Permutation matrices play a key role in matching and assignment problems across the fields, especially in computer vision and robotics. However, memory for explicitly representing permutation matrices grows quadratically with the size of the problem, prohibiting large problem instances. In this work, we propose to tackle the curse of dimensionality of large permutation matrices by approximating them using low-rank matrix factorization, followed by a nonlinearity. To this end, we rely on the Kissing number theory to infer the minimal rank required for representing a permutation matrix of a given size, which is significantly smaller than the problem size. This leads to a drastic reduction in computation and memory costs, e.g., up to $3$ orders of magnitude less memory for a problem of size $n=20000$, represented using $8.4\times10^5$ elements in two small matrices instead of using a single huge matrix with $4\times 10^8$ elements. The proposed representation allows for accurate representations of large permutation matrices, which in turn enables handling large problems that would have been infeasible otherwise. We demonstrate the applicability and merits of the proposed approach through a series of experiments on a range of problems that involve predicting permutation matrices, from linear and quadratic assignment to shape matching problems.

翻译：置换矩阵在多个领域的匹配与分配问题中扮演关键角色，尤其在计算机视觉与机器人领域。然而，显式表示置换矩阵所需的内存随问题规模呈二次增长，阻碍了大规模问题的求解。本文提出通过低秩矩阵分解结合非线性变换来逼近大置换矩阵，以克服其维数灾难。为此，我们利用亲吻数理论推导表示给定规模置换矩阵所需的最小秩（该秩远小于问题规模），从而大幅降低计算与内存成本：例如，对于规模 $n=20000$ 的问题，使用两个包含 $8.4\times10^5$ 元素的小矩阵而非单个含 $4\times10^8$ 元素的巨大矩阵表示，可节省高达3个数量级的内存。所提出的表示方法能够精确表示大置换矩阵，进而处理原本无法解决的大规模问题。我们通过一系列涉及置换矩阵预测的问题实验（涵盖线性/二次分配与形状匹配问题），验证了该方法的适用性与优势。