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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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 | import Mathlib
import OAI.LinearAlgebra.MatrixFields.Entropy.ZeroLeafRates
import OAI.LinearAlgebra.MatrixFields.Tensors.ComplexSeparatedTensorDecomposition
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
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
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 MatrixAllFields
end OAI
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