Trainable at Depth, Solvable at Small Size: What the Loss Surface of a Polynomial Activation Dictates for Designing, Initializing and Training Networks
cs.LG, cs.AI
Submitted: 2026-10-14
Updated: 2026-10-14
Code: https://github.com/lfwin/Polynomial-Activation-and-projectiv
License: http://creativecommons.org/licenses/by-nc-sa/4.0/
The gist: 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
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.
Sources
- Layer Normalization
- Gaussian Error Linear Units (GELUs)
- ParseNet: Looking Wider to See Better
- KAN: Kolmogorov-Arnold Networks
- nGPT: Normalized Transformer with Representation Learning on the Hypersphere
- Spherical Perspective on Learning with Normalization Layers
- GLU Variants Improve Transformer
- The Landscape of Multi-Layer Linear Neural Network From the Perspective of Algebraic Geometry
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