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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 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 | import OAI.LinearAlgebra.MatrixMultiplication.Entropy.ComplexSeparationAsymptotics
import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexSeparatedTensorDecomposition
namespace OAI
/-! Finite type counts, hierarchy separation and tensor execution bounds. -/
namespace MatrixMultiplication.Foundation
instance auxiliarySeparation_modulus_neZero (K : ℕ) :
NeZero (4 * Separation.gridWidth K) :=
⟨Nat.mul_ne_zero (by decide)
(Nat.ne_of_gt (lt_of_lt_of_le (by decide : 0 < 2)
(Separation.gridWidth_ge_two K)))⟩
section AuxiliarySeparation
variable {X Y Z S : Type*} [Fintype S] [DecidableEq S]
theorem auxiliary_separation_mem_restrictionOrbit
[Fintype X] [Fintype Y] [Fintype Z]
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
separatedTensor (U := Fin (Fintype.card S)) lx T ∈ Tensor.restrictionOrbit
(Tensor.product T (groupTensor (Separation.AuxiliaryGroup
(Separation.gridDimension (Fintype.card S))
(Separation.gridWidth (Fintype.card S))))) := by
exact finite_grid_separation_mem_restrictionOrbit T lx ly
(Separation.grid_population_bound (Fintype.card S))
(Separation.grid_population_bound (Fintype.card S)) hlabels
theorem auxiliary_separation_rankAtMost
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
(hT : Tensor.RankAtMost T r)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.RankAtMost (separatedTensor (U := Fin (Fintype.card S)) lx T)
(r * Separation.gridGroupOrder (Fintype.card S)) := by
exact finite_grid_separation_rankAtMost T lx ly
(lt_of_lt_of_le (by decide : 0 < 2)
(Separation.gridWidth_ge_two (Fintype.card S)))
(Separation.grid_population_bound (Fintype.card S))
(Separation.grid_population_bound (Fintype.card S)) hT hlabels
theorem auxiliary_separation_borderRankAtMost
[Fintype X] [Fintype Y] [Fintype Z]
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
(hT : Tensor.BorderRankAtMost T r)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.BorderRankAtMost (separatedTensor (U := Fin (Fintype.card S)) lx T)
(r * Separation.gridGroupOrder (Fintype.card S)) := by
exact finite_grid_separation_borderRankAtMost T lx ly
(lt_of_lt_of_le (by decide : 0 < 2)
(Separation.gridWidth_ge_two (Fintype.card S)))
(Separation.grid_population_bound (Fintype.card S))
(Separation.grid_population_bound (Fintype.card S)) hT hlabels
theorem auxiliary_separation_directSum_rankAtMost
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
(hT : Tensor.RankAtMost T r)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.RankAtMost (Tensor.dependentDirectSum (fun s =>
Tensor.product (labelFiberTensor lx ly T s)
(Tensor.dotPairing (Fin (Fintype.card S)))))
(r * Separation.gridGroupOrder (Fintype.card S)) := by
exact separatedTensor_fiber_directSum_rankAtMost T lx ly hlabels
(auxiliary_separation_rankAtMost T lx ly hT hlabels)
theorem auxiliary_separation_directSum_borderRankAtMost
[Fintype X] [Fintype Y] [Fintype Z]
(T : Tensor ℂ X Y Z) (lx : X → S) (ly : Y → S) {r : ℕ}
(hT : Tensor.BorderRankAtMost T r)
(hlabels : ∀ x y z, T x y z ≠ 0 → lx x = ly y) :
Tensor.BorderRankAtMost (Tensor.dependentDirectSum (fun s =>
Tensor.product (labelFiberTensor lx ly T s)
(Tensor.dotPairing (Fin (Fintype.card S)))))
(r * Separation.gridGroupOrder (Fintype.card S)) := by
exact separatedTensor_fiber_directSum_borderRankAtMost T lx ly hlabels
(auxiliary_separation_borderRankAtMost T lx ly hT hlabels)
end AuxiliarySeparation
theorem auxiliary_separation_logarithmic_cost :
Filter.Tendsto
(fun K : ℕ => Real.log (Separation.gridGroupOrder K : ℝ) / Real.log (K : ℝ))
Filter.atTop (nhds 1) :=
Separation.tendsto_log_gridGroupOrder_div_log_population
end MatrixMultiplication.Foundation
end OAI
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