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Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...

Linear Approximation in Frequency Domain

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear....

Accuracy, limits, and approximation

Accuracy, limits, and approximation

Accuracy, limits, and approximations are common in many fields, especially in engineering calculations. These concepts are imperative for ensuring that a given value is as close as possible to its true value.
Accuracy is defined as the closeness of the measured value to the true or actual value. In engineering mechanics, repeated measurements are taken during theoretical or experimental analyses to ensure that the result is precise and accurate.
The accuracy of any solution is based on the...

Linear time-invariant Systems

Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...

Comparison between RL and RC circuits

Comparison between RL and RC circuits

An RC circuit consists of resistance and capacitance, while in an RL circuit, capacitance is replaced by an inductor. RL and RC circuits are first-order differential circuits that store energy. An RC circuit stores energy in the electric field, while an RL circuit stores energy in the magnetic field. When connected to a battery, an RC circuit charges the capacitor, causing the current to decrease from maximum to zero upon being fully charged. This increases the voltage across the capacitor from...

State Space Representation

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...

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Updated: Sep 6, 2025

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

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Published on: December 15, 2023

Approximation in shift-invariant spaces with deep ReLU neural networks.

Yunfei Yang¹, Zhen Li², Yang Wang³

¹Department of Mathematics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; Theory Lab, Huawei Technologies Co., Ltd., Shenzhen, China.

Neural Networks : the Official Journal of the International Neural Network Society

|June 28, 2022

Summary

Deep ReLU neural networks efficiently approximate functions in signal and image processing spaces. Their construction achieves near-optimal approximation for Sobolev and Besov spaces.

Keywords:

Approximation complexity Besov spaces Deep neural networks Shift-invariant spaces Sobolev spaces

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Area of Science:

Computational mathematics
Deep learning theory
Functional analysis

Background:

Deep ReLU neural networks are powerful function approximators.
Dilated shift-invariant spaces are crucial in signal and image processing.
Understanding approximation capabilities is key for network design.

Purpose of the Study:

To analyze the expressive power of deep ReLU neural networks.
To estimate approximation error bounds concerning network width and depth.
To apply these findings to classical function spaces.

Main Methods:

Network construction leveraging bit extraction and data-fitting.
Derivation of approximation error bounds for deep neural networks.
Analysis of function approximation in dilated shift-invariant spaces.

Main Results:

Established approximation error bounds for deep ReLU networks.
Obtained approximation rates for Sobolev and Besov spaces.
Derived lower bounds for L^p approximation error in Sobolev spaces.

Conclusions:

Deep ReLU networks offer significant function approximation capabilities.
The proposed network construction is nearly optimal for Sobolev spaces.
These findings advance the theoretical understanding of deep learning for signal processing applications.