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Related Concept Videos

Downsampling01:20

Downsampling

540
When considering a sampled sequence with zero values between sampling instants, one can replace it by taking every N-th value of the sequence. At these integer multiples of N, the original and sampled sequences coincide. This process, known as decimation, involves extracting every N-th sample from a sequence, thereby creating a more efficient sequence.
The Fourier transform of the decimated sequence reveals a combination of scaled and shifted versions of the original spectrum. This...
540
Upsampling01:22

Upsampling

543
Managing signal sampling rates is essential in digital signal processing to maintain signal integrity. A decimated signal, characterized by a reduced frequency range due to its lower sampling rate, can be upsampled by inserting zeros between each sample. This upsampling process expands the original spectrum and introduces repeated spectral replicas at intervals dictated by the new Nyquist frequency. To refine this zero-inserted sequence, it is passed through a lowpass filter with a cutoff...
543
Deconvolution01:20

Deconvolution

495
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...
495
Reducing Line Loss01:18

Reducing Line Loss

315
In a three-phase circuit, line loss is an indicator of energy dissipated as heat due to the resistance of transmission lines. To address this, incorporating transformers into the system—a step-up transformer at the source and a step-down transformer at the load—is a strategic solution. Two three-phase transformers are introduced to improve this.
With a step-up transformer at the source, the voltage is increased, thereby reducing the current in the transmission lines since power loss in...
315
Convolution: Math, Graphics, and Discrete Signals01:24

Convolution: Math, Graphics, and Discrete Signals

755
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...
755
Convolution Properties II01:17

Convolution Properties II

519
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...
519

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Theory of deep convolutional neural networks: Downsampling.

Ding-Xuan Zhou1

  • 1School of Data Science and Department of Mathematics, City University of Hong Kong, Kowloon, Hong Kong.

Neural Networks : the Official Journal of the International Neural Network Society
|February 10, 2020
PubMed
Summary

Deep convolutional neural networks (CNNs) show promise for approximating complex functions and learning data manifold features. Downsampling techniques enable theoretical analysis, demonstrating CNNs match fully-connected network capabilities.

Keywords:
Approximation theoryConvolutional neural networksDeep learningDownsamplingFilter masks

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Area of Science:

  • Machine Learning
  • Artificial Intelligence
  • Theoretical Computer Science

Background:

  • Deep learning's practical success necessitates a strong theoretical foundation for structured deep neural networks.
  • Convolutional neural networks (CNNs) are widely applied but lack comprehensive theoretical understanding.

Purpose of the Study:

  • To develop an approximation theory for deep CNNs.
  • To analyze the approximation capabilities of CNNs induced by convolutions.
  • To demonstrate CNNs' potential for learning manifold features.

Main Methods:

  • Introduction of a downsampling operator to manage network widths for theoretical analysis.
  • Proof of CNNs' ability to approximate ridge functions.
  • Demonstration that CNNs can realize the output of fully-connected networks with comparable parameters.

Main Results:

  • Downsampled CNNs effectively approximate ridge functions, indicating modeling advantages.
  • CNNs exhibit approximation capabilities at least as strong as fully-connected networks.
  • A theorem is presented for approximating functions on Riemannian manifolds using CNNs.

Conclusions:

  • Deep CNNs possess significant approximation power, comparable to or exceeding fully-connected networks.
  • The proposed theoretical framework, using downsampling, facilitates analysis of CNNs.
  • CNNs are capable of learning intricate manifold features within data.