Barriers for rectangular matrix multiplication

TL;DR

This work addresses the barrier landscape for rectangular matrix multiplication by formulating a unifying tensor-based framework that uses adequate tensor parameters and virtual matrix multiplication tensors to bound the effectiveness of current methods. It demonstrates that CW-based intermediate tensors cannot push the rectangular MM exponent $\omega(p)$ below certain thresholds and yields the sharp numerical barrier $\alpha \le 0.6218$ for the dual exponent, strengthening prior barriers and extending them beyond $p\in[0,1]$. The paper also explains catalyticity as a structural aspect of practical algorithms and provides a rigorous, numerically grounded procedure to compute barriers via upper support functionals, including explicit results for CW$_q$ and related tensors. Taken together, these results illuminate the limitations of existing approaches to speeding rectangular matrix multiplication and guide where new ideas are needed to break the current barriers, with implications for related problems that depend on the dual exponent $\alpha$.

Abstract

We study the algorithmic problem of multiplying large matrices that are rectangular. We prove that the method that has been used to construct the fastest algorithms for rectangular matrix multiplication cannot give algorithms with complexity $n^{p + 1}$ for $n \times n$ by $n \times n^p$ matrix multiplication. In fact, we prove a precise numerical barrier for this method. Our barrier improves the previously known barriers, both in the numerical sense, as well as in its generality. In particular, we prove that any lower bound on the dual exponent of matrix multiplication $α$ via the big Coppersmith-Winograd tensors cannot exceed 0.6218.

Figures (3)

• Figure 1: The blue line is the upper bound on $\omega(p)$ obtained via $\operatorname{CW}_6$ as in le2012faster where $p \in [0,2]$ in on the horizontal axis. The yellow line is our barrier for upper bounds on $\omega(p)$ via degeneration and the intermediate tensor $\operatorname{CW}_6$. The red line is the lower bound on $\omega(p)$.
• Figure 2: The blue points are the lower bounds on $\alpha$ obtained via $\operatorname{CW}_q$ as in le2012faster for all $q \in \{2, \ldots, 8\}$. The yellow points are our barriers for the best lower bound on $\alpha$ obtainable via degeneration and the intermediate tensor $\operatorname{CW}_q$. The red points are the best upper bounds on $\alpha$, namely $1$. The lower bound $\alpha>0.3029$ in le2012faster is attained using $q=5$. Any lower bound on $\alpha$ using degeneration and $\operatorname{CW}_q$ for any $q$, cannot exceed $0.6218$, the highest yellow point in the graph.
• Figure 3: This is the graph from \ref{['fig1']} with arrows that indicate the influence of catalyticity. Roughly speaking, the barrier for $\operatorname{CW}_6$ (the yellow line) moves upwards when more catalyticity is used.

Theorems & Definitions (46)

• Theorem 1
• Theorem 2
• Remark 3
• Theorem 4
• Definition 5
• Lemma 6
• proof
• Definition 7
• Lemma 8
• Corollary 9