We prove identifiability results for constant-width residual networks with polynomial activations and bias terms. For activations of degree at least five with generic coefficients, the fiber over a generic parameter is a singleton. For activations of degree at least three with no linear term after a suitable translation of the input, the generic fiber is given by translating biases and then scaling weights and translated biases by real roots of unity. These results specialize to networks without bias terms. With the same proof technique, we further derive necessary conditions for critical points of the ResNet parametrization, that are central in singular learning theory.
Figures & tables
Figure 1. Scalar quadratic ResNets with two layers. Left: the real image M for 0≤c2≤2 , with its singular half-line in red. Right: the section c2=1 ; blue is the real image, and dashed gray branches belong to its Zariski closure but not to the image. The red point lies on the singular line of V(F) .
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.
Kathlén Kohn, Giovanni Luca Marchetti, Alex Massarenti +1
KTH Stockholm · AI4S AB Stockholm · University of Ferrara
We prove the weak composition-independence conjectures of Kohn, Marchetti, Massarenti and Mella. The proof uses deformations of rational curves and a single kernel of the linearized equation. Riemann-Roch gives a contradiction from tangent fields with controlled poles. By settling these conjectures, we prove that a neural network with a general polynomial activation function of sufficiently large degree is identifiable.
We study the realization map of deep ReLU networks, focusing on when a function determines its parameters up to scaling and permutation. To analyze hidden redundancies beyond these standard symmetries, we introduce a framework based on weighted polyhedral complexes. Our main result shows that for every architecture whose input and hidden layers have width at least two, there exists an open set of identifiable parameters. This implies that the functional dimension of every such architecture is exactly the number of parameters minus the number of hidden neurons. We further show that minimal functional representations can still have non-trivial parameter redundancies. Finally, we establish a generic depth hierarchy, whereby for an open set of parameters the realized function cannot be represented generically by any shallower network.
Moritz Grillo, Guido Montúfar
Max Planck Institute for Mathematics in the Sciences, Leipzig · Departments of Mathematics and Statistics & Data Science, UCLA