A Derivation of Feedforward Neural Network Gradients Using Fr\'echet Calculus

• URL: http://arxiv.org/abs/2209.13234v1
• Date: Tue, 27 Sep 2022 08:14:00 GMT
• Title: A Derivation of Feedforward Neural Network Gradients Using Fr\'echet Calculus
• Authors: Thomas Hamm
• Abstract summary: We show a derivation of the gradients of feedforward neural networks using Fr'teche calculus. We show how our analysis generalizes to more general neural network architectures including, but not limited to, convolutional networks.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present a derivation of the gradients of feedforward neural networks using Fr\'echet calculus which is arguably more compact than the ones usually presented in the literature. We first derive the gradients for ordinary neural networks working on vectorial data and show how these derived formulas can be used to derive a simple and efficient algorithm for calculating a neural networks gradients. Subsequently we show how our analysis generalizes to more general neural network architectures including, but not limited to, convolutional networks.

Related papers

• Graph Neural Networks for Learning Equivariant Representations of Neural Networks [55.04145324152541]
  We propose to represent neural networks as computational graphs of parameters. Our approach enables a single model to encode neural computational graphs with diverse architectures. We showcase the effectiveness of our method on a wide range of tasks, including classification and editing of implicit neural representations.
  arXiv  Detail & Related papers  (2024-03-18T18:01:01Z)
• When Deep Learning Meets Polyhedral Theory: A Survey [6.899761345257773]
  In the past decade, deep became the prevalent methodology for predictive modeling thanks to the remarkable accuracy of deep neural learning. Meanwhile, the structure of neural networks converged back to simplerwise and linear functions.
  arXiv  Detail & Related papers  (2023-04-29T11:46:53Z)
• Globally Optimal Training of Neural Networks with Threshold Activation Functions [63.03759813952481]
  We study weight decay regularized training problems of deep neural networks with threshold activations. We derive a simplified convex optimization formulation when the dataset can be shattered at a certain layer of the network.
  arXiv  Detail & Related papers  (2023-03-06T18:59:13Z)
• Neural Networks with Sparse Activation Induced by Large Bias: Tighter Analysis with Bias-Generalized NTK [86.45209429863858]
  We study training one-hidden-layer ReLU networks in the neural tangent kernel (NTK) regime. We show that the neural networks possess a different limiting kernel which we call textitbias-generalized NTK We also study various properties of the neural networks with this new kernel.
  arXiv  Detail & Related papers  (2023-01-01T02:11:39Z)
• Simple initialization and parametrization of sinusoidal networks via their kernel bandwidth [92.25666446274188]
  sinusoidal neural networks with activations have been proposed as an alternative to networks with traditional activation functions. We first propose a simplified version of such sinusoidal neural networks, which allows both for easier practical implementation and simpler theoretical analysis. We then analyze the behavior of these networks from the neural tangent kernel perspective and demonstrate that their kernel approximates a low-pass filter with an adjustable bandwidth.
  arXiv  Detail & Related papers  (2022-11-26T07:41:48Z)
• Correlation between entropy and generalizability in a neural network [9.223853439465582]
  We use Wang-Landau Mote Carlo algorithm to calculate the entropy at a given test accuracy. Our results show that entropical forces help generalizability.
  arXiv  Detail & Related papers  (2022-07-05T12:28:13Z)
• Generalization Error Bounds for Iterative Recovery Algorithms Unfolded as Neural Networks [6.173968909465726]
  We introduce a general class of neural networks suitable for sparse reconstruction from few linear measurements. By allowing a wide range of degrees of weight-sharing between the layers, we enable a unified analysis for very different neural network types.
  arXiv  Detail & Related papers  (2021-12-08T16:17:33Z)
• Why Lottery Ticket Wins? A Theoretical Perspective of Sample Complexity on Pruned Neural Networks [79.74580058178594]
  We analyze the performance of training a pruned neural network by analyzing the geometric structure of the objective function. We show that the convex region near a desirable model with guaranteed generalization enlarges as the neural network model is pruned.
  arXiv  Detail & Related papers  (2021-10-12T01:11:07Z)
• Expressivity of Deep Neural Networks [2.7909470193274593]
  In this review paper, we give a comprehensive overview of the large variety of approximation results for neural networks. While the mainbody of existing results is for general feedforward architectures, we also depict approximation results for convolutional, residual and recurrent neural networks.
  arXiv  Detail & Related papers  (2020-07-09T13:08:01Z)
• RicciNets: Curvature-guided Pruning of High-performance Neural Networks Using Ricci Flow [0.0]
  We use the definition of Ricci curvature to remove edges of low importance before mapping the computational graph to a neural network. We show a reduction of almost $35%$ in the number of floating-point operations (FLOPs) per pass, with no degradation in performance.
  arXiv  Detail & Related papers  (2020-07-08T15:56:02Z)
• Understanding Generalization in Deep Learning via Tensor Methods [53.808840694241]
  We advance the understanding of the relations between the network's architecture and its generalizability from the compression perspective. We propose a series of intuitive, data-dependent and easily-measurable properties that tightly characterize the compressibility and generalizability of neural networks.
  arXiv  Detail & Related papers  (2020-01-14T22:26:57Z)

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.