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import OAI.LinearAlgebra.MatrixMultiplication.Tensor.CompleteWordPair
import Mathlib.Algebra.Group.Equiv.Basic
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
namespace MatrixMultiplication.PermutationMatching
open scoped BigOperators
open MatrixMultiplication.InheritedMasks
open Classical
noncomputable section
variable {C : Type*} {P X Y : C → Type*}
variable [Fintype C] [DecidableEq C]
[∀ c, Fintype (P c)] [∀ c, DecidableEq (P c)]
variable {S T : C → Type*}
def wordPairLeftPositions (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (s : ∀ c, S c) : ∀ c, Finset (P c) :=
fun c => statisticPositions (w.left c) (sl c) (s c)
def wordPairRightPositions (w : CompleteWordPair P X Y)
(sr : ∀ c, Y c → T c) (t : ∀ c, T c) : ∀ c, Finset (P c) :=
fun c => statisticPositions (w.right c) (sr c) (t c)
def wordPairCount (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) : ℝ :=
∑ c, ∑ i : P c,
if sl c (w.left c i) = s c ∧ sr c (w.right c i) = t c then 1 else 0
omit [DecidableEq C] in
theorem wordPairCount_smul (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (g : HalfClassPermutations P) :
wordPairCount (g • w) sl sr s t =
totalTwoHalfCount (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g⁻¹ := by
unfold wordPairCount totalTwoHalfCount wordPairLeftPositions wordPairRightPositions
apply Finset.sum_congr rfl
intro c _
rw [twoHalfCount_statisticPositions]
rfl
theorem average_wordPairCount_shift (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (f : ℝ → ℝ) :
average (fun g : HalfClassPermutations P => f (wordPairCount (g • w) sl sr s t)) =
average (fun g : HalfClassPermutations P =>
f (totalTwoHalfCount (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g)) := by
simp_rw [wordPairCount_smul]
exact average_equiv (Equiv.inv (HalfClassPermutations P))
(fun g => f (totalTwoHalfCount (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g))
def wordPairMaskBad (sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (η ν : ℝ)
(w : CompleteWordPair P X Y) : Prop :=
η < |wordPairCount w sl sr s t / populationSize P - ν|
theorem wordPair_orbit_fraction_eq (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (η ν : ℝ) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(wordPairMaskBad sl sr s t η ν)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card =
average (fun g : HalfClassPermutations P =>
if η < |totalTwoHalfCount (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g / populationSize P - ν|
then 1 else 0) := by
rw [← average_bad_shift_eq_orbit_fraction]
exact average_wordPairCount_shift w sl sr s t
(fun k => if η < |k / populationSize P - ν| then 1 else 0)
theorem wordPair_orbit_rejection_le (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c)
(hn : ∀ c, 2 ≤ Fintype.card (P c)) (hm : 0 < populationSize P)
(η ν : ℝ) (hη : 0 < η)
(hmargin : |totalMatchingMean (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) / populationSize P - ν| ≤ η / 2) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(wordPairMaskBad sl sr s t η ν)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
1 / (2 * populationSize P * η ^ 2) := by
rw [wordPair_orbit_fraction_eq]
exact twoHalf_inheritedMask_rejection_le _ _ hn hm η ν hη hmargin
theorem wordPair_orbit_rejection_of_windows_le [∀ c, Nonempty (P c)]
(w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T 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 (wordPairLeftPositions w sl s) c - left c| ≤ τ)
(hright : ∀ c, |classDensity (wordPairRightPositions w sr t) c - right c| ≤ τ)
(happrox : |classProductMixture P left right - ν| ≤ α)
(hτ : τ ≤ η / 8) (hα : α ≤ η / 4) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(wordPairMaskBad sl sr s t η ν)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
1 / (2 * populationSize P * η ^ 2) := by
exact wordPair_orbit_rejection_le w sl sr s t hn hm η ν hη
(totalMatchingMean_inherited_margin _ _ hm left right η τ α ν
hright0 hright1 hleft hright happrox hτ hα)
def selectedWordPairCount (D : Finset C) (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) : ℝ :=
∑ c : D, ∑ i : P c,
if sl c (w.left c i) = s c ∧ sr c (w.right c i) = t c then 1 else 0
omit [Fintype C] [DecidableEq C] in
theorem selectedWordPairCount_smul (D : Finset C) (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (g : HalfClassPermutations P) :
selectedWordPairCount D (g • w) sl sr s t =
selectedTwoHalfCount D (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g⁻¹ := by
unfold selectedWordPairCount selectedTwoHalfCount totalTwoHalfCount
wordPairLeftPositions wordPairRightPositions
apply Finset.sum_congr rfl
intro c _
rw [twoHalfCount_statisticPositions]
rfl
theorem average_selectedWordPairCount_shift (D : Finset C)
(w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (f : ℝ → ℝ) :
average (fun g : HalfClassPermutations P =>
f (selectedWordPairCount D (g • w) sl sr s t)) =
average (fun g : HalfClassPermutations P =>
f (selectedTwoHalfCount D (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g)) := by
simp_rw [selectedWordPairCount_smul]
exact average_equiv (Equiv.inv (HalfClassPermutations P))
(fun g => f (selectedTwoHalfCount D (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g))
def selectedWordPairMaskBad (D : Finset C)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (η ν : ℝ)
(w : CompleteWordPair P X Y) : Prop :=
η < |selectedWordPairCount D w sl sr s t /
populationSize (fun c : D => P c) - ν|
theorem selectedWordPair_orbit_fraction_eq (D : Finset C)
(w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ c, Y c → T c)
(s : ∀ c, S c) (t : ∀ c, T c) (η ν : ℝ) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(selectedWordPairMaskBad D sl sr s t η ν)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card =
average (fun g : HalfClassPermutations P =>
if η < |selectedTwoHalfCount D (wordPairLeftPositions w sl s)
(wordPairRightPositions w sr t) g /
populationSize (fun c : D => P c) - ν| then 1 else 0) := by
rw [← average_bad_shift_eq_orbit_fraction]
exact average_selectedWordPairCount_shift D w sl sr s t
(fun k => if η < |k / populationSize (fun c : D => P c) - ν| then 1 else 0)
theorem selectedWordPair_orbit_rejection_le (D : Finset C)
(w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ 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)) (η ν : ℝ) (hη : 0 < η)
(hmargin : |totalMatchingMean
(fun c : D => wordPairLeftPositions w sl s c)
(fun c : D => wordPairRightPositions w sr t c) /
populationSize (fun c : D => P c) - ν| ≤ η / 2) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(selectedWordPairMaskBad D sl sr s t η ν)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
1 / (2 * populationSize (fun c : D => P c) * η ^ 2) := by
rw [selectedWordPair_orbit_fraction_eq]
exact selectedTwoHalfMask_rejection_le D _ _ hn hm η ν hη hmargin
theorem selectedWordPair_orbit_rejection_of_windows_le (D : Finset C)
(w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ 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))
(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 (wordPairLeftPositions w sl s) c - left c| ≤ τ)
(hright : ∀ c ∈ D, |classDensity (wordPairRightPositions w sr t) 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) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(selectedWordPairMaskBad D sl sr s t η ν)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
1 / (2 * populationSize (fun c : D => P c) * η ^ 2) := by
rw [selectedWordPair_orbit_fraction_eq]
exact selectedTwoHalfMask_rejection_of_windows_le D _ _ hn hm left right
η τ α ν hη hright0 hright1 hleft hright happrox hτ hα
theorem selectedWordPair_orbit_rejection_of_completeWindows_le
[∀ c, Fintype (X c)] [∀ c, DecidableEq (X c)]
[∀ c, Fintype (Y c)] [∀ c, DecidableEq (Y c)]
[∀ c, Fintype (S c)] [∀ c, DecidableEq (S c)]
[∀ c, Fintype (T c)] [∀ c, DecidableEq (T c)]
(D : Finset C) (w : CompleteWordPair P X Y)
(sl : ∀ c, X c → S c) (sr : ∀ 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.left c))
(hright : ∀ c ∈ D, typeWindow (rightLaw c) (rightε c) (w.right 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 (sl c) (leftLaw c) (s c))
(fun c => pushforwardLaw (sr c) (rightLaw c) (t c)) - ν| ≤ α)
(hα : α ≤ η / 4) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(selectedWordPairMaskBad D sl sr s t η ν)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
1 / (2 * populationSize (fun c : D => P c) * η ^ 2) := by
rw [selectedWordPair_orbit_fraction_eq]
exact selectedTwoHalf_statistic_rejection_of_completeWindows_le D w.left w.right
sl sr s t hn hm leftLaw rightLaw leftε rightε η α ν hη hright0 hrightSum
hleftε hrightε hleft hright hleftFactor hrightFactor happrox hα
section FiniteMasks
variable {I : Type*} [Fintype I] {SL SR : I → C → Type*}
theorem finite_selectedWordPair_sum_orbit_fraction_le
(D : I → Finset C) (w : CompleteWordPair P X Y)
(sl : ∀ i c, X c → SL i c) (sr : ∀ i c, Y c → SR i c)
(s : ∀ i c, SL i c) (t : ∀ i c, SR i c) (η : ℝ) (ν : I → ℝ)
(hη : 0 ≤ η) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(fun v => η < |(∑ i, selectedWordPairCount (D i) v
(sl i) (sr i) (s i) (t i) / populationSize (fun c : D i => P c)) -
∑ i, ν i|)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(fun v => ∃ i, selectedWordPairMaskBad (D i) (sl i) (sr i)
(s i) (t i) (η / Fintype.card I) (ν i) v)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card := by
by_cases hI : Fintype.card I = 0
· have : IsEmpty I := Fintype.card_eq_zero_iff.mp hI
simp [not_lt_of_ge hη]
have : Nonempty I := Fintype.card_pos_iff.mp (Nat.pos_of_ne_zero hI)
have hcard : (0 : ℝ) < Fintype.card I := by
exact_mod_cast (Fintype.card_pos (α := I))
let z (v : CompleteWordPair P X Y) (i : I) : ℝ :=
selectedWordPairCount (D i) v (sl i) (sr i) (s i) (t i) /
populationSize (fun c : D i => P c)
let R := OrbitCounting.orbitSet (G := HalfClassPermutations P) w
let sumBad (v : CompleteWordPair P X Y) : Prop :=
η < |(∑ i, z v i) - ∑ i, ν i|
let unionBad (v : CompleteWordPair P X Y) : Prop :=
∃ i, selectedWordPairMaskBad (D i) (sl i) (sr i)
(s i) (t i) (η / Fintype.card I) (ν i) v
have hEvent (v) (hv : sumBad v) : unionBad v := by
change ∃ i, η / Fintype.card I < |z v i - ν i|
change η < |(∑ i, z v i) - ∑ i, ν i| at hv
by_contra h
push Not at h
have he : |(∑ i, z v i) - ∑ i, ν i| ≤ η := by
rw [← Finset.sum_sub_distrib]
calc
_ ≤ ∑ i, |z v i - ν i| := Finset.abs_sum_le_sum_abs _ _
_ ≤ ∑ _i : I, η / Fintype.card I := Finset.sum_le_sum fun i _ => h i
_ = η := by
simp only [Finset.sum_const, Finset.card_univ, nsmul_eq_mul]
field_simp [ne_of_gt hcard]
linarith
have hsub : R.filter sumBad ⊆ R.filter unionBad :=
Finset.monotone_filter_right R (fun v _ => hEvent v)
have hcards : ((R.filter sumBad).card : ℝ) ≤ (R.filter unionBad).card := by
exact_mod_cast Finset.card_le_card hsub
exact div_le_div_of_nonneg_right hcards (Nat.cast_nonneg R.card)
theorem finite_selectedWordPair_orbit_rejection_of_windows_le
(D : I → Finset C) (w : CompleteWordPair P X Y)
(sl : ∀ i c, X c → SL i c) (sr : ∀ i c, Y c → SR i c)
(s : ∀ i c, SL i c) (t : ∀ i c, SR i c)
(hn : ∀ i c, c ∈ D i → 2 ≤ Fintype.card (P c))
(hm : ∀ i, 0 < populationSize (fun c : D i => P c))
(left right : I → C → ℝ) (η τ α ν : I → ℝ) (hη : ∀ i, 0 < η i)
(hright0 : ∀ i c, c ∈ D i → 0 ≤ right i c)
(hright1 : ∀ i c, c ∈ D i → right i c ≤ 1)
(hleft : ∀ i c, c ∈ D i →
|classDensity (wordPairLeftPositions w (sl i) (s i)) c - left i c| ≤ τ i)
(hright : ∀ i c, c ∈ D i →
|classDensity (wordPairRightPositions w (sr i) (t i)) c - right i c| ≤ τ i)
(happrox : ∀ i, |classProductMixture (fun c : D i => P c)
(fun c => left i c) (fun c => right i c) - ν i| ≤ α i)
(hτ : ∀ i, τ i ≤ η i / 8) (hα : ∀ i, α i ≤ η i / 4) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(fun v => ∃ i, selectedWordPairMaskBad (D i) (sl i) (sr i)
(s i) (t i) (η i) (ν i) v)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
∑ i, 1 / (2 * populationSize (fun c : D i => P c) * (η i) ^ 2) := by
rw [← average_bad_shift_eq_orbit_fraction]
refine le_trans (average_indicator_exists_le_sum _) ?_
apply Finset.sum_le_sum
intro i _
change average (fun g : HalfClassPermutations P =>
if η i < |selectedWordPairCount (D i) (g • w) (sl i) (sr i) (s i) (t i) /
populationSize (fun c : D i => P c) - ν i| then 1 else 0) ≤ _
rw [average_selectedWordPairCount_shift (D i) w (sl i) (sr i) (s i) (t i)
(fun k => if η i < |k / populationSize (fun c : D i => P c) - ν i|
then 1 else 0)]
exact selectedTwoHalfMask_rejection_of_windows_le (D i) _ _ (hn i) (hm i)
(left i) (right i) (η i) (τ i) (α i) (ν i) (hη i)
(hright0 i) (hright1 i) (hleft i) (hright i) (happrox i) (hτ i) (hα i)
theorem finite_selectedWordPair_orbit_inverse_linear_of_windows
(D : I → Finset C) (w : CompleteWordPair P X Y)
(sl : ∀ i c, X c → SL i c) (sr : ∀ i c, Y c → SR i c)
(s : ∀ i c, SL i c) (t : ∀ i c, SR i c)
(hn : ∀ i c, c ∈ D i → 2 ≤ Fintype.card (P c))
(mass : I → ℝ) (N : ℝ) (hN : 0 < N) (hmass : ∀ i, 0 < mass i)
(hsize : ∀ i, populationSize (fun c : D i => P c) = mass i * N)
(left right : I → C → ℝ) (η τ α ν : I → ℝ) (hη : ∀ i, 0 < η i)
(hright0 : ∀ i c, c ∈ D i → 0 ≤ right i c)
(hright1 : ∀ i c, c ∈ D i → right i c ≤ 1)
(hleft : ∀ i c, c ∈ D i →
|classDensity (wordPairLeftPositions w (sl i) (s i)) c - left i c| ≤ τ i)
(hright : ∀ i c, c ∈ D i →
|classDensity (wordPairRightPositions w (sr i) (t i)) c - right i c| ≤ τ i)
(happrox : ∀ i, |classProductMixture (fun c : D i => P c)
(fun c => left i c) (fun c => right i c) - ν i| ≤ α i)
(hτ : ∀ i, τ i ≤ η i / 8) (hα : ∀ i, α i ≤ η i / 4) :
(((OrbitCounting.orbitSet (G := HalfClassPermutations P) w).filter
(fun v => ∃ i, selectedWordPairMaskBad (D i) (sl i) (sr i)
(s i) (t i) (η i) (ν i) v)).card : ℝ) /
(OrbitCounting.orbitSet (G := HalfClassPermutations P) w).card ≤
(∑ i, 1 / (2 * mass i * (η i) ^ 2)) / N := by
have hm (i : I) : 0 < populationSize (fun c : D i => P c) := by
rw [hsize]
exact mul_pos (hmass i) hN
have h := finite_selectedWordPair_orbit_rejection_of_windows_le D w sl sr s t hn hm
left right η τ α ν hη hright0 hright1 hleft hright happrox hτ hα
simpa only [hsize, inverse_linear_mask_constant] using h
end FiniteMasks
end
end MatrixMultiplication.PermutationMatching
end OAI
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