Faster Convolutions: Yates and Strassen Revisited

TL;DR

This work reframes product-domain convolutions in parameterized algorithms as bilinear maps and analyzes them via tensor rank. By interpreting convolutions as $u\circledast_t v = \sum_{x,y} u(x)v(y)t(x,y)$ with a product-structured $t$, it shows how low-rank decompositions and known results from fast matrix multiplication yield substantial speedups over naive DP joins at join nodes. Two core results—Strassen-style bilinear splitting and Yates-style rank-based aggregation—provide $O^\ast(|D|^{2\omega/3\cdot k})=O(|D|^{1.582})$-type runtimes for fixed product-structure, and, under Strassen’s asymptotic rank conjecture, near-linear $|D|^{1+o(1)}$ time. The note offers a self-contained, algorithmic-expository treatment that connects classic convolution formulations (covering, subset, and XOR-convolutions) to contemporary algebraic techniques, with implications for faster join-node processing in tree-decomposition algorithms and related parameterized problems.

Abstract

Given two vectors $u,v \in \mathbb{Q}^D$ over a finite domain $D$ and a function $f : D\times D\to D$, the convolution problem asks to compute the vector $w \in \mathbb{Q}^D$ whose entries are defined by $w(d) = \sum_{\substack{x,y \in D \\ f(x,y)=d}} u(x)v(y).$ In parameterized and exponential-time algorithms, convolutions on product domains are particularly prominent: Here, a finite domain $B$ and a function $h : B \times B \to B$ are fixed, and convolution is done over the product domain $D = B^k$, using the function $h^k :D \times D\to D$ that applies $h$ coordinate-wise to its input tuples. We present a new perspective on product-domain convolutions through multilinear algebra. This viewpoint streamlines the presentation and analysis of existing algorithms, such as those by van Rooij et al. (ESA 2009). Moreover, using established results from the theory of fast matrix multiplication, we derive improved $O^\ast(|B|^{2ω/3 \cdot k}) = O(|D|^{1.582})$ time algorithms, improving upon previous upper bounds by Esmer et al. (Algorithmica 86(1), 2024) of the form $c^k |B|^{2k}$ for $c < 1$. Using the setup described in this note, Strassen's asymptotic rank conjecture from algebraic complexity theory would imply quasi-linear $|D|^{1+o(1)}$ time algorithms. This conjecture has recently gained attention in the algorithms community. (Björklund-Kaski and Pratt, STOC 2024, Björklund et al., SODA 2025) Our paper is intended as a self-contained exposition for an algorithms audience, and it includes all essential mathematical prerequisites with explicit coordinate-based notation. In particular, we assume no knowledge in abstract algebra.

Figures (1)

• Figure 1: Fix the base domain $B=\{0,1\}$ and one of the three base functions $h: B\times B\to B$ shown in the first row, where blue, pink and white squares correspond to the function values $0$, $1$, and undefined, respectively. The covering product, XOR and subset convolutions are powers $f = h^k$ of the respective base function that induce vector-valued functions $t:D\times D\to \mathbb \{0,1\}^D$ for $D=B^k$. The functions $t$ are depicted as tensors, i.e., $3$-dimensional arrays with $0/1$ entries, where $t(x,y)$ is stored in the vertical column $(x,y,\star)$, and a block at position $(x,y,z)$ indicates that the $z$-th entry of $t(x,y)$ equals $1$. (For aesthetic purposes, the block color encodes the Hamming weight of $z$; it does not encode the array value, which is $0$ or $1$.)

Theorems & Definitions (14)

• Remark 1.1
• Remark 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Remark 2.4
• Theorem 2.5
• Theorem 2.6
• Theorem 2.7
• Lemma 3.1