# 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 ⊗_f h)(v) := ∑_v_g,v_h ∈ D^n s.t. 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|^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 := 5/6 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 ⊗_f h)(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 npolylog(M)) time, where ω∈ [2,2.373) is the exponent of currently fastest matrix multiplication algorithm.

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