On the Universal Statistical Consistency of Expansive Hyperbolic Deep Convolutional Neural Networks

• URL: http://arxiv.org/abs/2411.10128v1
• Date: Fri, 15 Nov 2024 12:01:03 GMT
• Title: On the Universal Statistical Consistency of Expansive Hyperbolic Deep Convolutional Neural Networks
• Authors: Sagar Ghosh, Kushal Bose, Swagatam Das,
• Abstract summary: In this work, we propose Hyperbolic DCNN based on the Poincar'e Disc. We offer extensive theoretical insights pertaining to the universal consistency of the expansive convolution in the hyperbolic space. Results reveal that the hyperbolic convolutional architecture outperforms the Euclidean ones by a commendable margin.
• Score: 14.904264782690639
• License:
• Abstract: The emergence of Deep Convolutional Neural Networks (DCNNs) has been a pervasive tool for accomplishing widespread applications in computer vision. Despite its potential capability to capture intricate patterns inside the data, the underlying embedding space remains Euclidean and primarily pursues contractive convolution. Several instances can serve as a precedent for the exacerbating performance of DCNNs. The recent advancement of neural networks in the hyperbolic spaces gained traction, incentivizing the development of convolutional deep neural networks in the hyperbolic space. In this work, we propose Hyperbolic DCNN based on the Poincar\'{e} Disc. The work predominantly revolves around analyzing the nature of expansive convolution in the context of the non-Euclidean domain. We further offer extensive theoretical insights pertaining to the universal consistency of the expansive convolution in the hyperbolic space. Several simulations were performed not only on the synthetic datasets but also on some real-world datasets. The experimental results reveal that the hyperbolic convolutional architecture outperforms the Euclidean ones by a commendable margin.

Related papers

• From Alexnet to Transformers: Measuring the Non-linearity of Deep Neural Networks with Affine Optimal Transport [32.39176908225668]
  We introduce the concept of the non-linearity signature of DNN, the first theoretically sound solution for measuring the non-linearity of deep neural networks. We provide extensive experimental results that highlight the practical usefulness of the proposed non-linearity signature.
  arXiv  Detail & Related papers  (2023-10-17T17:50:22Z)
• Fully Hyperbolic Convolutional Neural Networks for Computer Vision [3.3964154468907486]
  We present HCNN, a fully hyperbolic convolutional neural network (CNN) designed for computer vision tasks. Based on the Lorentz model, we propose novel formulations of the convolutional layer, batch normalization, and multinomial logistic regression. Experiments on standard vision tasks demonstrate the promising performance of our HCNN framework in both hybrid and fully hyperbolic settings.
  arXiv  Detail & Related papers  (2023-03-28T12:20:52Z)
• Convolutional Neural Networks on Manifolds: From Graphs and Back [122.06927400759021]
  We propose a manifold neural network (MNN) composed of a bank of manifold convolutional filters and point-wise nonlinearities. To sum up, we focus on the manifold model as the limit of large graphs and construct MNNs, while we can still bring back graph neural networks by the discretization of MNNs.
  arXiv  Detail & Related papers  (2022-10-01T21:17:39Z)
• Deep Architecture Connectivity Matters for Its Convergence: A Fine-Grained Analysis [94.64007376939735]
  We theoretically characterize the impact of connectivity patterns on the convergence of deep neural networks (DNNs) under gradient descent training. We show that by a simple filtration on "unpromising" connectivity patterns, we can trim down the number of models to evaluate.
  arXiv  Detail & Related papers  (2022-05-11T17:43:54Z)
• Hyperbolic Graph Neural Networks: A Review of Methods and Applications [55.5502008501764]
  Graph neural networks generalize conventional neural networks to graph-structured data. The performance of Euclidean models in graph-related learning is still bounded and limited by the representation ability of Euclidean geometry. Recently, hyperbolic space has gained increasing popularity in processing graph data with tree-like structure and power-law distribution.
  arXiv  Detail & Related papers  (2022-02-28T15:08:48Z)
• Nested Hyperbolic Spaces for Dimensionality Reduction and Hyperbolic NN Design [8.250374560598493]
  Hyperbolic neural networks have been popular in the recent past due to their ability to represent hierarchical data sets effectively and efficiently. The challenge in developing these networks lies in the nonlinearity of the embedding space namely, the Hyperbolic space. We present a novel fully hyperbolic neural network which uses the concept of projections (embeddings) followed by an intrinsic aggregation and a nonlinearity all within the hyperbolic space.
  arXiv  Detail & Related papers  (2021-12-03T03:20:27Z)
• Improvising the Learning of Neural Networks on Hyperspherical Manifold [0.0]
  The impact of convolution neural networks (CNNs) in the supervised settings provided tremendous increment in performance. The representation learned from CNN's operated on hyperspherical manifold led to insightful outcomes in face recognition. A broad range of activation functions is developed with hypersphere intuition which performs superior to softmax in euclidean space.
  arXiv  Detail & Related papers  (2021-09-29T22:39:07Z)
• Fully Hyperbolic Neural Networks [63.22521652077353]
  We propose a fully hyperbolic framework to build hyperbolic networks based on the Lorentz model. We show that our method has better performance for building both shallow and deep networks.
  arXiv  Detail & Related papers  (2021-05-31T03:36:49Z)
• Hyperbolic Deep Neural Networks: A Survey [31.04110049167551]
  We refer to the model as hyperbolic deep neural network in this paper. To stimulate future research, this paper presents acoherent and comprehensive review of the literature around the neural components in the construction of hyperbolic deep neuralnetworks.
  arXiv  Detail & Related papers  (2021-01-12T15:55:16Z)
• Combining Differentiable PDE Solvers and Graph Neural Networks for Fluid Flow Prediction [79.81193813215872]
  We develop a hybrid (graph) neural network that combines a traditional graph convolutional network with an embedded differentiable fluid dynamics simulator inside the network itself. We show that we can both generalize well to new situations and benefit from the substantial speedup of neural network CFD predictions.
  arXiv  Detail & Related papers  (2020-07-08T21:23:19Z)
• Hyperbolic Neural Networks++ [66.16106727715061]
  We generalize the fundamental components of neural networks in a single hyperbolic geometry model, namely, the Poincar'e ball model. Experiments show the superior parameter efficiency of our methods compared to conventional hyperbolic components, and stability and outperformance over their Euclidean counterparts.
  arXiv  Detail & Related papers  (2020-06-15T08:23:20Z)

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.