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Convolution: Math, Graphics, and Discrete Signals

Convolution: Math, Graphics, and Discrete Signals

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...

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The positive direction of the t-axis aligns with the increasing position of the car along the curved path, denoted by the unit vector ut. Simultaneously, the n-axis, perpendicular to the t-axis, dissects the curved path into differential arc segments, each forming the arc of a circle with a radius of...

Convolution Properties I

Convolution Properties I

Convolution computations can be simplified by utilizing their inherent properties.
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Convolution Properties II

Convolution Properties II

The important convolution properties include width, area, differentiation, and integration properties.
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Vector Algebra: Graphical Method

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Related Experiment Video

Updated: Aug 24, 2025

Evidence-based Knowledge Synthesis and Hypothesis Validation: Navigating Biomedical Knowledge Bases via Explainable AI and Agentic Systems

Evidence-based Knowledge Synthesis and Hypothesis Validation: Navigating Biomedical Knowledge Bases via Explainable AI and Agentic Systems

Published on: June 13, 2025

On the Explainability of Graph Convolutional Network With GCN Tangent Kernel.

Xianchen Zhou¹, Hongxia Wang²

¹National University of Defense Technology, Changsha 410073, P.R.C. zhouxianchen13@nudt.edu.cn.

Neural Computation

|October 25, 2022

Summary

We show that Graph Convolutional Networks (GCNs) with wide hidden layers can be explained by differential equations. The Graph Convolutional Neural Tangent Kernel (GCNTK) governs their stable dynamics for node classification tasks.

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Last Updated: Aug 24, 2025

Evidence-based Knowledge Synthesis and Hypothesis Validation: Navigating Biomedical Knowledge Bases via Explainable AI and Agentic Systems

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Revealing Neural Circuit Topography in Multi-Color

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Published on: November 14, 2011

Area of Science:

Artificial Intelligence
Machine Learning
Graph Neural Networks

Background:

Graph Convolutional Networks (GCNs) are effective for graph data but lack explainability.
Understanding the training dynamics of GCNs is challenging.

Purpose of the Study:

To provide an explainable framework for GCNs.
To analyze the behavior of GCNs in semisupervised learning settings.

Main Methods:

We analyze GCNs with a wide hidden feature dimension.
We derive a differential equation to describe GCN output.
We introduce the Graph Convolutional Neural Tangent Kernel (GCNTK) to model dynamics.

Main Results:

GCN output for semisupervised problems can be described by a differential equation.
The GCNTK determines stable dynamics as hidden dimension width approaches infinity.
Node classification solutions are directly explained by the derived differential equation.

Conclusions:

The GCNTK model offers a consistent and explainable framework for GCNs.
This approach enhances the interpretability of deep learning on graph data.