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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 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.ExactRecovery
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.OrbitCounting
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Positivity
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
open scoped BigOperators
open MatrixMultiplication.Foundation
namespace MatrixMultiplication.OrbitRecovery
variable {G X Y Z : Type*} [Group G] [Fintype G]
[MulAction G X] [MulAction G Y] [MulAction G Z]
[DecidableEq X] [DecidableEq Y] [DecidableEq Z]
def badShifts (px : X → Prop) (py : Y → Prop) (pz : Z → Prop)
[DecidablePred px] [DecidablePred py] [DecidablePred pz]
(x : X) (y : Y) (z : Z) : Finset G :=
Finset.univ.filter fun g => ¬(px (g • x) ∧ py (g • y) ∧ pz (g • z))
omit [DecidableEq X] [DecidableEq Y] [DecidableEq Z] in
theorem badShifts_card_le (px : X → Prop) (py : Y → Prop) (pz : Z → Prop)
[DecidablePred px] [DecidablePred py] [DecidablePred pz]
(x : X) (y : Y) (z : Z) :
(badShifts (G := G) px py pz x y z).card ≤
(Finset.univ.filter fun g : G => ¬px (g • x)).card +
(Finset.univ.filter fun g : G => ¬py (g • y)).card +
(Finset.univ.filter fun g : G => ¬pz (g • z)).card := by
classical
have heq : badShifts (G := G) px py pz x y z =
(Finset.univ.filter fun g : G => ¬px (g • x)) ∪
((Finset.univ.filter fun g : G => ¬py (g • y)) ∪
(Finset.univ.filter fun g : G => ¬pz (g • z))) := by
ext g
by_cases hx : px (g • x) <;> by_cases hy : py (g • y) <;>
by_cases hz : pz (g • z) <;> simp [badShifts, hx, hy, hz]
rw [heq]
exact (Finset.card_union_le _ _).trans
((Nat.add_le_add_left (Finset.card_union_le _ _) _).trans (le_of_eq (Nat.add_assoc _ _ _).symm))
theorem badShifts_fraction_le_at (px : X → Prop) (py : Y → Prop) (pz : Z → Prop)
[DecidablePred px] [DecidablePred py] [DecidablePred pz]
(γ : ℝ) (x : X) (y : Y) (z : Z)
(hx : (((OrbitCounting.orbitSet (G := G) x).filter fun v => ¬px v).card : ℝ) /
(OrbitCounting.orbitSet (G := G) x).card ≤ γ)
(hy : (((OrbitCounting.orbitSet (G := G) y).filter fun v => ¬py v).card : ℝ) /
(OrbitCounting.orbitSet (G := G) y).card ≤ γ)
(hz : (((OrbitCounting.orbitSet (G := G) z).filter fun v => ¬pz v).card : ℝ) /
(OrbitCounting.orbitSet (G := G) z).card ≤ γ)
:
((badShifts (G := G) px py pz x y z).card : ℝ) / Fintype.card G ≤ 3 * γ := by
have hg : (0 : ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
have hx' := hx
have hy' := hy
have hz' := hz
rw [← OrbitCounting.bad_shift_fraction (G := G) x (fun v => ¬px v)] at hx'
rw [← OrbitCounting.bad_shift_fraction (G := G) y (fun v => ¬py v)] at hy'
rw [← OrbitCounting.bad_shift_fraction (G := G) z (fun v => ¬pz v)] at hz'
have hxc := (div_le_iff₀ hg).mp hx'
have hyc := (div_le_iff₀ hg).mp hy'
have hzc := (div_le_iff₀ hg).mp hz'
have hc : ((badShifts (G := G) px py pz x y z).card : ℝ) ≤
((Finset.univ.filter fun g : G => ¬px (g • x)).card : ℝ) +
((Finset.univ.filter fun g : G => ¬py (g • y)).card : ℝ) +
((Finset.univ.filter fun g : G => ¬pz (g • z)).card : ℝ) := by
exact_mod_cast badShifts_card_le (G := G) px py pz x y z
apply (div_le_iff₀ hg).mpr
linarith
theorem floor_loss_fraction (γ : ℝ) (hγ : 0 ≤ γ) :
(Nat.floor (3 * γ * Fintype.card G) : ℝ) / Fintype.card G ≤ 3 * γ := by
have hg : (0 : ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
apply (div_le_iff₀ hg).mpr
exact Nat.floor_le (by positivity)
theorem repair_from_orbit_loss
{K : Type*} [CommSemiring K] [Fintype X] [Fintype Y] [Fintype Z]
(Q : Tensor K X Y Z)
(px : X → Prop) (py : Y → Prop) (pz : Z → Prop)
[DecidablePred px] [DecidablePred py] [DecidablePred pz]
(hpreserve : ∀ (g : G) x y z, Q (g • x) (g • y) (g • z) = Q x y z)
(γ : ℝ)
(hx : ∀ x, (∃ y z, Q x y z ≠ 0) → (((OrbitCounting.orbitSet (G := G) x).filter fun v => ¬px v).card : ℝ) /
(OrbitCounting.orbitSet (G := G) x).card ≤ γ)
(hy : ∀ y, (∃ x z, Q x y z ≠ 0) → (((OrbitCounting.orbitSet (G := G) y).filter fun v => ¬py v).card : ℝ) /
(OrbitCounting.orbitSet (G := G) y).card ≤ γ)
(hz : ∀ z, (∃ x y, Q x y z ≠ 0) → (((OrbitCounting.orbitSet (G := G) z).filter fun v => ¬pz v).card : ℝ) /
(OrbitCounting.orbitSet (G := G) z).card ≤ γ)
(L : ℕ) (hL : 0 < L)
(hbudget : (@Finset.filter (X × Y × Z)
(fun p => Q p.1 p.2.1 p.2.2 ≠ 0) (Classical.decPred _) Finset.univ).card *
Nat.floor (3 * γ * Fintype.card G) ^ L < Fintype.card G ^ L) :
∃ (a : X → ((Finset (Fin L) × Finset (Fin L) × Finset (Fin L)) × X) → K)
(b : Y → ((Finset (Fin L) × Finset (Fin L) × Finset (Fin L)) × Y) → K)
(c : Z → ((Finset (Fin L) × Finset (Fin L) × Finset (Fin L)) × Z) → K),
Tensor.restrict a b c (Tensor.directSum
(fun _ : Finset (Fin L) × Finset (Fin L) × Finset (Fin L) =>
ExactRecovery.delete Q px py pz)) = Q := by
classical
apply ExactRecovery.repair_of_counting (I := G) Q (ExactRecovery.delete Q px py pz)
(fun g x => px (g • x)) (fun g y => py (g • y)) (fun g z => pz (g • z))
(fun g x => g • x) (fun g y => g • y) (fun g z => g • z)
(fun g x y z => by simp [ExactRecovery.delete, hpreserve])
(Nat.floor (3 * γ * Fintype.card G)) L hL ?_ hbudget
intro x y z hq
have heq : (@Finset.filter G
(fun g => ¬(px (g • x) ∧ py (g • y) ∧ pz (g • z)))
(Classical.decPred _) Finset.univ) = badShifts (G := G) px py pz x y z := by
ext g
simp only [badShifts, Finset.mem_filter]
rw [heq]
apply Nat.le_floor
have hg : (0 : ℝ) < Fintype.card G := Nat.cast_pos.mpr Fintype.card_pos
exact (div_le_iff₀ hg).mp (badShifts_fraction_le_at (G := G) px py pz γ x y z
(hx x ⟨y, z, hq⟩) (hy y ⟨x, z, hq⟩) (hz z ⟨x, y, hq⟩))
end MatrixMultiplication.OrbitRecovery
end OAI
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