发表机构
Flatiron Institute; Courant Institute, NYU; Google Research（弗莱堡研究所; 纽约大学库朗数学研究所; 谷歌研究院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对稀疏非负矩阵补全问题，提出可扩展的随机交替最小二乘算法，结合稀疏优化与CUDA内核加速，成功应用于果蝇连接组突触权重矩阵分析，揭示其潜在低秩结构的预测能力。
AI 中文摘要
我们研究当一个稀疏非负矩阵通过将其负元素置零，能否从一个低得多秩的实值矩阵中恢复出来。这类分解的潜力表明稀疏性与秩之间存在数学关联；我们分析了多个具有潜在低秩结构的稀疏矩阵，并用它们阐明这种关联的几何起源。此前的算法通过对低秩矩阵的因子进行交替最小化来发现这些分解，但为实现这一点，它们还需要计算并存储另一个与该乘积同规模的既非稀疏也非低秩的矩阵。我们开发了一种随机交替最小二乘算法，该算法对这个稠密矩阵的较小块进行操作，因此可扩展到大得多的问题。我们还展示了如何通过稀疏优化和定制的CUDA内核进一步加速该算法。作为示例，我们使用该算法分析最近发布的果蝇（Drosphilia）连接组的突触权重稀疏矩阵。这个矩阵有139255行和列，其非零元素记录了雌性果蝇神经系统中细胞间的突触数量。尽管奇异值谱缓慢衰减，该矩阵仍呈现出潜在低秩结构，可跨多个特异性级别预测细胞类别。
英文摘要
We investigate when a sparse nonnegative matrix can be recovered from a real-valued matrix of much lower rank by zeroing out its negative elements. The potential for such decompositions suggests a mathematical connection between sparsity and rank; we analyze a number of sparse matrices with this latent low-rank structure and use them to illustrate the geometric origins of this connection. Previous algorithms have discovered these decompositions via an alternating minimization over the factors of a low-rank matrix, but to do so, they have also needed to compute and store another matrix, neither sparse nor low-rank, that is the size of their product. We develop a stochastic, alternating least-squares algorithm that operates on smaller blocks of this dense matrix and scales as a result to much larger problems. We also show how to further accelerate this algorithm with sparse optimizations and customized CUDA kernels. As one example, we use the algorithm to analyze the sparse matrix of synaptic weights for the recently published $\textit{Drosphilia}$ connectome. The nonzero elements of this matrix, with 139,255 rows and columns, record the number of synapses between cells in the nervous system of a female fruit fly. Despite a slowly decaying spectrum of singular values, this matrix exhibits a latent low-rank structure that is predictive of cell categories across multiple levels of specificity.
