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cdec422 | 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 | 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
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