Topological Expressivity of ReLU Neural Networks

• URL: http://arxiv.org/abs/2310.11130v2
• Date: Mon, 10 Jun 2024 08:58:42 GMT
• Title: Topological Expressivity of ReLU Neural Networks
• Authors: Ekin Ergen, Moritz Grillo,
• Abstract summary: We study the expressivity of ReLU neural networks in the setting of a binary classification problem from a topological perspective. Results show that deep ReLU neural networks are exponentially more powerful than shallow ones in terms of topological simplification.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We study the expressivity of ReLU neural networks in the setting of a binary classification problem from a topological perspective. Recently, empirical studies showed that neural networks operate by changing topology, transforming a topologically complicated data set into a topologically simpler one as it passes through the layers. This topological simplification has been measured by Betti numbers, which are algebraic invariants of a topological space. We use the same measure to establish lower and upper bounds on the topological simplification a ReLU neural network can achieve with a given architecture. We therefore contribute to a better understanding of the expressivity of ReLU neural networks in the context of binary classification problems by shedding light on their ability to capture the underlying topological structure of the data. In particular the results show that deep ReLU neural networks are exponentially more powerful than shallow ones in terms of topological simplification. This provides a mathematically rigorous explanation why deeper networks are better equipped to handle complex and topologically rich data sets.

Related papers

• Defining Neural Network Architecture through Polytope Structures of Dataset [53.512432492636236]
  This paper defines upper and lower bounds for neural network widths, which are informed by the polytope structure of the dataset in question. We develop an algorithm to investigate a converse situation where the polytope structure of a dataset can be inferred from its corresponding trained neural networks. It is established that popular datasets such as MNIST, Fashion-MNIST, and CIFAR10 can be efficiently encapsulated using no more than two polytopes with a small number of faces.
  arXiv  Detail & Related papers  (2024-02-04T08:57:42Z)
• 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)
• Topological Data Analysis of Neural Network Layer Representations [0.0]
  topological features of a simple feedforward neural network's layer representations of a modified torus with a Klein bottle-like twist were computed. The resulting noise hampered the ability of persistent homology to compute these features.
  arXiv  Detail & Related papers  (2022-07-01T00:51:19Z)
• Dive into Layers: Neural Network Capacity Bounding using Algebraic Geometry [55.57953219617467]
  We show that the learnability of a neural network is directly related to its size. We use Betti numbers to measure the topological geometric complexity of input data and the neural network. We perform the experiments on a real-world dataset MNIST and the results verify our analysis and conclusion.
  arXiv  Detail & Related papers  (2021-09-03T11:45:51Z)
• A neural anisotropic view of underspecification in deep learning [60.119023683371736]
  We show that the way neural networks handle the underspecification of problems is highly dependent on the data representation. Our results highlight that understanding the architectural inductive bias in deep learning is fundamental to address the fairness, robustness, and generalization of these systems.
  arXiv  Detail & Related papers  (2021-04-29T14:31:09Z)
• Topological obstructions in neural networks learning [67.8848058842671]
  We study global properties of the loss gradient function flow. We use topological data analysis of the loss function and its Morse complex to relate local behavior along gradient trajectories with global properties of the loss surface.
  arXiv  Detail & Related papers  (2020-12-31T18:53:25Z)
• Depth-Width Trade-offs for Neural Networks via Topological Entropy [0.0]
  We show a new connection between the expressivity of deep neural networks and topological entropy from dynamical system. We discuss the relationship between topological entropy, the number of oscillations, periods and Lipschitz constant.
  arXiv  Detail & Related papers  (2020-10-15T08:14:44Z)
• Learning Connectivity of Neural Networks from a Topological Perspective [80.35103711638548]
  We propose a topological perspective to represent a network into a complete graph for analysis. By assigning learnable parameters to the edges which reflect the magnitude of connections, the learning process can be performed in a differentiable manner. This learning process is compatible with existing networks and owns adaptability to larger search spaces and different tasks.
  arXiv  Detail & Related papers  (2020-08-19T04:53:31Z)
• Topological Insights into Sparse Neural Networks [16.515620374178535]
  We introduce an approach to understand and compare sparse neural network topologies from the perspective of graph theory. We first propose Neural Network Sparse Topology Distance (NNSTD) to measure the distance between different sparse neural networks. We show that adaptive sparse connectivity can always unveil a plenitude of sparse sub-networks with very different topologies which outperform the dense model.
  arXiv  Detail & Related papers  (2020-06-24T22:27:21Z)

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.