Nonlinear mapping

As explained in Graph Neural Networks in Action

A parameterized operation that maps inputs to outputs without preserving linearity, allowing a neural network to model nonlinear relationships.

A nonlinear mapping transforms aggregated node or edge information through a neural network and activation function. Unlike a purely linear transformation, it lets stacked message-passing layers model more than a linear or logistic regression model. section 3.4.2

A message-passing layer applies the mapping in a short pipeline:

• Aggregate information from neighboring nodes with an order-independent operation such as summation, averaging, or maximum. section 3.4.2
• Pass the aggregated messages through learnable neural-network weights and an activation function to produce embeddings. section 3.4.2
• Combine the transformed embeddings with node or edge data and previous embeddings, then use the resulting representations for prediction or graph embeddings. section 2.4.4

ReLU in a graph neural network

ReLU is a nonlinear mapping that returns the maximum of zero and its input. Applying it after message aggregation prevents a multilayer model from collapsing into an overall linear transformation. section 3.4.2

Figure 2.14 Structure of a multilayer perceptron

Use the multilayer perceptron diagram to visualize the neural-network transformation applied after graph information is gathered. section 2.4.4

How Graph Neural Networks in Action covers it

How deep it goes

Explains aggregation, learnable transformation, activation, and update steps, including ReLU and a general embedding formula.

Examples

Uses ReLU, multilayer perceptrons, neighbor aggregation, and employee recommendation or company-joining prediction.

What you can do after

Apply it through activation functions in message-passing layers to construct nonlinear graph embeddings and predictions.

How it connects

Connects activation functions with message passing, neural networks, graph embeddings, permutation equivariance, and pooling.

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Also explained in

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  Math and Architectures of Deep LearningA property of a function or layer whose output is not a linear transformation of its inputs.
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  Machine Learning for Tabular Data: XGBoost, Deep Learning, and AIA nonlinear transformation maps inputs to outputs in a way that cannot be represented as a linear function.
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  Deep Learning with PyTorch, Second EditionA process whose behavior cannot be represented as a simple linear relation between inputs and outputs.
• 

  liveBookmanning.com
  Statistics Slam DunkNonlinearity is the property that a relationship cannot be represented adequately by a straight-line pattern or linear function.
• 

  liveBookmanning.com
  Statistics Every Programmer NeedsA nonlinear relationship is a relationship in which changes in one quantity are not represented by a linear function of another.
• 

  liveBookmanning.com
  Inside Deep Learning: Math, Algorithms, ModelsA transformation inserted between linear steps to let a neural network represent functions that are not purely linear.

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Related concepts

• activation functionproduced by · An activation function maps a neural-network unit’s or layer’s input to an output, often introducing nonlinearity into the network’s computation.
• 80-20 ruleequivalent to · The 80-20 rule is an informal heuristic that roughly 80 percent of an outcome may be associated with 20 percent of its contributing causes or effort.
• linear regressioncontrasts with · Linear regression is a statistical modeling method that estimates a continuous response as a linear combination of predictor variables and fitted coefficients.
• mathematical functionkind of · A mathematical function is a mapping that assigns exactly one output to each permitted input in its domain.
• expressive powerproduces · Expressive power is the capacity of a model, language, or system to represent or convey a range and complexity of functions, patterns, distinctions, or ideas.
• perceptroncontrasts with · A perceptron is a single-layer computational model and linear classifier that combines weighted inputs with a bias and applies a threshold or activation rule.

This explainer was generated by Manning's knowledge graph from the text of Graph Neural Networks in Action. The section links point to the parts of the book it draws on.