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cdec422 | 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.MaskCore
import OAI.LinearAlgebra.MatrixMultiplication.Recovery.InverseLinearRecovery
namespace OAI
/-! Finite extraction histories, inherited masks and recovery bounds. -/
noncomputable section
namespace MatrixMultiplication.AllFieldHistoryRecovery
open AllFieldHistory AllFieldHistorySupport
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
def minimumDilation (allocation : Allocation) (ε : ℝ) : ℕ :=
Nat.ceil (3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6)) + 2
theorem minimumDilation_spec (allocation : Allocation) (ε : ℝ) {m : ℕ}
(hm : minimumDilation (K := K) (tick := tick) allocation ε ≤ m) :
2 ≤ m ∧
3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6) <
(m : ℝ) := by
constructor
· exact (Nat.le_add_left 2 _).trans hm
· have hc := Nat.le_ceil
(3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6))
have hm' : (Nat.ceil
(3 * (AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6)) : ℝ) + 2 ≤ m := by
exact_mod_cast hm
linarith
def shiftCount (allocation : Allocation) (ε : ℝ) (m : ℕ) : ℕ :=
RecoveryGrowth.shiftCount (supportRate (K := K) (tick := tick) allocation)
(AllFieldHistoryMasks.lossConstant (K := K) (tick := tick) allocation ε + 6) m
theorem replication_subexponential (allocation : Allocation) (ε : ℝ) :
Filter.Tendsto (fun m : ℕ =>
Real.log (Fintype.card (InverseLinearRecovery.MaskRectangles
(shiftCount (K := K) (tick := tick) allocation ε m))) / m)
Filter.atTop (nhds 0) :=
InverseLinearRecovery.subexponential_copies (supportRate_pos allocation)
(by have := AllFieldHistoryMasks.lossConstant_nonneg (K := K) (tick := tick) allocation ε
linarith)
end MatrixMultiplication.AllFieldHistoryRecovery
end
end OAI
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