℝ) * ((Fintype.card P : ℝ) - A.card) :=
mul_nonneg ha0 (sub_nonneg.mpr ha)
have hnb : 0 ≤ (B.card : ℝ) * ((Fintype.card P : ℝ) - B.card) :=
mul_nonneg hb0 (sub_nonneg.mpr hb)
have hqa : (A.card : ℝ) * ((Fintype.card P : ℝ) - A.card) ≤
(Fintype.card P : ℝ) ^ 2 / 4 := by
nlinarith [sq_nonneg ((A.card : ℝ) - (Fintype.card P : ℝ) / 2)]
have hqb : (B.card : ℝ) * ((Fintype.card P : ℝ) - B.card) ≤
(Fintype.card P : ℝ) ^ 2 / 4 := by
nlinarith [sq_nonneg ((B.card : ℝ) - (Fintype.card P : ℝ) / 2)]
have hprod := mul_le_mul hqa hqb hnb (by positivity :
(0 : ℝ) ≤ (Fintype.card P : ℝ) ^ 2 / 4)
have hn3 : 0 ≤ (Fintype.card P : ℝ) ^ 3 * ((Fintype.card P : ℝ) - 2) :=
mul_nonneg (by positivity) (by linarith)
have hn1pos : (0 : ℝ) < (Fintype.card P : ℝ) - 1 := by linarith
have hnpos : (0 : ℝ) < Fintype.card P := by linarith
apply (div_le_iff₀ (show (0 : ℝ) <
(Fintype.card P : ℝ) ^ 2 * ((Fintype.card P : ℝ) - 1) by positivity)).mpr
nlinarith
noncomputable def statisticPositions {A B : Type*} (w : P → A)
(statistic : A → B) (b : B) : Finset P := by
classical
exact Finset.univ.filter fun i => statistic (w i) = b
noncomputable def statisticMatchingCount {A B S T : Type*}
(left : P → A) (right : P → B) (sl : A → S) (sr : B → T)
(s : S) (t : T) : Equiv.Perm P → ℝ :=
matchingCount (statisticPositions left sl s) (statisticPositions right sr t)
theorem statisticMatchingCount_variance_le {A B S T : Type*}
(left : P → A) (right : P → B) (sl : A → S) (sr : B → T)
(s : S) (t : T) (hn : 2 ≤ Fintype.card P) :
average (fun p =>
(statisticMatchingCount left right sl sr s t p -
average (statisticMatchingCount left right sl sr s t)) ^ 2) ≤
(Fintype.card P : ℝ) / 8 :=
matchingCount_variance_le _ _ hn
end MatrixMultiplication.PermutationMatching
end
end OAI
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