openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Convolution /Rank.lean
| import OAI.LinearAlgebra.MatrixMultiplication.AuxiliarySeparation.Convolution.Basic | |
| import Mathlib.LinearAlgebra.Lagrange | |
| namespace OAI | |
| /-! | |
| # Polynomial convolution by evaluation and interpolation | |
| Evaluating the two input polynomials at `a + b - 1` distinct complex points, | |
| then interpolating their product, expresses the convolution tensor as a sum | |
| of `a + b - 1` rank-one tensors. This is the polynomial multiplication rank | |
| upper bound used in Section 5. | |
| -/ | |
| open scoped BigOperators | |
| open Polynomial | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open MatrixMultiplication.Foundation | |
| /-- Recover each monomial coefficient from evaluations at distinct nodes. -/ | |
| lemma monomial_coeff_eq_sum_interpolation {n m : ℕ} (hm : m < n) | |
| (v : Fin n → ℂ) (hv : Function.Injective v) (k : ℕ) : | |
| (if m = k then (1 : ℂ) else 0) = | |
| ∑ r : Fin n, v r ^ m * (Lagrange.basis Finset.univ v r).coeff k := by | |
| have hdeg : (X ^ m : ℂ[X]).degree < (Finset.univ : Finset (Fin n)).card := by | |
| simpa using hm | |
| have hpoly := Lagrange.eq_interpolate (f := (X ^ m : ℂ[X])) hv.injOn hdeg | |
| have hcoeff := congrArg (fun p : ℂ[X] => p.coeff k) hpoly | |
| simpa only [Lagrange.interpolate_apply, Polynomial.finsetSum_coeff, | |
| Polynomial.coeff_C_mul, Polynomial.eval_pow, Polynomial.eval_X, | |
| Polynomial.coeff_X_pow, eq_comm] using hcoeff | |
| /-- Evaluation at distinct nodes and Lagrange interpolation give an explicit | |
| rank-one decomposition of polynomial convolution. -/ | |
| theorem convolution_eq_sum_rankOne {a b : ℕ} | |
| (v : Fin (a + b - 1) → ℂ) (hv : Function.Injective v) : | |
| convolution a b = fun i j k => ∑ r : Fin (a + b - 1), | |
| Tensor.rankOne (fun i : Fin a => v r ^ i.val) | |
| (fun j : Fin b => v r ^ j.val) | |
| (fun k : Fin (a + b - 1) => (Lagrange.basis Finset.univ v r).coeff k.val) | |
| i j k := by | |
| funext i j k | |
| have hij : i.val + j.val < a + b - 1 := (convolutionOutput i j).isLt | |
| simpa only [convolution_apply, Tensor.rankOne, pow_add] using | |
| monomial_coeff_eq_sum_interpolation hij v hv k.val | |
| /-- The convolution tensor `C(a,b)` has rank at most `a + b - 1` over `ℂ`. -/ | |
| theorem convolution_rankAtMost (a b : ℕ) : | |
| Tensor.RankAtMost (convolution a b) (a + b - 1) := by | |
| let v : Fin (a + b - 1) → ℂ := fun r => r.val | |
| have hv : Function.Injective v := by | |
| intro i j h | |
| apply Fin.ext | |
| change (i.val : ℂ) = j.val at h | |
| exact_mod_cast h | |
| rw [convolution_eq_sum_rankOne v hv] | |
| simpa using Tensor.rankAtMost_sum_rankOne | |
| (fun r (i : Fin a) => v r ^ i.val) | |
| (fun r (j : Fin b) => v r ^ j.val) | |
| (fun r (k : Fin (a + b - 1)) => (Lagrange.basis Finset.univ v r).coeff k.val) | |
| /-- The interpolation algorithm also bounds border rank. -/ | |
| theorem convolution_borderRankAtMost (a b : ℕ) : | |
| Tensor.BorderRankAtMost (convolution a b) (a + b - 1) := | |
| (convolution_rankAtMost a b).borderRankAtMost | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |