Partition Rank and Algebraic Circuit Lower Bounds

2026-07-02

AI Summary

This work investigates the connection between multiplication complexity in algebraic circuits and tensor-rank–type measures in the high-order multilinear setting. Introducing Naslund’s partition rank as a generalized notion of tensor rank, we establish for the first time a direct link between partition rank and the multiplication complexity of algebraic circuits, thereby extending classical rank-based lower-bound techniques—originally developed for bilinear computation, such as Strassen’s seminal results—to arbitrary constant-order multilinear computations. This framework not only yields a streamlined proof of the NP-hardness of symmetric slice rank but also elucidates intrinsic relationships among partition rank, slice rank, and symmetric slice rank, offering a new rank-based toolkit for fine-grained complexity questions such as the hyperclique conjecture.

Abstract

Strassen's theory of bilinear complexity provides a mathematical characterization of the arithmetic complexity of primitives such as matrix multiplication via the rank of tensors. However, the connection to tensor rank is known to break down in higher degrees of multilinearity. In this work, we highlight an unexplored connection between a generalized notion of tensor rank, which can be defined in Naslund's framework of partition ranks (JCTA 2020), and multiplicative complexity. These partition ranks allow us to control the multiplicative complexity, and thus arithmetic complexity, in any constant degree of multilinearity from below, while recovering Strassen's seminal characterization in the bilinear case. This enables novel potential applications of the rank-based approaches to problems in fine-grained algorithms and complexity, such as the hyperclique conjecture of Lincoln-Williams-Vassilevska Williams (SODA 2018). Moreover, we exhibit connections to established notions of rank, such as tensor slice rank (in the sense of Tao and Sawin), as well as its symmetric variant. For computing the latter symmetric variant, we point out a simple NP-hardness proof, contrasting the rather involved NP-hardness proof for ordinary, non-symmetric tensor slice rank by Bläser et al. (SODA 2021).

Problem

• tensor rank
• multiplicative complexity
• partition rank
• arithmetic complexity
• multilinear algebra

Innovation

• partition rank
• tensor rank
• multiplicative complexity
• slice rank
• arithmetic complexity