Linear Independence of Polynomial Compositions and Identifiability of Deep Neural Networks

Motivated by theoretical problems in deep learning, we conjecture that post-composing a fixed number of pairwise distinct nonconstant polynomials with a generic polynomial of sufficiently large degree yields linearly independent polynomials. This generalizes Newman--Slater's theorem on powers of polynomials. We establish several cases of this conjecture and its origin-passing variant: We prove the result for two polynomials, and for an arbitrary number of polynomials when their degrees are bounded. Furthermore, we show how the conjecture implies a complete understanding of the identifiability (i.e., parameter symmetries) of deep fully connected neural network architectures with generic polynomial activation functions. In particular, for network architectures with layer-specific activations of increasing degree, our established versions of the conjecture fully characterize the set of parameters yielding the same end-to-end network function. As a special case, we fully resolve the identifiability of shallow polynomial networks.

Activation-Free Backbones for Image Recognition: Polynomial Alternatives within MetaFormer-Style Vision Models

Modern vision backbones treat pointwise activations (e.g., ReLU, GELU) and exponential softmax as essential sources of nonlinearity, but we demonstrate they are not required within MetaFormer-style vision backbones. We design activation-free polynomial alternatives for three core primitives (MLPs, convolutions, and attention), where Hadamard products replace standard nonlinearities to yield polynomial functions of the input. These modules integrate seamlessly into existing architectures: instantiated within MetaFormer, a modular framework for vision backbones, our PolyNeXt models match or exceed activation-based counterparts across model scales on ImageNet classification, ADE20K semantic segmentation, and out-of-distribution robustness. We also substantially outperform prior polynomial networks at reduced computational cost, showing that polynomial variants of standard modules beat complex custom architectures.