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 Dn 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:Dn→{−M,…,M} is

(g⊗fh)(v):=∑vg,vh∈Dns.t. vg⊕fvhg(vg)⋅h(vh)

for every v∈Dn. 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|2npolylog(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)npolylog(M)) for constant c:=3/4 when D has even cardinality. Our main observation is that a \emph{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:Dn→{−M,…,M} alongside with a vector v∈Dn 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|ω2npolylog(M)) time, where ω∈[2,2.372) is the exponent of currently fastest matrix multiplication algorithm.