File size: 6,136 Bytes
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 | import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexEqualSummandASI
namespace OAI
/-! Rectangular matrix multiplication algorithms and their asymptotic exponents. -/
noncomputable section
namespace MatrixMultiplication
open MatrixMultiplication.Foundation
namespace Tensor
open MatrixMultiplication.Foundation.Tensor
theorem rectangular_rank_of_injective
{K A B C A' B' C' : Type*} [CommSemiring K]
[DecidableEq A] [DecidableEq B] [DecidableEq C]
[DecidableEq A'] [DecidableEq B'] [DecidableEq C'] {R : ℕ}
(h : RankAtMost (matrixCoefficients (K := K) A B C) R)
(f : A' → A) (hf : Function.Injective f)
(g : B' → B) (hg : Function.Injective g)
(j : C' → C) (hj : Function.Injective j) :
RankAtMost (matrixCoefficients (K := K) A' B' C') R := by
have hp := h.pullback (fun x : A' × B' => (f x.1, g x.2))
(fun y : B' × C' => (g y.1, j y.2))
(fun z : C' × A' => (j z.1, f z.2))
convert hp using 1
funext x y z
simp only [pullback, matrixCoefficients, hf.eq_iff, hg.eq_iff, hj.eq_iff]
theorem rectangular_directSum_rank_of_injective
{K ι κ A B C A' B' C' : Type*} [CommSemiring K]
[DecidableEq ι] [DecidableEq κ]
[DecidableEq A] [DecidableEq B] [DecidableEq C]
[DecidableEq A'] [DecidableEq B'] [DecidableEq C'] {R : ℕ}
(h : RankAtMost (directSum (fun _ : ι => matrixCoefficients (K := K) A B C)) R)
(e : κ → ι) (he : Function.Injective e)
(f : A' → A) (hf : Function.Injective f)
(g : B' → B) (hg : Function.Injective g)
(j : C' → C) (hj : Function.Injective j) :
RankAtMost (directSum (fun _ : κ => matrixCoefficients (K := K) A' B' C')) R := by
have hp := h.pullback
(fun x : κ × (A' × B') => (e x.1, (f x.2.1, g x.2.2)))
(fun y : κ × (B' × C') => (e y.1, (g y.2.1, j y.2.2)))
(fun z : κ × (C' × A') => (e z.1, (j z.2.1, f z.2.2)))
convert hp using 1
funext x y z
simp only [pullback, directSum, matrixCoefficients,
he.eq_iff, hf.eq_iff, hg.eq_iff, hj.eq_iff]
theorem rectangular_directSum_power_rank {K : Type*} [CommSemiring K]
{a b c L R t : ℕ}
(h : RankAtMost (power (directSum (fun _ : Fin L =>
matrixCoefficients (K := K) (Fin a) (Fin b) (Fin c))) t) R) :
RankAtMost (directSum (fun _ : Fin (L ^ t) =>
matrixCoefficients (K := K) (Fin (a ^ t)) (Fin (b ^ t)) (Fin (c ^ t)))) R := by
have hmatrix := matrixCoefficients_directSum_rank_of_power h
let tags : Fin (L ^ t) ≃ (Fin t → Fin L) := Fintype.equivOfCardEq (by simp)
let rows : Fin (a ^ t) ≃ (Fin t → Fin a) := Fintype.equivOfCardEq (by simp)
let inner : Fin (b ^ t) ≃ (Fin t → Fin b) := Fintype.equivOfCardEq (by simp)
let cols : Fin (c ^ t) ≃ (Fin t → Fin c) := Fintype.equivOfCardEq (by simp)
exact rectangular_directSum_rank_of_injective hmatrix tags tags.injective
rows rows.injective inner inner.injective cols cols.injective
theorem rectangular_pow_batchRank {K : Type*} [CommSemiring K]
{a b c L r : ℕ} (hL : 0 < L)
(hbatch : RankAtMost (directSum (fun _ : Fin L =>
matrixCoefficients (K := K) (Fin a) (Fin b) (Fin c))) r) :
∀ t : ℕ, RankAtMost
(matrixCoefficients (K := K) (Fin (a ^ t)) (Fin (b ^ t)) (Fin (c ^ t)))
(batchRank L r t) := by
intro t
induction t with
| zero => exact matrixCoefficients_fin_one_rank (K := K)
| succ t ih =>
let rows : Fin (a ^ (t + 1)) ≃ Fin (a ^ t) × Fin a :=
Fintype.equivOfCardEq (by simp [pow_succ])
let inner : Fin (b ^ (t + 1)) ≃ Fin (b ^ t) × Fin b :=
Fintype.equivOfCardEq (by simp [pow_succ])
let cols : Fin (c ^ (t + 1)) ≃ Fin (c ^ t) × Fin c :=
Fintype.equivOfCardEq (by simp [pow_succ])
have hp := matrixCoefficients_rank_of_product (ih.batch_product_ceiling hbatch hL)
have hr := rectangular_rank_of_injective hp
rows rows.injective inner inner.injective cols cols.injective
simpa only [batchRank_succ] using hr
theorem rectangular_polynomial_power_rank {K : Type*} [Field K] [CharZero K]
{a b c L r d D : ℕ}
(P : PolynomialApproximation (directSum (fun _ : Fin L =>
matrixCoefficients (K := K) (Fin a) (Fin b) (Fin c))) r d D) (t : ℕ) :
RankAtMost (directSum (fun _ : Fin (L ^ t) =>
matrixCoefficients (K := K) (Fin (a ^ t)) (Fin (b ^ t)) (Fin (c ^ t))))
((D * t + 1) * r ^ t) :=
rectangular_directSum_power_rank (P.rank_power t)
theorem rectangular_directSum_cyclic_rank
{K ι A B C : Type*} [CommSemiring K]
[DecidableEq ι] [DecidableEq A] [DecidableEq B] [DecidableEq C] {r : ℕ}
(h : RankAtMost (directSum (fun _ : ι => matrixCoefficients (K := K) A B C)) r) :
RankAtMost (directSum (fun _ : ι => matrixCoefficients (K := K) B C A)) r := by
rcases h with ⟨a, b, c, heq⟩
refine ⟨b, c, a, ?_⟩
funext x y z
have hcyc : directSum (fun _ : ι => matrixCoefficients (K := K) B C A) x y z =
directSum (fun _ : ι => matrixCoefficients (K := K) A B C) z x y := by
have htags : (x.1 = y.1 ∧ x.1 = z.1) ↔ (z.1 = x.1 ∧ z.1 = y.1) := by
constructor
· rintro ⟨hxy, hxz⟩
exact ⟨hxz.symm, hxz.symm.trans hxy⟩
· rintro ⟨hzx, hzy⟩
exact ⟨hzx.symm.trans hzy, hzx.symm⟩
simp only [directSum, htags]
split_ifs
· exact (matrixCoefficients_cyclic (K := K) x.2 y.2 z.2).symm
· rfl
rw [hcyc, congrFun (congrFun (congrFun heq z) x) y]
apply Finset.sum_congr rfl
intro i hi
simp only [rankOne]
ring
theorem rectangular_directSum_rank_lower {K : Type*} [Field K]
{a b L r : ℕ} (ha : 0 < a) (hb : 0 < b)
(h : RankAtMost (directSum (fun _ : Fin L =>
matrixCoefficients (K := K) (Fin a) (Fin b) (Fin a))) r) :
L * a ^ 2 ≤ r ∧ L * (a * b) ≤ r := by
let : Nonempty (Fin a) := ⟨⟨0, ha⟩⟩
let : Nonempty (Fin b) := ⟨⟨0, hb⟩⟩
constructor
· simpa only [Fintype.card_fin, pow_two] using
directSum_matrixCoefficients_rank_lower h
· simpa only [Fintype.card_fin] using
directSum_matrixCoefficients_rank_lower (rectangular_directSum_cyclic_rank h)
end Tensor
end MatrixMultiplication
end
end OAI
|