On the Topology of Neural Network Superlevel Sets

• URL: http://arxiv.org/abs/2603.02973v1
• Date: Tue, 03 Mar 2026 13:30:06 GMT
• Title: On the Topology of Neural Network Superlevel Sets
• Authors: Bahman Gharesifard,
• Abstract summary: We show that neural networks with activations satisfying a Riccati-type ordinary differential equation condition, produce Pfaffian outputs on analytic domains with format controlled only by the architecture.
• Score: 1.827510863075184
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We show that neural networks with activations satisfying a Riccati-type ordinary differential equation condition, an assumption arising in recent universal approximation results in the uniform topology, produce Pfaffian outputs on analytic domains with format controlled only by the architecture. Consequently, superlevel sets, as well as Lie bracket rank drop loci for neural network parameterized vector fields, admit architecture-only bounds on topological complexity, in particular on total Betti numbers, uniformly over all weights.

Related papers

• Spectral complexity of deep neural networks [2.099922236065961]
  We use the angular power spectrum of the limiting field to characterize the complexity of the network architecture.<n>On this basis, we classify neural networks as low-disorder, sparse, or high-disorder.<n>We show how this classification highlights a number of distinct features for standard activation functions, and in particular, sparsity properties of ReLU networks.
  arXiv  Detail & Related papers  (2024-05-15T17:55:05Z)
• Generalization of Scaled Deep ResNets in the Mean-Field Regime [55.77054255101667]
  We investigate emphscaled ResNet in the limit of infinitely deep and wide neural networks. Our results offer new insights into the generalization ability of deep ResNet beyond the lazy training regime.
  arXiv  Detail & Related papers  (2024-03-14T21:48:00Z)
• On Characterizing the Evolution of Embedding Space of Neural Networks using Algebraic Topology [9.537910170141467]
  We study how the topology of feature embedding space changes as it passes through the layers of a well-trained deep neural network (DNN) through Betti numbers. We demonstrate that as depth increases, a topologically complicated dataset is transformed into a simple one, resulting in Betti numbers attaining their lowest possible value.
  arXiv  Detail & Related papers  (2023-11-08T10:45:12Z)
• Data Topology-Dependent Upper Bounds of Neural Network Widths [52.58441144171022]
  We first show that a three-layer neural network can be designed to approximate an indicator function over a compact set. This is then extended to a simplicial complex, deriving width upper bounds based on its topological structure. We prove the universal approximation property of three-layer ReLU networks using our topological approach.
  arXiv  Detail & Related papers  (2023-05-25T14:17:15Z)
• Norm-based Generalization Bounds for Compositionally Sparse Neural Networks [11.987589603961622]
  We prove generalization bounds for multilayered sparse ReLU neural networks, including convolutional neural networks. Taken together, these results suggest that compositional sparsity of the underlying target function is critical to the success of deep neural networks.
  arXiv  Detail & Related papers  (2023-01-28T00:06:22Z)
• On the Effective Number of Linear Regions in Shallow Univariate ReLU Networks: Convergence Guarantees and Implicit Bias [50.84569563188485]
  We show that gradient flow converges in direction when labels are determined by the sign of a target network with $r$ neurons. Our result may already hold for mild over- parameterization, where the width is $tildemathcalO(r)$ and independent of the sample size.
  arXiv  Detail & Related papers  (2022-05-18T16:57:10Z)
• The Sample Complexity of One-Hidden-Layer Neural Networks [57.6421258363243]
  We study a class of scalar-valued one-hidden-layer networks, and inputs bounded in Euclidean norm. We prove that controlling the spectral norm of the hidden layer weight matrix is insufficient to get uniform convergence guarantees. We analyze two important settings where a mere spectral norm control turns out to be sufficient.
  arXiv  Detail & Related papers  (2022-02-13T07:12:02Z)
• Dist2Cycle: A Simplicial Neural Network for Homology Localization [66.15805004725809]
  Simplicial complexes can be viewed as high dimensional generalizations of graphs that explicitly encode multi-way ordered relations. We propose a graph convolutional model for learning functions parametrized by the $k$-homological features of simplicial complexes.
  arXiv  Detail & Related papers  (2021-10-28T14:59:41Z)
• Revealing the Structure of Deep Neural Networks via Convex Duality [70.15611146583068]
  We study regularized deep neural networks (DNNs) and introduce a convex analytic framework to characterize the structure of hidden layers. We show that a set of optimal hidden layer weights for a norm regularized training problem can be explicitly found as the extreme points of a convex set. We apply the same characterization to deep ReLU networks with whitened data and prove the same weight alignment holds.
  arXiv  Detail & Related papers  (2020-02-22T21:13:44Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.