GPU上的吉咖规模核矩阵向量乘法
Giga-scale Kernel Matrix Vector Multiplication on GPU
AI总结：
本文提出Faster-Fast and Free Memory Method（$\fthreem$），在GPU上实现吉咖规模核矩阵向量乘法的线性时间和内存复杂度，相对误差约$10^{-3}$，并作为FALKON的即插即用组件，以<1%精度损失换取1.5-5.5倍加速。
摘要：
核矩阵向量乘法（KMVM）是机器学习和科学计算中的基础操作。然而，由于KMVM在内存和时间上往往呈二次方扩展，应用常常受到这些计算约束的限制。在本文中，我们提出了一种新颖的近似过程，称为“更快-快速且免内存方法”（$\fthreem$），以解决KMVM在细长（$10^8 \thicksim 10^9$）和瘦小（$D \thicksim 7$）数据上的扩展问题。大量实验表明，$\fthreem$具有经验上的线性时间和内存复杂度，相对误差约为$10^{-3}$，并且可以在高端GPU上在一分钟内计算十亿点的完整KMVM，与现有CPU方法相比，实现了显著的加速。我们通过将其作为最先进的基于GPU的线性求解器FALKON的即插即用组件来展示我们过程的实用性，以不到1%的精度下降为代价，将速度提高了1.5-5.5倍。我们进一步在多种真实世界数据集上展示了与高斯过程回归相结合的竞争性结果，并实现了显著的加速。
英文摘要：
Kernel matrix-vector multiplication (KMVM) is a foundational operation in machine learning and scientific computing. However, as KMVM tends to scale quadratically in both memory and time, applications are often limited by these computational constraints. In this paper, we propose a novel approximation procedure coined \textit{Faster-Fast and Free Memory Method} ($\fthreem$) to address these scaling issues of KMVM for tall~($10^8\sim 10^9$) and skinny~($D\leq7$) data. Extensive experiments demonstrate that $\fthreem$ has empirical \emph{linear time and memory} complexity with a relative error of order $10^{-3}$ and can compute a full KMVM for a billion points \emph{in under a minute} on a high-end GPU, leading to a significant speed-up in comparison to existing CPU methods. We demonstrate the utility of our procedure by applying it as a drop-in for the state-of-the-art GPU-based linear solver FALKON, \emph{improving speed 1.5-5.5 times} at the cost of $<1\%$ drop in accuracy. We further demonstrate competitive results on \emph{Gaussian Process regression} coupled with significant speedups on a variety of real-world datasets.
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