openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Convolution /RankLowerBound.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Convolution.Rank | |
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Arithmetic.RankExponent | |
| namespace OAI | |
| /-! | |
| # Exact rank of polynomial convolution | |
| Each output coefficient of polynomial multiplication occurs in some product | |
| of two input monomials. Choosing one such pair for each output gives an | |
| identity minor in the output flattening. Together with evaluation and | |
| interpolation, this proves the exact rank of `C(a,b)` for positive dimensions. | |
| -/ | |
| noncomputable section | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open MatrixMultiplication.Foundation | |
| /-- Choosing one product for every output monomial gives an identity minor | |
| of the output flattening. -/ | |
| theorem convolution_identity_minor {a b : ℕ} (ha : 0 < a) (hb : 0 < b) : | |
| ∃ (x : Fin (a + b - 1) → Fin a) (y : Fin (a + b - 1) → Fin b), | |
| ∀ k l, convolution a b (x k) (y k) l = if k = l then 1 else 0 := by | |
| choose x y hxy using fun k : Fin (a + b - 1) => convolution_output_supported ha hb k | |
| have hsum (k : Fin (a + b - 1)) : (x k).val + (y k).val = k.val := | |
| (convolution_eq_one_iff _ _ _).mp (hxy k) | |
| refine ⟨x, y, ?_⟩ | |
| intro k l | |
| simp only [convolution_apply, hsum, Fin.val_inj] | |
| /-- Every rank decomposition of convolution has at least one term per output | |
| coefficient. -/ | |
| theorem convolution_rank_lower {a b r : ℕ} (ha : 0 < a) (hb : 0 < b) | |
| (hRank : Tensor.RankAtMost (convolution a b) r) : a + b - 1 ≤ r := by | |
| obtain ⟨x, y, hxy⟩ := convolution_identity_minor ha hb | |
| simpa using hRank.card_le_of_identity x y id hxy | |
| /-- Polynomial multiplication has exact rank `a+b-1` when both inputs have | |
| positive dimension. -/ | |
| theorem exactRank_convolution {a b : ℕ} (ha : 0 < a) (hb : 0 < b) : | |
| exactRank (convolution a b) = a + b - 1 := by | |
| apply le_antisymm (exactRank_le (convolution_rankAtMost a b)) | |
| exact convolution_rank_lower ha hb (exactRank_spec _) | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end | |
| end OAI | |