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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 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 | import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.ConstructionLabels
import OAI.LinearAlgebra.MatrixMultiplication.FieldHistory.RecoveryBudget
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.SourceRates
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
namespace MatrixMultiplication.AllFieldConstructionRates
open MatrixMultiplication.Foundation AllFieldHistory Filter
open scoped BigOperators Topology Classical
variable {K : ℕ}
abbrev Copies (allocation : Allocation) (ε : ℝ) (m : ℕ) :=
∀ tick : Fin (K + 2), InverseLinearRecovery.MaskRectangles
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick.val) allocation ε m)
theorem card_copies (allocation : Allocation) (ε : ℝ) (m : ℕ) :
Fintype.card (Copies (K := K) allocation ε m) =
∏ tick : Fin (K + 2), RecoveryGrowth.replicationCount
(AllFieldHistorySupport.supportRate (K := K) (tick := tick.val) allocation)
(AllFieldHistoryMasks.lossConstant (K := K) (tick := tick.val) allocation ε + 6) m := by
rw [Fintype.card_pi]
apply Finset.prod_congr rfl
intro tick _
exact InverseLinearRecovery.card_recovery_copies
(AllFieldHistorySupport.supportRate (K := K) (tick := tick.val) allocation)
(AllFieldHistoryMasks.lossConstant (K := K) (tick := tick.val) allocation ε + 6) m
theorem card_copies_range (allocation : Allocation) (ε : ℝ) (m : ℕ) :
Fintype.card (Copies (K := K) allocation ε m) =
∏ tick ∈ Finset.range (K + 2), Fintype.card
(InverseLinearRecovery.MaskRectangles
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m)) := by
rw [Fintype.card_pi]
exact Fin.prod_univ_eq_prod_range (fun tick : ℕ => Fintype.card
(InverseLinearRecovery.MaskRectangles
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick) allocation ε m))) (K + 2)
theorem card_copies_pos (allocation : Allocation) (ε : ℝ) (m : ℕ) :
0 < Fintype.card (Copies (K := K) allocation ε m) := Fintype.card_pos
theorem copies_subexponential (allocation : Allocation) (ε : ℝ) :
Tendsto (fun m : ℕ => Real.log (Fintype.card (Copies (K := K) allocation ε m) : ℝ) /
(m : ℝ)) atTop (𝓝 0) := by
have h := RecoveryGrowth.tendsto_log_prod_div_nat (Finset.univ : Finset (Fin (K + 2)))
(r := fun tick m => Fintype.card (InverseLinearRecovery.MaskRectangles
(AllFieldHistoryRecovery.shiftCount (K := K) (tick := tick.val) allocation ε m)))
(fun _ _ _ => Fintype.card_pos)
(fun tick _ => AllFieldHistoryRecovery.replication_subexponential
(K := K) (tick := tick.val) allocation ε)
simpa only [Copies, Fintype.card_pi] using h
theorem copies_rate (hK : 0 < K) (allocation : Allocation) (ε : ℝ) :
Tendsto (fun m : ℕ => Real.log (Fintype.card (Copies (K := K) allocation ε m) : ℝ) /
((K : ℝ) * populationLength (K := K) allocation m)) atTop (𝓝 0) := by
have hden : (K : ℝ) * denominator (K := K) allocation ≠ 0 :=
mul_ne_zero (Nat.cast_ne_zero.mpr hK.ne')
(Nat.cast_ne_zero.mpr (denominator_pos allocation).ne')
have h := (copies_subexponential (K := K) allocation ε).div tendsto_const_nhds hden
simp only [zero_div] at h
apply h.congr
intro m
rw [Pi.div_apply, population_length, Nat.cast_mul]
ring
theorem source_rank_rate (hK : 0 < K) (allocation : Allocation) (ε : ℝ) :
Tendsto (fun m : ℕ => Real.log
(AllFieldSource.rankBudget (Copies (K := K) allocation ε m) K
(populationLength (K := K) allocation m) : ℝ) /
((K : ℝ) * populationLength (K := K) allocation m))
atTop (𝓝 (8 * Real.log 7)) := by
have hp := (AllFieldSourceRates.tendsto_log_polynomial_overhead K hK).comp
(populationLength_tendsto (K := K) allocation)
have he := (AllFieldSourceRates.tendsto_log_exponential_source K hK).comp
(populationLength_tendsto (K := K) allocation)
have h := ((copies_rate hK allocation ε).add hp).add he
simp only [zero_add] at h
apply h.congr
intro m
rw [AllFieldSourceRates.log_rankBudget _ _ _ (card_copies_pos allocation ε m), add_div, add_div]
rfl
theorem source_rank_terminal_rate (hK : 0 < K) (allocation : Allocation) (ε : ℝ) :
Tendsto (fun m : ℕ => Real.log
(AllFieldSource.rankBudget
(Copies (K := K) allocation ε (AllFieldTerminalRates.terminalDilation K m)) K
(AllFieldTerminalRates.terminalLength allocation K m) : ℝ) /
((K : ℝ) * AllFieldTerminalRates.terminalLength allocation K m))
atTop (𝓝 (8 * Real.log 7)) :=
(source_rank_rate hK allocation ε).comp (terminalDilation_tendsto K)
theorem source_rank_terminal_rate_of_cardinality
(ActualCopies : ℕ → Type*) [∀ m, Fintype (ActualCopies m)]
(hK : 0 < K) (allocation : Allocation) (ε : ℝ)
(cards : ∀ᶠ m : ℕ in atTop, Fintype.card (ActualCopies m) = Fintype.card
(Copies (K := K) allocation ε (AllFieldTerminalRates.terminalDilation K m))) :
Tendsto (fun m : ℕ => Real.log (AllFieldSource.rankBudget (ActualCopies m) K
(AllFieldTerminalRates.terminalLength allocation K m) : ℝ) /
((K : ℝ) * AllFieldTerminalRates.terminalLength allocation K m))
atTop (𝓝 (8 * Real.log 7)) := by
apply (source_rank_terminal_rate hK allocation ε).congr'
filter_upwards [cards] with m hm
simp only [AllFieldSource.rankBudget, hm]
end MatrixMultiplication.AllFieldConstructionRates
end
end OAI
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