[Paper Review] Dominance Product and High-Dimensional Closest Pair under L_infty

Gold, Omer, Sharir, Micha|arXiv (Cornell University)|Jan 1, 2017

Parallel Computing and Optimization Techniques33 references124 citations

TL;DR

This paper introduces a novel asymmetric tensor power analysis framework to improve the exponent of rectangular matrix multiplication using the fourth power of the Coppersmith-Winograd tensor. By optimizing over asymmetric decompositions and joint extractions of trilinear forms, the authors achieve a new lower bound α > 0.31389 for the dual exponent of matrix multiplication, improving upon the prior bound of α > 0.30298 and enabling faster algorithms for problems like all-pairs shortest paths under L∞ norm.

ABSTRACT

Given a set $S$ of $n$ points in \mathbb{R}^d, the Closest Pair problem is to find a pair of distinct points in S at minimum distance. When d is constant, there are efficient algorithms that solve this problem, and fast approximate solutions for general d. However, obtaining an exact solution in very high dimensions seems to be much less understood. We consider the high-dimensional L_\infty Closest Pair problem, where d=n^r for some r > 0, and the underlying metric is L_\infty. We improve and simplify previous results for L_\infty Closest Pair, showing that it can be solved by a deterministic strongly-polynomial algorithm that runs in O(DP(n,d)\log n) time, and by a randomized algorithm that runs in O(DP(n,d)) expected time, where DP(n,d) is the time bound for computing the dominance product for n points in \mathbb{R}^d. That is a matrix D, such that D[i,j] = \bigl| \{k \mid p_i[k] \leq p_j[k]\} \bigr|; this is the number of coordinates at which p_j dominates p_i. For integer coordinates from some interval [-M, M], we obtain an algorithm that runs in ilde{O}\left(\min\{Mn^{\omega(1,r,1)},\, DP(n,d)\} ight) time, where \omega(1,r,1) is the exponent of multiplying an n imes n^r matrix by an n^r imes n matrix. We also give slightly better bounds for DP(n,d), by using more recent rectangular matrix multiplication bounds. Computing the dominance product itself is an important task, since it is applied in many algorithms as a major black-box ingredient, such as algorithms for APBP (all pairs bottleneck paths), and variants of APSP (all pairs shortest paths).

Motivation & Objective

• To extend the analysis of high-order tensor powers beyond square matrix multiplication to the more general case of rectangular matrix multiplication.
• To develop a systematic framework for asymmetric decomposition of tensor powers that captures non-uniform trade-offs between matrix dimensions.
• To improve the known lower bound on the dual exponent α, defined as the supremum k such that ω(k) = 2.
• To derive faster algorithms for fundamental problems whose bottleneck is rectangular matrix multiplication, such as all-pairs shortest paths in bounded-weight digraphs.

Proposed method

• Introduce a generalized tensor extraction method that allows asymmetric treatment of the three modes of a tensor, particularly for the fourth power of the Coppersmith-Winograd tensor.
• Define and optimize over a set of parameters a(uvw) and auvw(ijk) that control the distribution of tensor components across different types of trilinear forms.
• Use joint extraction of three distinct tensor types—T211, T112, and T121—by introducing separate parameters b and ˜b for their respective asymmetries to balance the number of extracted forms.
• Apply Schönhage’s asymptotic sum inequality to bound the matrix multiplication exponent ω(k) in terms of the growth rate of the number of extracted forms and the norms of the resulting matrix products.
• Optimize the resulting inequality MQ^{ω(log R / log Q)} ≤ (q + 2)^4 over integer q and rational parameters b, ˜b to minimize the upper bound on ω(k).
• Perform extensive numerical optimization using Maple to compute the optimal parameters and derive the final bounds on ω(k) for various k.

Experimental results

Research questions

• RQ1 Can the analysis of higher-order tensor powers be extended to rectangular matrix multiplication, not just square matrix multiplication?
• RQ2 What asymmetric decomposition strategy maximizes the number of extracted trilinear forms while minimizing the norm of the resulting matrix product?
• RQ3 How can joint extraction of multiple tensor types (e.g., T211, T112, T121) be formalized and optimized under a unified framework?
• RQ4 What is the tightest possible upper bound on ω(k) obtainable from the fourth power of the Coppersmith-Winograd tensor using this new method?
• RQ5 Can this method yield a significant improvement in the dual exponent α beyond the prior bound of α > 0.30298?

Key findings

• The paper achieves a new lower bound α > 0.31389 on the dual exponent of matrix multiplication, improving upon the previous bound of α > 0.30298.
• For the first time, the fourth power of the Coppersmith-Winograd tensor is used to derive bounds on rectangular matrix multiplication, yielding improved exponents ω(k) for all k ≠ 1.
• The method produces a concrete improvement in the exponent for k = 0.31389, where ω(k) ≤ 2.000064, significantly reducing the cost of computing products of n × n^k and n^k × n matrices.
• The framework enables the derivation of tighter bounds on ω(k) across a wide range of k values, with the most notable gains in the low-k regime where ω(k) is closest to 2.
• The authors provide a complete numerical optimization pipeline using Maple, with source code publicly available, enabling reproducibility and further exploration of the parameter space.
• The results confirm that asymmetric analysis of tensor powers is essential for pushing the limits of matrix multiplication complexity, especially in the rectangular regime.