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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 | import OAI.LinearAlgebra.MatrixMultiplication.CoppersmithWinograd.FieldCW
import OAI.LinearAlgebra.MatrixMultiplication.FieldConstruction.Witness
import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialRestriction
import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexTensorBatching
namespace OAI
/-! Tensor extraction over arbitrary fields and its asymptotic rate. -/
noncomputable section
open scoped BigOperators
open MatrixMultiplication.Foundation
namespace MatrixMultiplication.AllFieldSource
abbrev Word (K N : ℕ) := Fin (8 * K * N) → Fin 7
abbrev Index (Copies : Type*) (K N : ℕ) := Copies × Word K N
def tensor (F Copies : Type*) [CommRing F] [DecidableEq Copies] (K N : ℕ) :
Tensor F (Index Copies K N) (Index Copies K N) (Index Copies K N) :=
Tensor.directSum (fun _ : Copies => Tensor.power (FieldCW.tensor F 5) (8 * K * N))
def rankBudget (Copies : Type*) [Fintype Copies] (K N : ℕ) : ℕ :=
Fintype.card Copies * ((3 * (8 * K * N) + 1) ^ 2 * 7 ^ (8 * K * N))
theorem rank_bound (F Copies : Type*) [Field F]
[Fintype Copies] [DecidableEq Copies] (K N : ℕ) :
Tensor.RankAtMost (tensor F Copies K N) (rankBudget Copies K N) := by
have h := (Tensor.diagonal_rankAtMost (K := F) Copies).product
(FieldCW.rank_power F 5 (8 * K * N))
convert h using 1
funext x y z
simp [tensor, Tensor.directSum, Tensor.product, Tensor.diagonal, ite_mul]
rfl
theorem rankBudget_pos (Copies : Type*) [Fintype Copies] (K N : ℕ)
(hcopies : 0 < Fintype.card Copies) : 0 < rankBudget Copies K N := by
unfold rankBudget
exact Nat.mul_pos hcopies
(Nat.mul_pos (pow_pos (Nat.succ_pos _) _) (pow_pos (by decide) _))
theorem rankBudget_prod (Slots Copies : Type*) [Fintype Slots] [Fintype Copies]
(K N : ℕ) :
rankBudget (Slots × Copies) K N = Fintype.card Slots * rankBudget Copies K N := by
simp only [rankBudget, Fintype.card_prod, Nat.mul_assoc]
def matrixTarget (F : Type*) [CommSemiring F] (a b c L : ℕ) :=
Tensor.directSum (fun _ : Fin L =>
Tensor.matrixCoefficients (K := F) (Fin a) (Fin b) (Fin c))
section Consumer
variable {F Copies : Type*} [Field F] [Fintype Copies] [DecidableEq Copies]
variable {K N a b c L : ℕ}
theorem restricted_rank_bound
(A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F)
(B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F)
(C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F)
(hcoeff : ∀ x y z,
Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z) :
Tensor.RankAtMost (matrixTarget F a b c L) (rankBudget Copies K N) := by
have heq : Tensor.restrict A B C (tensor F Copies K N) = matrixTarget F a b c L :=
funext fun x => funext fun y => funext fun z => hcoeff x y z
rw [← heq]
exact (rank_bound F Copies K N).restrict A B C
def restrictedApproximation
(A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F)
(B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F)
(C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F)
(hcoeff : ∀ x y z,
Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z) :
Tensor.PolynomialApproximation (matrixTarget F a b c L) (rankBudget Copies K N) 0 0 :=
Tensor.PolynomialApproximation.ofRankAtMost _ _ (restricted_rank_bound A B C hcoeff)
def ofRestriction
(A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F)
(B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F)
(C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F)
(hcoeff : ∀ x y z,
Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z)
(hcopies : 0 < Fintype.card Copies) (ha : 0 < a) (hb : 0 < b)
(hc : 0 < c) (hL : 0 < L) : AllFieldWitness F where
rows := a
inner := b
columns := c
multiplicity := L
rankBound := rankBudget Copies K N
order := 0
degree := 0
rows_pos := ha
inner_pos := hb
columns_pos := hc
multiplicity_pos := hL
rankBound_pos := rankBudget_pos Copies K N hcopies
approximation := restrictedApproximation A B C hcoeff
theorem ofRestriction_polynomial
(A : (Fin L × (Fin a × Fin b)) → Index Copies K N → F)
(B : (Fin L × (Fin b × Fin c)) → Index Copies K N → F)
(C : (Fin L × (Fin c × Fin a)) → Index Copies K N → F)
(hcoeff : ∀ x y z,
Tensor.restrict A B C (tensor F Copies K N) x y z = matrixTarget F a b c L x y z)
(hcopies : 0 < Fintype.card Copies) (ha : 0 < a) (hb : 0 < b)
(hc : 0 < c) (hL : 0 < L) :
(ofRestriction A B C hcoeff hcopies ha hb hc hL).approximation.polynomial =
fun x y z => Polynomial.C (matrixTarget F a b c L x y z) := rfl
end Consumer
end MatrixMultiplication.AllFieldSource
end
end OAI
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