Jove
Visualize
Contact Us
JoVE
x logofacebook logolinkedin logoyoutube logo
ABOUT JoVE
OverviewLeadershipBlogJoVE Help Center
AUTHORS
Publishing ProcessEditorial BoardScope & PoliciesPeer ReviewFAQSubmit
LIBRARIANS
TestimonialsSubscriptionsAccessResourcesLibrary Advisory BoardFAQ
RESEARCH
JoVE JournalMethods CollectionsJoVE Encyclopedia of ExperimentsArchive
EDUCATION
JoVE CoreJoVE BusinessJoVE Science EducationJoVE Lab ManualFaculty Resource CenterFaculty Site
Terms & Conditions of Use
Privacy Policy
Policies

Related Concept Videos

Deconvolution01:20

Deconvolution

244
Deconvolution, also known as inverse filtering, is the process of extracting the impulse response from known input and output signals. This technique is vital in scenarios where the system's characteristics are unknown, and they must be inferred from the observable signals.
Deconvolution involves several mathematical techniques to derive the impulse response. One common approach is polynomial division. In this method, the input and output sequences are treated as coefficients of...
244
Convolution: Math, Graphics, and Discrete Signals01:24

Convolution: Math, Graphics, and Discrete Signals

377
In any LTI (Linear Time-Invariant) system, the convolution of two signals is denoted using a convolution operator, assuming all initial conditions are zero. The convolution integral can be divided into two parts: the zero-input or natural response and the zero-state or forced response, with t0 indicating the initial time.
To simplify the convolution integral, it is assumed that both the input signal and impulse response are zero for negative time values. The graphical convolution process...
377
Convolution Properties II01:17

Convolution Properties II

275
The important convolution properties include width, area, differentiation, and integration properties.
The width property indicates that if the durations of input signals are T1 and T2, then the width of the output response equals the sum of both durations, irrespective of the shapes of the two functions. For instance, convolving two rectangular pulses with durations of 2 seconds and 1 second results in a function with a width of 3 seconds.
The area property asserts that the area under the...
275
Convolution Properties I01:20

Convolution Properties I

229
Convolution computations can be simplified by utilizing their inherent properties.
The commutative property reveals that the input and the impulse response of an LTI (Linear Time-Invariant) system can be interchanged without affecting the output:
229
Neural Circuits01:25

Neural Circuits

1.5K
Neural circuits and neuronal pools are two of the main structures found in the nervous system. Neural circuits are networks of neurons that work together to carry out a specific task or process. They consist of interconnected neurons and glial cells, which provide structural and metabolic support.
Neuronal pools are collections of nerve cells with similar functions and interact through chemical and electrical signals. These pools include both interneurons (the central neural circuit nodes that...
1.5K
Region of Convergence01:17

Region of Convergence

536
The z-transform is a powerful mathematical tool used in the analysis of discrete-time signals and systems. It is a crucial tool in the analysis of discrete-time systems, but its convergence is limited to specific values of the complex variable z. This range of values, known as the Region of Convergence (ROC), is fundamental in determining the behavior and stability of a system or signal. The ROC defines the region in the complex plane where the z-transform converges, which can take various...
536

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by
Same author

Computed Quantitative Planar Imaging for Targeted Alpha Therapy: Model-Based Sparse Reconstruction Validated With a Novel <sup>225</sup>Ac Epoxy Phantom.

IEEE transactions on medical imaging·2026
Same author

Hypothesis spaces for deep learning.

Neural networks : the official journal of the International Neural Network Society·2025
Same author

Uniform Convergence of Deep Neural Networks With Lipschitz Continuous Activation Functions and Variable Widths.

IEEE transactions on information theory·2025
Same author

Motif-aware curriculum learning for node classification.

Neural networks : the official journal of the International Neural Network Society·2025
Same author

Dual Information Enhanced Multiview Attributed Graph Clustering.

IEEE transactions on neural networks and learning systems·2024
Same author

Sparse Machine Learning in Banach Spaces.

Applied numerical mathematics : transactions of IMACS·2023

Related Experiment Video

Updated: Sep 4, 2025

Author Spotlight: Enhancement of Salient Object Detection for Smart Grid Applications
03:31

Author Spotlight: Enhancement of Salient Object Detection for Smart Grid Applications

Published on: December 15, 2023

625

Convergence of deep convolutional neural networks.

Yuesheng Xu1, Haizhang Zhang2

  • 1Department of Mathematics & Statistics, Old Dominion University, Norfolk, VA 23529, USA.

Neural Networks : the Official Journal of the International Neural Network Society
|July 15, 2022
PubMed
Summary

This study explores the convergence of deep neural networks with increasing widths, a crucial aspect for deep learning foundations. We establish conditions for the convergence of matrix products, enabling analysis of deep Rectified Linear Unit (ReLU) networks and convolutional networks.

Keywords:
Activation domainsDeep convolutional neural networksDeep learningInfinite product of matricesReLU networks

More Related Videos

Deep Neural Networks for Image-Based Dietary Assessment
13:19

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

9.3K
Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique
04:48

Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique

Published on: July 5, 2024

488

Related Experiment Videos

Last Updated: Sep 4, 2025

Author Spotlight: Enhancement of Salient Object Detection for Smart Grid Applications
03:31

Author Spotlight: Enhancement of Salient Object Detection for Smart Grid Applications

Published on: December 15, 2023

625
Deep Neural Networks for Image-Based Dietary Assessment
13:19

Deep Neural Networks for Image-Based Dietary Assessment

Published on: March 13, 2021

9.3K
Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique
04:48

Swin-PSAxialNet: An Efficient Multi-Organ Segmentation Technique

Published on: July 5, 2024

488

Area of Science:

  • Deep Learning Theory
  • Artificial Intelligence
  • Mathematical Foundations of Machine Learning

Background:

  • Convergence of deep neural networks is vital for theoretical understanding.
  • Previous work focused on fixed-width Rectified Linear Unit (ReLU) networks.
  • Convolutional neural networks with increasing widths require further investigation.

Purpose of the Study:

  • To investigate the convergence of general deep ReLU networks with increasing widths.
  • To extend convergence analysis to deep convolutional neural networks.
  • To establish a mathematical foundation for these network architectures.

Main Methods:

  • Analysis of general deep ReLU networks with layer-wise increasing widths.
  • Reduction of network convergence to the convergence of infinite products of matrices with increasing sizes.
  • Establishment of sufficient conditions for matrix product convergence.

Main Results:

  • Sufficient conditions for the convergence of infinite products of matrices with increasing sizes were established.
  • Sufficient conditions for the pointwise convergence of general deep ReLU networks with increasing widths were derived.
  • Sufficient conditions for the pointwise convergence of deep ReLU convolutional neural networks were presented.

Conclusions:

  • The study provides a theoretical framework for understanding the convergence of deep networks with increasing widths.
  • The findings are applicable to both general deep ReLU networks and deep convolutional neural networks.
  • This work contributes to the mathematical foundations of deep learning, particularly for architectures with variable layer widths.