Multi-Layer Networks

Definition

A multi-layer network is a neural network with more than one transformation layer.

Basic form:

input -> layer 1 -> layer 2 -> layer 3 -> output

Each layer receives values from the previous layer and produces new values.

A multi-layer network composes learned transformations.

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Layer Composition

Each layer can be understood as a function.

layer 1 = f1
layer 2 = f2
layer 3 = f3

The full network is a composition:

output = f3(f2(f1(input)))

This means later layers operate on transformed representations, not directly on the original input.

Multi-layer networks build outputs through chained functions.

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Why Multiple Layers Help

Multiple layers let the network build intermediate structure.

Example:

input values
-> simple signals
-> combined signals
-> task-specific output

Each layer can prepare a representation for the next layer.

Multiple layers let the model solve a problem in stages.

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Parameters Across Layers

Each layer has its own parameters.

layer 1 weights and biases
layer 2 weights and biases
layer 3 weights and biases

Training adjusts parameters across all layers.

Backpropagation computes how each layer's parameters affected the final loss.

Multi-layer training requires gradients across the whole chain.

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Final Summary

• A multi-layer network has several transformation layers.
• Each layer is a function.
• The network composes these functions.
• Later layers operate on earlier layer outputs.
• Multiple layers allow staged representation building.
• Training adjusts parameters across all layers.