Neural Network as Function Composition

Definition

A neural network can be understood as a composition of functions.

Each layer is a function.

The network connects them in sequence.

input -> function 1 -> function 2 -> function 3 -> output

A neural network is a layered input-output function.

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Function Composition

Function composition means the output of one function becomes the input to the next.

Example:

first function: f
second function: g

input -> f -> g -> output

Written compactly:

g(f(input))

Composition chains transformations together.

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Layers as Functions

Each layer transforms values.

layer 1: input values -> hidden values
layer 2: hidden values -> new hidden values
layer 3: hidden values -> output values

The whole network is the result of composing these layer functions.

Layers are transformation steps inside a larger function.

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Parameters Inside the Functions

The functions are not fixed in every detail.

They contain parameters.

weights
biases

Training changes these parameters.

So the network is a function whose behavior can be adjusted.

Neural networks are parameterized function compositions.

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Why Composition Matters

Composition lets the network build transformations step by step.

raw input
-> first representation
-> second representation
-> final output

Each stage can reshape the information before passing it forward.

Function composition lets networks build layered representations.

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Final Summary

• A neural network is a composition of functions.
• Each layer is a function.
• The output of one layer becomes the input to the next.
• The network contains parameters inside its functions.
• Training adjusts those parameters.
• Composition lets the network build representations step by step.