What is it about?
Conventional neural networks use vectors to represent information and training data to optimise those vectors, yet they struggle to reason reliably. By treating vectors as spheres with zero radius, this paper argues that allowing a nonzero radius enables neural networks to perform reliable logical reasoning without training data.
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Why is it important?
LLM-based AI systems are changing our society, but remain blackboxs, unreliable, and extremely expensive, as training demands huge amount of data and computational power. We show the reason behind is the vector embedding, a sphere missing a radius. With this deformity, more training data and more GPUs cannot guarantee reliable reasoning. We show that restoring the missing radius to neural networks enables them to reason reliably on founamental logical reasoning without training data.
Perspectives
We wonder why conventional neural networks use vector embeddings and trace their origins to the 1943 McCulloch–Pitts model -- The first artifical neural network exactly used vector embedding for classic logical reasoning. At that time, Norbert Wiener considered the model overly simplistic, yet its basic design has shaped neural network research for more than eight decades. When we successfully developed Sphere Neural Netowrk that achieves rigorous syllogistic reasoning without using training data, we noticed the original design defect, and that restoring the radius offers an efficient path for reliable and safe AI.
Dr. Tiansi Dong
The University of Cambridge & The Alan Turing Institute
Read the Original
This page is a summary of: Non-Zero Radius: The Missing Component of Neural Networks, August 2026, ACM (Association for Computing Machinery),
DOI: 10.1145/3806096.3844817.
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