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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 92 93 94 95 96 | import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialKernelExecution
namespace OAI
/-! Finite type counts, hierarchy separation and tensor execution bounds. -/
noncomputable section
namespace MatrixMultiplication.ExecutionApproximation
open MatrixMultiplication.Foundation
open Tensor LocalMaps PolynomialLocalConstruction PolynomialKernelExecution
variable {X Y Z Prefix PX PY PZ AX AY AZ : Type*}
variable [Fintype AX] [Fintype AY] [Fintype AZ]
variable {support : X → Y → Z → Prop}
variable (E : Execution X Y Z Prefix PX PY PZ AX AY AZ support)
theorem polynomial_substitution (original : Tensor ℂ X Y Z) :
fiberTransform E.leftMap E.middleMap E.rightMap
(fun x y z => Polynomial.C (Tensor.product original E.auxiliary x y z)) =
originalScale (fun x y z => Polynomial.C (original x y z))
(kernel E.auxiliary E.leftMap E.middleMap E.rightMap) := by
have h := fiberTransform_originalScale E.leftMap E.middleMap E.rightMap
(fun x y z => Polynomial.C (original x y z))
(fun x y z => Polynomial.C (E.auxiliary x.2 y.2 z.2))
have hinput :
(fun x y z => Polynomial.C (Tensor.product original E.auxiliary x y z)) =
originalScale (fun x y z => Polynomial.C (original x y z))
(fun x y z => Polynomial.C (E.auxiliary x.2 y.2 z.2)) := by
funext x y z
simp only [Tensor.product, originalScale, map_mul]
rw [hinput]
simpa only [kernel] using h
variable [Fintype X] [Fintype Y] [Fintype Z]
def degeneration (original : Tensor ℂ X Y Z)
(supported : ∀ x y z, ¬ support x y z → original x y z = 0) :
PolynomialRestrictionDegeneration (Tensor.product original E.auxiliary)
(originalScale original E.value) E.order E.leftDegree E.middleDegree E.rightDegree := by
classical
refine {
leftMap := fiberMatrix E.leftMap
middleMap := fiberMatrix E.middleMap
rightMap := fiberMatrix E.rightMap
left_degree := ?_
middle_degree := ?_
right_degree := ?_
vanishes := ?_
leading := ?_
}
· intro x i
by_cases h : x.1 = i.1
· simpa only [fiberMatrix, ite_eq_left h] using E.left_degree x.1 x.2 i.2
· simp only [fiberMatrix, ite_eq_right h, Polynomial.degree_zero]
exact bot_le
· intro y j
by_cases h : y.1 = j.1
· simpa only [fiberMatrix, ite_eq_left h] using E.middle_degree y.1 y.2 j.2
· simp only [fiberMatrix, ite_eq_right h, Polynomial.degree_zero]
exact bot_le
· intro z k
by_cases h : z.1 = k.1
· simpa only [fiberMatrix, ite_eq_left h] using E.right_degree z.1 z.2 k.2
· simp only [fiberMatrix, ite_eq_right h, Polynomial.degree_zero]
exact bot_le
· intro x y z j hj
rw [← fiberTransform_eq_restrict, polynomial_substitution E original]
simp only [originalScale, Polynomial.coeff_C_mul]
by_cases hs : support x.1 y.1 z.1
· rw [E.vanishes x y z hs j hj, mul_zero]
· rw [supported x.1 y.1 z.1 hs, zero_mul]
· intro x y z
rw [← fiberTransform_eq_restrict, polynomial_substitution E original]
simp only [originalScale, Polynomial.coeff_C_mul]
by_cases hs : support x.1 y.1 z.1
· rw [E.leading x y z hs]
· rw [supported x.1 y.1 z.1 hs]
simp only [zero_mul]
def approximation {original : Tensor ℂ X Y Z} {r d D : ℕ}
(source : PolynomialApproximation original r d D)
(supported : ∀ x y z, ¬ support x y z → original x y z = 0) :
PolynomialApproximation (originalScale original E.value)
(r * E.rankBound) (d * (E.order + 1) + E.order)
(D * (E.order + 1) + E.leftDegree + E.middleDegree + E.rightDegree) := by
have combined := source.productExact E.auxiliary E.rankBound E.auxiliary_rank
exact (degeneration E original supported).compose combined
end MatrixMultiplication.ExecutionApproximation
end
end OAI
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