| 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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|
