Trainable at Depth, Solvable at Small Size: What the Loss Surface of a Polynomial Activation Dictates for Designing, Initializing and Training Networks

Xiuyi Yang

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arXiv · 2610.20069

Trainable at Depth, Solvable at Small Size: What the Loss Surface of a Polynomial Activation Dictates for Designing, Initializing and Training Networks

Xiuyi Yang

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Abstract

What makes a deep network train, and what shape is its loss surface? We answer both with one construction at two scales: a polynomial activation f (z) = z + λz 3 , clamped and projected onto the unit sphere, plus an unnormalized identity path. As a deep stack it trains to one hundred layers on MNIST with no normalization machinery; as a small model its square loss is a polynomial system we solve exactly. The small scale explains the large one: it prescribes how to design, initialize and train the stack.

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BibTeX RIS

Xiuyi Yang. 2026-10-14. Trainable at Depth, Solvable at Small Size: What the Loss Surface of a Polynomial Activation Dictates for Designing, Initializing and Training Networks. https://arxiv.org/abs/2610.20069

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