{"slug": "defeating-vanishing-gradients-in-deep-neural-networks-quotient-tree-arithmetic", "title": "Defeating vanishing gradients in deep neural networks: Quotient Tree Arithmetic", "summary": "A new computational framework called Quotient Tree Arithmetic (QTA) prevents gradient underflow in deep neural network training by representing values as deferred quotient pairs evaluated lazily at a materialization boundary, according to a paper submitted to arXiv on 15 Jun 2026. QTA provides O(1)-cost gradient computation via chain-rule tape collapse and reduces DAG storage from O(BLd) to O(L+Bd) for a batch of B examples through L layers, with a single IEEE division introducing at most one-half ULP of rounding error versus O(m) ULP for eager evaluation.", "body_md": "# Computer Science > Symbolic Computation\n\n[Submitted on 15 Jun 2026]\n\n# Title:Quotient Tree Arithmetic: Deferred-Division Computation with Bounded Symbolic Depth and Cross-Subtree Cancellation\n\n[View PDF](/pdf/2607.22612)\n\n[HTML (experimental)](https://arxiv.org/html/2607.22612v1)\n\nAbstract:We introduce Quotient Tree Arithmetic (QTA), a computational substrate in which values are represented as deferred quotient pairs (N, D) whose ratio is evaluated lazily at a designated materialization boundary. The framework applies to any domain: IEEE 754 doubles used as exact integer containers give exact rational arithmetic within the 2^53 exactness window; arbitrary IEEE doubles extend coverage to transcendental values including machine learning activations such as exp(x) and sqrt(x).\n\nThree structural theorems underpin QTA. (1) Bounded Depth Growth: each arithmetic operation increases tree depth by at most 1, giving O(m) tree size after m operations with no combinatorial explosion. (2) Cross-Subtree Cancellation: subtrees appearing in both numerator and denominator positions cancel via reference identity without arithmetic, including transcendental values computed once and shared. (3) Deferred Stability: a single IEEE division at the materialization boundary introduces at most one-half ULP of rounding error, versus O(m) ULP for eager evaluation.\n\nFor machine learning training, QTA provides: structural prevention of gradient underflow to zero; O(1)-cost gradient computation via chain-rule tape collapse when intermediate activations are reference-identical; shared-weight batch compression reducing DAG storage from O(BLd) to O(L+Bd) for a batch of B examples through L layers; and tracked factor cancellation replacing O(log n) GCD with O(1) trial division when denominators are known. We propose a vectorized hardware normalization instruction (RatCleanup) for SIMD-parallel rational pair reduction. The algebraic foundation is the localization of a ring at its multiplicative set, connecting QTA to algebraic structure theory while grounding it in hardware-native IEEE arithmetic.\n\n### Current browse context:\n\ncs.SC\n\n### References & Citations\n\nLoading...\n\n# Bibliographic and Citation Tools\n\nBibliographic Explorer\n\n*(*[What is the Explorer?](https://info.arxiv.org/labs/showcase.html#arxiv-bibliographic-explorer))\nConnected Papers\n\n*(*[What is Connected Papers?](https://www.connectedpapers.com/about))\nLitmaps\n\n*(*[What is Litmaps?](https://www.litmaps.co/))\nscite Smart Citations\n\n*(*[What are Smart Citations?](https://www.scite.ai/))# Code, Data and Media Associated with this Article\n\nalphaXiv\n\n*(*[What is alphaXiv?](https://alphaxiv.org/))\nCatalyzeX Code Finder for Papers\n\n*(*[What is CatalyzeX?](https://www.catalyzex.com))\nDagsHub\n\n*(*[What is DagsHub?](https://dagshub.com/))\nGotit.pub\n\n*(*[What is GotitPub?](http://gotit.pub/faq))\nHugging Face\n\n*(*[What is Huggingface?](https://huggingface.co/huggingface))\nScienceCast\n\n*(*[What is ScienceCast?](https://sciencecast.org/welcome))# Demos\n\n# Recommenders and Search Tools\n\nInfluence Flower\n\n*(*[What are Influence Flowers?](https://influencemap.cmlab.dev/))\nCORE Recommender\n\n*(*[What is CORE?](https://core.ac.uk/services/recommender))# arXivLabs: experimental projects with community collaborators\n\narXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.\n\nBoth individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.\n\nHave an idea for a project that will add value for arXiv's community? [ Learn more about arXivLabs](https://info.arxiv.org/labs/index.html).", "url": "https://wpnews.pro/news/defeating-vanishing-gradients-in-deep-neural-networks-quotient-tree-arithmetic", "canonical_source": "https://arxiv.org/abs/2607.22612", "published_at": "2026-07-30 05:44:11+00:00", "updated_at": "2026-07-30 05:52:14.601649+00:00", "lang": "en", "topics": ["artificial-intelligence", "machine-learning", "neural-networks", "ai-research"], "entities": ["Quotient Tree Arithmetic", "arXiv", "IEEE 754"], "alternates": {"html": "https://wpnews.pro/news/defeating-vanishing-gradients-in-deep-neural-networks-quotient-tree-arithmetic", "markdown": "https://wpnews.pro/news/defeating-vanishing-gradients-in-deep-neural-networks-quotient-tree-arithmetic.md", "text": "https://wpnews.pro/news/defeating-vanishing-gradients-in-deep-neural-networks-quotient-tree-arithmetic.txt", "jsonld": "https://wpnews.pro/news/defeating-vanishing-gradients-in-deep-neural-networks-quotient-tree-arithmetic.jsonld"}}