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# Single layer limitations

This page presents with a simple example the main limitation of single layer neural networks.

## Network architecture

Let's consider the following single-layer network architecture with two inputs ( $$a, b$$ ) and one output ( $$y$$ ).

## Logic OR function

Let's assume we want to train an artificial single-layer neural network to learn logic functions. Let's start with the OR logic function:

a b y = a + b
0 0 0
0 1 1
1 0 1
1 1 1

The space of the OR fonction can be drawn. X-axis and Y-axis are respectively the $$a$$ and $$b$$ inputs. The green line is the separation line ( $$y=0$$ ). As illustrated below, the network can find an optimal solution:

## Logic XOR function

Assume we now want to train the network on the XOR logic function:

a b y = a ⊕ b
0 0 0
0 1 1
1 0 1
1 1 0

As for the OR function, space can be drawn. Unfortunatly, the network isn't able to disriminate ones from zeros.

## Conclusion

The transfert function of this single-layer network is given by:

$$\begin{equation} y= w_1a + w_2b +w_3 \label{eq:transfert-function} \end{equation}$$

The equation $$\eqref{eq:transfert-function}$$ is a linear model. This explain why the frontier between ones and zeros is necessary a line. The XOR function is a non-linear problem that can't be classified with a linear model. Fortunatly, multilayer perceptron (MLP) can deal with non-linear problems.