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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 | import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexAuxiliarySeparation
import OAI.LinearAlgebra.MatrixMultiplication.Separation.ComplexStageHierarchyResources
namespace OAI
/-! Finite type counts, hierarchy separation and tensor execution bounds. -/
noncomputable section
namespace MatrixMultiplication
open MatrixMultiplication.Foundation Filter
open scoped Topology BigOperators
namespace AuxiliarySeparation
variable {X Y Z Label : Type*} [Fintype Label] [DecidableEq Label]
theorem directSum_rankAtMost (T : Tensor ℂ X Y Z)
(readX : X → Label) (readY : Y → Label) {R : ℕ}
(hT : Tensor.RankAtMost T R)
(readable : ∀ x y z, T x y z ≠ 0 → readX x = readY y) :
Tensor.RankAtMost (Tensor.dependentDirectSum (fun label =>
Tensor.product (labelFiberTensor readX readY T label)
(Tensor.dotPairing (Fin (Fintype.card Label)))))
(R * Separation.gridGroupOrder (Fintype.card Label)) :=
auxiliary_separation_directSum_rankAtMost T readX readY hT readable
theorem directSum_borderRankAtMost [Fintype X] [Fintype Y] [Fintype Z]
(T : Tensor ℂ X Y Z) (readX : X → Label) (readY : Y → Label) {R : ℕ}
(hT : Tensor.BorderRankAtMost T R)
(readable : ∀ x y z, T x y z ≠ 0 → readX x = readY y) :
Tensor.BorderRankAtMost (Tensor.dependentDirectSum (fun label =>
Tensor.product (labelFiberTensor readX readY T label)
(Tensor.dotPairing (Fin (Fintype.card Label)))))
(R * Separation.gridGroupOrder (Fintype.card Label)) :=
auxiliary_separation_directSum_borderRankAtMost T readX readY hT readable
theorem logarithmic_cost :
Tendsto (fun K : ℕ => Real.log (Separation.gridGroupOrder K : ℝ) /
Real.log (K : ℝ)) atTop (𝓝 1) :=
auxiliary_separation_logarithmic_cost
theorem cost_minus_population_tendsto_zero {population : ℕ → ℕ} {rate : ℝ}
(positive : ∀ n, 0 < population n) (rate_nonneg : 0 ≤ rate)
(population_rate : Tendsto
(fun n => Real.log (population n : ℝ) / (n : ℝ)) atTop (𝓝 rate)) :
Tendsto (fun n =>
(Real.log (LabelHierarchySeparation.poolAuxiliaryCost (population n) : ℝ) -
Real.log (population n : ℝ)) / (n : ℝ)) atTop (𝓝 0) := by
have h := (LabelHierarchySeparation.tendsto_log_poolAuxiliaryCost_div_nat
positive rate_nonneg population_rate).sub population_rate
simpa only [sub_self, sub_div] using h
end AuxiliarySeparation
end MatrixMultiplication
end
end OAI
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