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arXiv 2510.03426cs.LGcs.AIcs.NAmath.NA
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广义数量级:用于可扩展、并行、高动态范围计算

Generalized Orders of Magnitude for Scalable, Parallel, High-Dynamic-Range Computation

GlassRoom Software LLC(GlassRoom软件公司);Thomas J. Watson Research Center, IBM Research(IBM研究院Watson研究中心);Brown University(布朗大学)

Franz A. Heinsen, Leo Kozachkov

AI总结:

针对长序列实数复合运算中的数值下溢/上溢问题,提出广义数量级(GOOMs)扩展浮点数,结合并行前缀扫描在GPU上实现,通过三个实验证明其在高动态范围计算中可扩展且数值稳健。

摘要:

从深度学习到金融等众多领域,都需要对长序列的实数进行复合运算,这常常导致灾难性的数值下溢或上溢。我们引入了广义数量级(GOOMs),这是传统数量级的一种原理性扩展,将浮点数作为特例包含在内,并且在实践中能够实现在比以往可能范围大得多的实数动态范围内进行稳定计算。我们实现了GOOMs,并配以高效的定制并行前缀扫描,以支持在GPU等并行硬件上的原生执行。我们通过三个具有代表性的实验证明了我们的GOOMs实现优于传统方法,这些实验以前被认为不切实际或不可能,现在变得可能且实用:(1)在远超标准浮点数极限的范围内复合实数矩阵乘积;(2)并行估计李雅普诺夫指数谱,比以往方法快数个数量级,并应用一种新颖的选择性重置方法以防止状态共线性;(3)在深度循环神经网络中捕获具有非对角循环状态的长程依赖,通过前缀扫描并行计算,无需任何形式的稳定化。我们的结果表明,我们的GOOMs实现结合高效的并行扫描,为高动态范围应用提供了一种可扩展且数值稳健的替代传统浮点数的方法。

英文摘要:

Many domains, from deep learning to finance, require compounding real numbers over long sequences, often leading to catastrophic numerical underflow or overflow. We introduce generalized orders of magnitude (GOOMs), a principled extension of traditional orders of magnitude that incorporates floating-point numbers as a special case, and which in practice enables stable computation over significantly larger dynamic ranges of real numbers than previously possible. We implement GOOMs, along with an efficient custom parallel prefix scan, to support native execution on parallel hardware such as GPUs. We demonstrate that our implementation of GOOMs outperforms traditional approaches with three representative experiments, all of which were previously considered impractical or impossible, and now become possible and practical: (1) compounding real matrix products far beyond standard floating-point limits; (2) estimating spectra of Lyapunov exponents in parallel, orders of magnitude faster than with previous methods, applying a novel selective-resetting method to prevent state colinearity; and (3) capturing long-range dependencies in deep recurrent neural networks with non-diagonal recurrent states, computed in parallel via a prefix scan, without requiring any form of stabilization. Our results show that our implementation of GOOMs, combined with efficient parallel scanning, offers a scalable and numerically robust alternative to conventional floating-point numbers for high-dynamic-range applications.

机构、总结、中文摘要均由AI分析而来,可能存在错误,请以论文正文为准。

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