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Computer Science > Machine Learning

arXiv:2510.03426 (cs)
[Submitted on 3 Oct 2025 (v1), last revised 9 Oct 2025 (this version,
v2)]

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

Authors:Franz A. Heinsen, Leo Kozachkov
View a PDF of the paper titled Generalized Orders of Magnitude for
Scalable, Parallel, High-Dynamic-Range Computation, by Franz A.
Heinsen and Leo Kozachkov
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    Abstract: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.

Comments:          18 pages, 4 figures (main text). 14 pages, 21
                   figures (appendix). Code is at this https URL
Subjects:          Machine Learning (cs.LG); Artificial Intelligence
                   (cs.AI); Numerical Analysis (math.NA)
Cite as:           arXiv:2510.03426 [cs.LG]
                   (or arXiv:2510.03426v2 [cs.LG] for this version)
                   https://doi.org/10.48550/arXiv.2510.03426
                   Focus to learn more
                   arXiv-issued DOI via DataCite
Journal reference: Transactions on Machine Learning Research (TMLR),
                   2025

Submission history

From: Franz Heinsen [view email]
[v1] Fri, 3 Oct 2025 18:38:26 UTC (6,504 KB)
[v2] Thu, 9 Oct 2025 13:23:43 UTC (6,504 KB)
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