发表机构
NVIDIA（英伟达）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究通过加速对称非负矩阵分解进行低秩相依性分解，在绝对皮尔逊相关和尾部成对相依矩阵上对七个算法家族超30种配置做大规模GPU研究，给出多节点扩展结果，比较不同算法在不同谱上表现，还将球形K - means作为基线测试。
AI 中文摘要
对称非负矩阵分解（SymNMF）从相依矩阵中恢复潜在的组结构，但其密集的二次内存目标使先前工作局限于中等规模。本文对七个算法家族（超30种配置）在绝对皮尔逊相关和极值理论中的尾部成对相依矩阵上进行大规模GPU研究，这是大型投资组合中经验风险因素估计的两个代理。通过迹恒等式重新表述消除了所有\(n×n\)中间变量，单GPU可达\(n≈10^5\)，多节点分布可扩展到\(n = 10^6\)及以上。在两阶段协议下，十一种方法在中等规模收敛；六种在\(n = 10^5\)时仍足够高效，五种AdaGrad家族方法在\(n = 10^6\)时仍收敛。在\(n = 10^6\)时，最快的求解器跟踪矩阵谱：Block - SVRG AdaptGrow在平坦、病态的尾部相依谱上获胜，全批量AdaGrad在占主导的低秩相关谱上获胜。还将球形K - means作为硬标签基线进行基准测试。
英文摘要
Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes. We present a large-scale GPU study of seven algorithm families (over 30 configurations) on absolute Pearson correlation and tail pairwise dependence matrices from Extreme Value Theory, two proxies for empirical risk-factor estimation on large portfolios. A trace-identity reformulation eliminates all $n \times n$ intermediates, so a single GPU reaches $n \approx 10^5$ and multi-node distribution scales to $n = 10^6$ and beyond. Under a two-phase protocol, eleven methods converge at moderate scale; six remain efficient enough at $n = 10^5$ (five AdaGrad-family plus ADMM), and five AdaGrad-family methods still converge at $n = 10^6$: AdaGrad, RMSprop, and three we introduce (Piecewise AdaGrad, Row-Stochastic SVRG, Block-SVRG AdaptGrow). At $n = 10^6$ the fastest solver tracks the matrix spectrum: Block-SVRG AdaptGrow wins on the flat, ill-conditioned tail-dependence spectrum, where its lower per-iteration cost decides a long factorization, and full-batch AdaGrad wins on the dominant-low-rank correlation spectrum, where the run is short. We also benchmark spherical K-means as a hard-label baseline: cheaper when angular cluster structure is present, yet provably degenerate once the matrix collapses toward a single common factor, where the soft factorization remains necessary.
Comments40 pages
