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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 | import OAI.LinearAlgebra.MatrixMultiplication.Recovery.OrbitRecovery
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.RecoveryGrowth
namespace OAI
/-! Finite orbit symmetries, masks and exact recovery operations. -/
open MatrixMultiplication.Foundation
namespace MatrixMultiplication.InverseLinearRecovery
abbrev MaskRectangles (L : ℕ) := Finset (Fin L) × Finset (Fin L) × Finset (Fin L)
def RecoveredBy {K X Y Z : Type*} [CommSemiring K]
[Fintype X] [Fintype Y] [Fintype Z]
(Q H : Tensor K X Y Z) (L : ℕ) : Prop :=
∃ (a : X → (MaskRectangles L × X) → K)
(b : Y → (MaskRectangles L × Y) → K)
(c : Z → (MaskRectangles L × Z) → K),
Tensor.restrict a b c (Tensor.directSum (fun _ : MaskRectangles L => H)) = Q
theorem repair_inverse_linear
{K G X Y Z : Type*} [CommSemiring K] [Group G] [Fintype G]
[MulAction G X] [MulAction G Y] [MulAction G Z]
[Fintype X] [Fintype Y] [Fintype Z]
[DecidableEq X] [DecidableEq Y] [DecidableEq 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)
{b C : ℝ} {N : ℕ} (hb : 0 ≤ b) (hC : 0 < C) (hN : 3 * C < (N : ℝ))
(hsupport : ((@Finset.filter (X × Y × Z)
(fun p => Q p.1 p.2.1 p.2.2 ≠ 0) (Classical.decPred _) Finset.univ).card : ℝ) ≤
Real.exp (b * N))
(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 ≤ C / N)
(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 ≤ C / N)
(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 ≤ C / N) :
RecoveredBy Q (ExactRecovery.delete Q px py pz) (RecoveryGrowth.shiftCount b C N) := by
classical
apply OrbitRecovery.repair_from_orbit_loss Q px py pz hpreserve (C / N) hx hy hz
(RecoveryGrowth.shiftCount b C N) (RecoveryGrowth.shiftCount_pos hb hC hN)
apply RecoveryGrowth.cardinal_budget hC hN Fintype.card_pos hsupport
simpa only [mul_div_assoc] using
OrbitRecovery.floor_loss_fraction (G := G) (C / N)
(div_nonneg hC.le (Nat.cast_nonneg N))
theorem card_recovery_copies (b C : ℝ) (N : ℕ) :
Fintype.card (MaskRectangles (RecoveryGrowth.shiftCount b C N)) =
RecoveryGrowth.replicationCount b C N := by
simpa only [MaskRectangles, Fintype.card_fin, RecoveryGrowth.replicationCount] using
(ExactRecovery.card_mask_rectangles (I := Fin (RecoveryGrowth.shiftCount b C N)))
theorem subexponential_copies {b C : ℝ} (hb : 0 < b) (hC : 0 < C) :
Filter.Tendsto (fun N : ℕ =>
Real.log (Fintype.card (MaskRectangles (RecoveryGrowth.shiftCount b C N))) / N)
Filter.atTop (nhds 0) := by
simp_rw [card_recovery_copies]
exact RecoveryGrowth.tendsto_log_replicationCount_div_nat hb hC
end MatrixMultiplication.InverseLinearRecovery
end OAI
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