A rank decomposition for the topological classification of neural representations

• URL: http://arxiv.org/abs/2404.19710v3
• Date: Tue, 4 Jun 2024 09:47:30 GMT
• Title: A rank decomposition for the topological classification of neural representations
• Authors: Kosio Beshkov, Gaute T. Einevoll,
• Abstract summary: In this work, we leverage the fact that neural networks are equivalent to continuous piecewise-affine maps. We study the homology groups of the quotient of a manifold $mathcalM$ and a subset $A$, assuming some minimal properties on these spaces. We show that in randomly narrow networks, there will be regions in which the (co)homology groups of a data manifold can change.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Neural networks can be thought of as applying a transformation to an input dataset. The way in which they change the topology of such a dataset often holds practical significance for many tasks, particularly those demanding non-homeomorphic mappings for optimal solutions, such as classification problems. In this work, we leverage the fact that neural networks are equivalent to continuous piecewise-affine maps, whose rank can be used to pinpoint regions in the input space that undergo non-homeomorphic transformations, leading to alterations in the topological structure of the input dataset. Our approach enables us to make use of the relative homology sequence, with which one can study the homology groups of the quotient of a manifold $\mathcal{M}$ and a subset $A$, assuming some minimal properties on these spaces. As a proof of principle, we empirically investigate the presence of low-rank (topology-changing) affine maps as a function of network width and mean weight. We show that in randomly initialized narrow networks, there will be regions in which the (co)homology groups of a data manifold can change. As the width increases, the homology groups of the input manifold become more likely to be preserved. We end this part of our work by constructing highly non-random wide networks that do not have this property and relating this non-random regime to Dale's principle, which is a defining characteristic of biological neural networks. Finally, we study simple feedforward networks trained on MNIST, as well as on toy classification and regression tasks, and show that networks manipulate the topology of data differently depending on the continuity of the task they are trained on.

Related papers

• Algebraic Topological Networks via the Persistent Local Homology Sheaf [15.17547132363788]
  We introduce a novel approach to enhance graph convolution and attention modules by incorporating local topological properties of the data. We consider the framework of sheaf neural networks, which has been previously leveraged to incorporate additional structure into graph neural networks' features.
  arXiv  Detail & Related papers  (2023-11-16T19:24:20Z)
• 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)
• 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)
• 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)
• Decomposing neural networks as mappings of correlation functions [57.52754806616669]
  We study the mapping between probability distributions implemented by a deep feed-forward network. We identify essential statistics in the data, as well as different information representations that can be used by neural networks.
  arXiv  Detail & Related papers  (2022-02-10T09:30:31Z)
• Data-driven emergence of convolutional structure in neural networks [83.4920717252233]
  We show how fully-connected neural networks solving a discrimination task can learn a convolutional structure directly from their inputs. By carefully designing data models, we show that the emergence of this pattern is triggered by the non-Gaussian, higher-order local structure of the inputs.
  arXiv  Detail & Related papers  (2022-02-01T17:11:13Z)
• A singular Riemannian geometry approach to Deep Neural Networks II. Reconstruction of 1-D equivalence classes [78.120734120667]
  We build the preimage of a point in the output manifold in the input space. We focus for simplicity on the case of neural networks maps from n-dimensional real spaces to (n - 1)-dimensional real spaces.
  arXiv  Detail & Related papers  (2021-12-17T11:47:45Z)
• The Separation Capacity of Random Neural Networks [78.25060223808936]
  We show that a sufficiently large two-layer ReLU-network with standard Gaussian weights and uniformly distributed biases can solve this problem with high probability. We quantify the relevant structure of the data in terms of a novel notion of mutual complexity.
  arXiv  Detail & Related papers  (2021-07-31T10:25:26Z)
• A Topological Framework for Deep Learning [0.7310043452300736]
  We show that the classification problem in machine learning is always solvable under very mild conditions. In particular, we show that a softmax classification network acts on an input topological space by a finite sequence of topological moves to achieve the classification task.
  arXiv  Detail & Related papers  (2020-08-31T15:56:42Z)
• 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)
• NeuroFabric: Identifying Ideal Topologies for Training A Priori Sparse Networks [2.398608007786179]
  Long training times of deep neural networks are a bottleneck in machine learning research. We provide a theoretical foundation for the choice of intra-layer topology. We show that seemingly similar topologies can often have a large difference in attainable accuracy.
  arXiv  Detail & Related papers  (2020-02-19T18:29:18Z)

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.