openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /Recovery /InverseLinearRecovery.lean
| 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 : ((.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 | |