File size: 13,255 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMaskConcentration
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedMasks
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InheritedStatisticWindows
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
namespace MatrixMultiplication.PermutationMatching
open scoped BigOperators
open MatrixMultiplication.InheritedMasks
open Classical
noncomputable section
section ClassMixtures
variable {C : Type*} [Fintype C] [DecidableEq C] {P : C → Type*}
[∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)]
def classWeight (P : C → Type*) [∀ c, Fintype (P c)] (c : C) : ℝ :=
(Fintype.card (P c) : ℝ) / populationSize P
def classDensity (A : ∀ c, Finset (P c)) (c : C) : ℝ :=
((A c).card : ℝ) / Fintype.card (P c)
def classProductMixture (P : C → Type*) [∀ c, Fintype (P c)]
(left right : C → ℝ) : ℝ :=
∑ c, classWeight P c * (left c * right c)
omit [DecidableEq C] [∀ c, DecidableEq (P c)] in
theorem classWeight_nonneg (hm : 0 < populationSize P) (c : C) :
0 ≤ classWeight P c := div_nonneg (Nat.cast_nonneg _) (le_of_lt hm)
omit [DecidableEq C] [∀ c, DecidableEq (P c)] in
theorem sum_classWeight (hm : 0 < populationSize P) :
∑ c, classWeight P c = 1 := by
unfold classWeight
rw [← Finset.sum_div]
exact div_self (ne_of_gt hm)
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (P c)] in
theorem classDensity_nonneg (A : ∀ c, Finset (P c)) (c : C) :
0 ≤ classDensity A c := div_nonneg (Nat.cast_nonneg _) (Nat.cast_nonneg _)
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (P c)] in
theorem classDensity_le_one [∀ c, Nonempty (P c)]
(A : ∀ c, Finset (P c)) (c : C) : classDensity A c ≤ 1 := by
have hn : (0 : ℝ) < Fintype.card (P c) := by exact_mod_cast Fintype.card_pos
apply (div_le_iff₀ hn).2
simpa using (Nat.cast_le.mpr (Finset.card_le_univ (A c)) :
((A c).card : ℝ) ≤ Fintype.card (P c))
omit [DecidableEq C] in
theorem totalMatchingMean_normalized_eq [∀ c, Nonempty (P c)]
(A B : ∀ c, Finset (P c)) (hm : 0 < populationSize P) :
totalMatchingMean A B / populationSize P =
classProductMixture P (classDensity A) (classDensity B) := by
classical
rw [totalMatchingMean_eq, Finset.sum_div]
unfold classProductMixture
apply Finset.sum_congr rfl
intro c _
have hn : (Fintype.card (P c) : ℝ) ≠ 0 := by
exact_mod_cast (ne_of_gt (Fintype.card_pos (α := P c)))
dsimp [classWeight, classDensity]
field_simp [hn, ne_of_gt hm]
omit [DecidableEq C] in
theorem totalMatchingMean_prescribed_error [∀ c, Nonempty (P c)]
(A B : ∀ c, Finset (P c)) (hm : 0 < populationSize P)
(left right : C → ℝ) (τ : ℝ)
(hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1)
(hleft : ∀ c, |classDensity A c - left c| ≤ τ)
(hright : ∀ c, |classDensity B c - right c| ≤ τ) :
|totalMatchingMean A B / populationSize P -
classProductMixture P left right| ≤ 2 * τ := by
rw [totalMatchingMean_normalized_eq A B hm]
have h := weighted_window (classWeight P)
(fun c => classDensity A c * classDensity B c - left c * right c)
0 (2 * τ) (classWeight_nonneg hm) (sum_classWeight hm) (by
intro c
simpa only [sub_zero] using product_coordinate_error
(classDensity_nonneg A c) (classDensity_le_one A c)
(hright0 c) (hright1 c) (hleft c) (hright c))
simpa only [classProductMixture, sub_zero, mul_sub, Finset.sum_sub_distrib] using h
omit [DecidableEq C] in
theorem totalMatchingMean_inherited_margin [∀ c, Nonempty (P c)]
(A B : ∀ c, Finset (P c)) (hm : 0 < populationSize P)
(left right : C → ℝ) (η τ α ν : ℝ)
(hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1)
(hleft : ∀ c, |classDensity A c - left c| ≤ τ)
(hright : ∀ c, |classDensity B c - right c| ≤ τ)
(happrox : |classProductMixture P left right - ν| ≤ α)
(hτ : τ ≤ η / 8) (hα : α ≤ η / 4) :
|totalMatchingMean A B / populationSize P - ν| ≤ η / 2 := by
have hchild := totalMatchingMean_prescribed_error A B hm left right τ
hright0 hright1 hleft hright
have h := inherited_margin (actual := ν)
(expected := classProductMixture P left right)
(prescribed := totalMatchingMean A B / populationSize P)
(by simpa only [abs_sub_comm] using hchild)
(by simpa only [abs_sub_comm] using happrox) hτ hα
simpa only [abs_sub_comm] using h
theorem inheritedMask_rejection_of_windows_le [∀ c, Nonempty (P c)]
(A B : ∀ c, Finset (P c)) (hn : ∀ c, 2 ≤ Fintype.card (P c))
(hm : 0 < populationSize P) (left right : C → ℝ) (η τ α ν : ℝ)
(hη : 0 < η)
(hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1)
(hleft : ∀ c, |classDensity A c - left c| ≤ τ)
(hright : ∀ c, |classDensity B c - right c| ≤ τ)
(happrox : |classProductMixture P left right - ν| ≤ α)
(hτ : τ ≤ η / 8) (hα : α ≤ η / 4) :
average (fun p : ClassPermutations P =>
if η < |totalMatchingCount A B p / populationSize P - ν| then 1 else 0) ≤
1 / (2 * populationSize P * η ^ 2) := by
exact inheritedMask_rejection_le A B hn hm η ν hη
(totalMatchingMean_inherited_margin A B hm left right η τ α ν
hright0 hright1 hleft hright happrox hτ hα)
theorem twoHalf_inheritedMask_rejection_of_windows_le [∀ c, Nonempty (P c)]
(A B : ∀ c, Finset (P c)) (hn : ∀ c, 2 ≤ Fintype.card (P c))
(hm : 0 < populationSize P) (left right : C → ℝ) (η τ α ν : ℝ)
(hη : 0 < η)
(hright0 : ∀ c, 0 ≤ right c) (hright1 : ∀ c, right c ≤ 1)
(hleft : ∀ c, |classDensity A c - left c| ≤ τ)
(hright : ∀ c, |classDensity B c - right c| ≤ τ)
(happrox : |classProductMixture P left right - ν| ≤ α)
(hτ : τ ≤ η / 8) (hα : α ≤ η / 4) :
average (fun p : ClassPermutations P × ClassPermutations P =>
if η < |totalTwoHalfCount A B p / populationSize P - ν| then 1 else 0) ≤
1 / (2 * populationSize P * η ^ 2) := by
exact twoHalf_inheritedMask_rejection_le A B hn hm η ν hη
(totalMatchingMean_inherited_margin A B hm left right η τ α ν
hright0 hright1 hleft hright happrox hτ hα)
theorem selectedTwoHalfMask_rejection_of_windows_le
(D : Finset C) (A B : ∀ c, Finset (P c))
(hn : ∀ c ∈ D, 2 ≤ Fintype.card (P c))
(hm : 0 < populationSize (fun c : D => P c))
(left right : C → ℝ) (η τ α ν : ℝ) (hη : 0 < η)
(hright0 : ∀ c ∈ D, 0 ≤ right c) (hright1 : ∀ c ∈ D, right c ≤ 1)
(hleft : ∀ c ∈ D, |classDensity A c - left c| ≤ τ)
(hright : ∀ c ∈ D, |classDensity B c - right c| ≤ τ)
(happrox : |classProductMixture (fun c : D => P c)
(fun c => left c) (fun c => right c) - ν| ≤ α)
(hτ : τ ≤ η / 8) (hα : α ≤ η / 4) :
average (fun p : ClassPermutations P × ClassPermutations P =>
if η < |selectedTwoHalfCount D A B p /
populationSize (fun c : D => P c) - ν| then 1 else 0) ≤
1 / (2 * populationSize (fun c : D => P c) * η ^ 2) := by
let : ∀ c : D, Nonempty (P c) := fun c =>
Fintype.card_pos_iff.mp (lt_of_lt_of_le (by decide : 0 < 2) (hn c c.property))
apply selectedTwoHalfMask_rejection_le D A B hn hm η ν hη
exact totalMatchingMean_inherited_margin (fun c : D => A c) (fun c : D => B c)
hm (fun c => left c) (fun c => right c) η τ α ν
(fun c => hright0 c c.property) (fun c => hright1 c c.property)
(fun c => hleft c c.property) (fun c => hright c c.property) happrox hτ hα
end ClassMixtures
section CompleteWordStatistics
variable {C : Type*} [Fintype C] [DecidableEq C] {P X Y S T : C → Type*}
[∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)]
[∀ c, Fintype (S c)] [∀ c, DecidableEq (S c)]
[∀ c, Fintype (T c)] [∀ c, DecidableEq (T c)]
omit [Fintype C] [DecidableEq C] [∀ c, DecidableEq (P c)] in
theorem classDensity_statisticPositions (w : ∀ c, P c → X c)
(statistic : ∀ c, X c → S c) (s : ∀ c, S c) (c : C) :
classDensity (fun c => statisticPositions (w c) (statistic c) (s c)) c =
empiricalLaw (statistic c ∘ w c) (s c) := by
rw [classDensity, statisticPositions_card, empiricalLaw]
omit [DecidableEq C] in
theorem totalMatchingMean_statistic_eq [∀ c, Nonempty (P c)]
(w : ∀ c, P c → X c) (v : ∀ c, P c → Y c)
(leftStat : ∀ c, X c → S c) (rightStat : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (hm : 0 < populationSize P) :
totalMatchingMean (fun c => statisticPositions (w c) (leftStat c) (s c))
(fun c => statisticPositions (v c) (rightStat c) (t c)) / populationSize P =
classProductMixture P
(fun c => empiricalLaw (leftStat c ∘ w c) (s c))
(fun c => empiricalLaw (rightStat c ∘ v c) (t c)) := by
rw [totalMatchingMean_normalized_eq _ _ hm]
simp only [classProductMixture, classDensity_statisticPositions]
theorem twoHalf_statistic_rejection_of_typeWindows_le [∀ c, Nonempty (P c)]
(w : ∀ c, P c → X c) (v : ∀ c, P c → Y c)
(leftStat : ∀ c, X c → S c) (rightStat : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c)
(hn : ∀ c, 2 ≤ Fintype.card (P c)) (hm : 0 < populationSize P)
(leftLaw : ∀ c, S c → ℝ) (rightLaw : ∀ c, T c → ℝ)
(η τ α ν : ℝ) (hη : 0 < η)
(hright0 : ∀ c, 0 ≤ rightLaw c (t c))
(hright1 : ∀ c, rightLaw c (t c) ≤ 1)
(hleft : ∀ c, typeWindow (leftLaw c) τ (leftStat c ∘ w c))
(hright : ∀ c, typeWindow (rightLaw c) τ (rightStat c ∘ v c))
(happrox : |classProductMixture P
(fun c => leftLaw c (s c)) (fun c => rightLaw c (t c)) - ν| ≤ α)
(hτ : τ ≤ η / 8) (hα : α ≤ η / 4) :
average (fun p : ClassPermutations P × ClassPermutations P =>
if η < |totalTwoHalfCount
(fun c => statisticPositions (w c) (leftStat c) (s c))
(fun c => statisticPositions (v c) (rightStat c) (t c)) p /
populationSize P - ν| then 1 else 0) ≤
1 / (2 * populationSize P * η ^ 2) := by
apply twoHalf_inheritedMask_rejection_of_windows_le _ _ hn hm
(fun c => leftLaw c (s c)) (fun c => rightLaw c (t c)) η τ α ν hη
hright0 hright1
· intro c
simpa only [classDensity_statisticPositions] using hleft c (s c)
· intro c
simpa only [classDensity_statisticPositions] using hright c (t c)
· exact happrox
· exact hτ
· exact hα
theorem selectedTwoHalf_statistic_rejection_of_completeWindows_le
[∀ c, Fintype (X c)] [∀ c, DecidableEq (X c)]
[∀ c, Fintype (Y c)] [∀ c, DecidableEq (Y c)]
(D : Finset C) (w : ∀ c, P c → X c) (v : ∀ c, P c → Y c)
(leftStat : ∀ c, X c → S c) (rightStat : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c)
(hn : ∀ c ∈ D, 2 ≤ Fintype.card (P c))
(hm : 0 < populationSize (fun c : D => P c))
(leftLaw : ∀ c, X c → ℝ) (rightLaw : ∀ c, Y c → ℝ)
(leftε rightε : C → ℝ) (η α ν : ℝ) (hη : 0 < η)
(hright0 : ∀ c ∈ D, ∀ y, 0 ≤ rightLaw c y)
(hrightSum : ∀ c ∈ D, ∑ y, rightLaw c y = 1)
(hleftε : ∀ c ∈ D, 0 ≤ leftε c) (hrightε : ∀ c ∈ D, 0 ≤ rightε c)
(hleft : ∀ c ∈ D, typeWindow (leftLaw c) (leftε c) (w c))
(hright : ∀ c ∈ D, typeWindow (rightLaw c) (rightε c) (v c))
(hleftFactor : ∀ c ∈ D, (Fintype.card (X c) : ℝ) * leftε c ≤ η / 8)
(hrightFactor : ∀ c ∈ D, (Fintype.card (Y c) : ℝ) * rightε c ≤ η / 8)
(happrox : |classProductMixture (fun c : D => P c)
(fun c => pushforwardLaw (leftStat c) (leftLaw c) (s c))
(fun c => pushforwardLaw (rightStat c) (rightLaw c) (t c)) - ν| ≤ α)
(hα : α ≤ η / 4) :
average (fun p : ClassPermutations P × ClassPermutations P =>
if η < |selectedTwoHalfCount D
(fun c => statisticPositions (w c) (leftStat c) (s c))
(fun c => statisticPositions (v c) (rightStat c) (t c)) p /
populationSize (fun c : D => P c) - ν| then 1 else 0) ≤
1 / (2 * populationSize (fun c : D => P c) * η ^ 2) := by
apply selectedTwoHalfMask_rejection_of_windows_le D _ _ hn hm
(fun c => pushforwardLaw (leftStat c) (leftLaw c) (s c))
(fun c => pushforwardLaw (rightStat c) (rightLaw c) (t c))
η (η / 8) α ν hη
· intro c hc
exact pushforwardLaw_nonneg (rightStat c) (rightLaw c) (hright0 c hc) (t c)
· intro c hc
exact pushforwardLaw_le_one (rightStat c) (rightLaw c)
(hright0 c hc) (hrightSum c hc) (t c)
· intro c hc
rw [classDensity_statisticPositions]
exact (typeWindow_statistic (leftLaw c) (leftε c) (w c) (leftStat c)
(hleftε c hc) (hleft c hc) (s c)).trans (hleftFactor c hc)
· intro c hc
rw [classDensity_statisticPositions]
exact (typeWindow_statistic (rightLaw c) (rightε c) (v c) (rightStat c)
(hrightε c hc) (hright c hc) (t c)).trans (hrightFactor c hc)
· exact happrox
· exact le_rfl
· exact hα
end CompleteWordStatistics
end
end MatrixMultiplication.PermutationMatching
end OAI
|