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 \(f :D\times D\rightarrow D\) be a function and let \(\oplus _f\) be a coordinate-wise application of f. The \(f\)-Convolution of two functions \(g,h :D^n \rightarrow \{-M,\ldots ,M\}\) is

$$\begin{aligned} (g \mathbin {\circledast _{f}}h)(\textbf{v}) {:}{=}\sum _{\begin{array}{c} \textbf{v}_g,\textbf{v}_h \in D^n\\ \text {s.t. } \textbf{v}= \textbf{v}_g \oplus _f \textbf{v}_h \end{array}} g(\textbf{v}_g) \cdot h(\textbf{v}_h) \end{aligned}$$

for every \(\textbf{v}\in D^n\). 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 \(\widetilde{{\mathcal {O}}}(|D|^{2n} \cdot \textrm{polylog}(M))\) time. Our main result is an improvement over this naive algorithm. We show that \(f\)-Convolution can be computed exactly in \(\widetilde{{\mathcal {O}}}( (c \cdot |D|^2)^{n} \cdot \textrm{polylog}(M))\) for constant \(c {:}{=}3/4\) when \(D\) has even cardinality. Our main observation is that a cyclic partition of a function \(f :D\times D\rightarrow D\) 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 \(g,h :D^n \rightarrow \{-M,\ldots ,M\}\) alongside with a vector \(\textbf{v}\in D^n\) and the task of the \(f\)-Query problem is to compute integer \((g \mathbin {\circledast _{f}}h)(\textbf{v})\). This is a generalization of the well-known Orthogonal Vectors problem. We show that \(f\)-Query can be computed in \(\widetilde{{\mathcal {O}}}(|D|^{\frac{\omega }{2} n} \cdot \textrm{polylog}(M))\) time, where \(\omega \in [2,2.372)\) is the exponent of currently fastest matrix multiplication algorithm.