| import OAI.LinearAlgebra.MatrixMultiplication.Tensor.ComplexEqualSummandASI |
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| namespace OAI |
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| /-! Rectangular matrix multiplication algorithms and their asymptotic exponents. -/ |
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| noncomputable section |
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| namespace MatrixMultiplication |
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| open MatrixMultiplication.Foundation |
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| namespace Tensor |
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| open MatrixMultiplication.Foundation.Tensor |
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| 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] |
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| 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] |
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| 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 |
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| 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 |
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| 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) |
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| 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 |
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| 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) |
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| end Tensor |
| end MatrixMultiplication |
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| end |
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| end OAI |
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