Computing Generalized Convolutions Faster Than Brute Force

In this paper, we consider a general notion of convolution. Let D be a finite domain and let D^n be the set of n-length vectors (tuples) of D . Let f :D× D→ D be a function and let ⊕ _f be a coordinate-wise application of f. The f -Convolution of two functions g,h :D^n →{-M,… ,M} is (g ⊛ _fh)(v) :=∑ _[ v_g,v_h ∈ D^n; s.t. v= v_g ⊕ _f v_h ] g(v_g) · h(v_h) for every v∈ D^n . This problem generalizes many fundamental convolutions such as Subset Convolution, XOR Product, Covering Product or Packing Product, etc. For arbitrary function f and domain D we can compute f -Convolution via brute-force enumeration in 𝒪(|D|^2n·polylog(M)) time. Our main result is an improvement over this naive algorithm. We show that f -Convolution can be computed exactly in 𝒪( (c · |D|^2)^n·polylog(M)) for constant c :=3/4 when D has even cardinality. Our main observation is that a cyclic partition of a function f :D× D→ D can be used to speed up the computation of f -Convolution, and we show that an appropriate cyclic partition exists for every f. Furthermore, we demonstrate that a single entry of the f -Convolution can be computed more efficiently. In this variant, we are given two functions g,h :D^n →{-M,… ,M} alongside with a vector v∈ D^n and the task of the f -Query problem is to compute integer (g ⊛ _fh)(v) . This is a generalization of the well-known Orthogonal Vectors problem. We show that f -Query can be computed in 𝒪(|D|^ω/2 n·polylog(M)) time, where ω∈ [2,2.372) is the exponent of currently fastest matrix multiplication algorithm.