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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 97 98 99 100 101 102 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexEqualSummandASI
import OAI.LinearAlgebra.MatrixMultiplication.Polynomial.ComplexPolynomialWitnessProduct
namespace OAI
/-! Rectangular matrix multiplication algorithms and their asymptotic exponents. -/
noncomputable section
namespace MatrixMultiplication.Foundation
namespace Tensor
private def rectangularTagEquiv (L : ℕ) :
Fin (L ^ 3) ≃ (Fin L × Fin L) × Fin L :=
Fintype.equivOfCardEq (by simp [pow_succ])
private def rectangularIndexEquiv (a b c : ℕ) :
Fin (a * b * c) ≃ Fin a × Fin b × Fin c :=
Fintype.equivOfCardEq (by simp [mul_assoc])
theorem matrixMultiplication_directSum_symmetrized_rank {a b c L r : ℕ}
(h : RankAtMost (directSum (fun _ : Fin L => matrixMultiplication a b c)) r) :
RankAtMost (directSum (fun _ : Fin (L ^ 3) =>
matrixMultiplication (a * b * c) (a * b * c) (a * b * c))) (r ^ 3) := by
have hs := matrixCoefficients_directSum_symmetrized_rank h
exact matrixCoefficients_directSum_rank_of_injective hs
(rectangularTagEquiv L) (rectangularTagEquiv L).injective
(rectangularIndexEquiv a b c) (rectangularIndexEquiv a b c).injective
def PolynomialApproximation.matrixCoefficients_directSum_of_injective
{F ι κ A B : Type*} [Field F]
[DecidableEq ι] [DecidableEq κ] [DecidableEq A] [DecidableEq B] {r d D : ℕ}
(P : PolynomialApproximation
(directSum (fun _ : ι => matrixCoefficients (K := F) A A A)) r d D)
(f : κ → ι) (hf : Function.Injective f) (g : B → A) (hg : Function.Injective g) :
PolynomialApproximation
(directSum (fun _ : κ => matrixCoefficients (K := F) B B B)) r d D := by
let coord : κ × (B × B) → ι × (A × A) :=
fun x => (f x.1, (g x.2.1, g x.2.2))
have heq : Tensor.pullback coord coord coord
(directSum (fun _ : ι => matrixCoefficients (K := F) A A A)) =
directSum (fun _ : κ => matrixCoefficients (K := F) B B B) := by
funext x y z
simp only [Tensor.pullback, coord, directSum, matrixCoefficients, hf.eq_iff, hg.eq_iff]
rw [← heq]
exact P.pullback coord coord coord
def PolynomialApproximation.matrixMultiplication_directSum_symmetrized
{a b c L r d D : ℕ}
(P : PolynomialApproximation
(directSum (fun _ : Fin L => matrixMultiplication a b c)) r d D) :
PolynomialApproximation (directSum (fun _ : Fin (L ^ 3) =>
matrixMultiplication (a * b * c) (a * b * c) (a * b * c)))
(r ^ 3) (3 * d) (3 * D) :=
P.matrixCoefficients_directSum_symmetrized.matrixCoefficients_directSum_of_injective
(rectangularTagEquiv L) (rectangularTagEquiv L).injective
(rectangularIndexEquiv a b c) (rectangularIndexEquiv a b c).injective
end Tensor
theorem mul_rpow_third_le_of_cube {u v w r : ℝ} (hv : 0 ≤ v) (hr : 0 ≤ r)
(h : u ^ 3 * v ^ w ≤ r ^ 3) : u * v ^ (w / 3) ≤ r := by
have hpow : (v ^ (w / 3)) ^ (3 : ℕ) = v ^ w := by
rw [← Real.rpow_mul_natCast hv]
congr 1
norm_num
apply le_of_pow_le_pow_left₀ (by decide : (3 : ℕ) ≠ 0) hr
simpa only [mul_pow, hpow] using h
namespace Arithmetic
theorem equalRectangular_inequality {a b c L r : ℕ}
(ha : 1 ≤ a) (hb : 1 ≤ b) (hc : 1 ≤ c)
(h : Tensor.RankAtMost
(Tensor.directSum (fun _ : Fin L => Tensor.matrixMultiplication a b c)) r) :
(L : ℝ) * ((a * b * c : ℕ) : ℝ) ^ (omega / 3) ≤ (r : ℝ) := by
have hv : 1 ≤ a * b * c := Nat.mul_pos (Nat.mul_pos ha hb) hc
have hs := equalSummand_inequality hv
(Tensor.matrixMultiplication_directSum_symmetrized_rank h)
apply mul_rpow_third_le_of_cube (Nat.cast_nonneg (a * b * c)) (Nat.cast_nonneg r)
simpa only [Nat.cast_pow] using hs
theorem equalRectangular_inequality_of_polynomialApproximation
{a b c L r d D : ℕ} (ha : 1 ≤ a) (hb : 1 ≤ b) (hc : 1 ≤ c)
(P : Tensor.PolynomialApproximation
(Tensor.directSum (fun _ : Fin L => Tensor.matrixMultiplication a b c)) r d D) :
(L : ℝ) * ((a * b * c : ℕ) : ℝ) ^ (omega / 3) ≤ (r : ℝ) := by
have hv : 1 ≤ a * b * c := Nat.mul_pos (Nat.mul_pos ha hb) hc
have hs := equalSummand_inequality_of_polynomialApproximation hv
P.matrixMultiplication_directSum_symmetrized
apply mul_rpow_third_le_of_cube (Nat.cast_nonneg (a * b * c)) (Nat.cast_nonneg r)
simpa only [Nat.cast_pow] using hs
end Arithmetic
end MatrixMultiplication.Foundation
end
end OAI
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