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# Limitations of a Single-Layer Perceptron

A perceptron is a simple algorithm that takes input values, applies a threshold, and divides these inputs into two groups, much like classifying students as `pass (1)` or `fail (0)` based on their grades.

For instance, if the test score is `50` or above, it's considered a `pass (1)`, while below `50` is considered a `fail (0)`.

Problems that can be split by a single `line (threshold)` can be easily solved by a single-layer perceptron.

However, not all problems are this straightforward.

<br/>

## Problems a Perceptron Cannot Solve

Consider a rule that outputs `1` only when the two input values are different `(0,1 or 1,0)`.

| Input 1 ($x_1$) | Input 2 ($x_2$) | Output ($y$) |
|----------------|----------------|-------------|
| 0             | 0             | 0           |
| 0             | 1             | 1           |
| 1             | 0             | 1           |
| 1             | 1             | 0           |

This is known as the **XOR problem**.

When represented in an (x, y) coordinate system, it looks like this:

<br/>

![thumbnail-public](https://assets.codefriends.net/images/ai/lectures/xor.png)

- `(0,0)` and `(1,1)` belong to the same group since they have an output of `0`

- `(0,1)` and `(1,0)` belong to the same group since their output is `1`

<br/>

What happens when you try to separate these two groups with a single straight line?

No matter how you draw the line, you cannot perfectly separate these groups.

A perceptron cannot solve this problem of classifying these groups with just one line.

> Problems that cannot be separated linearly by a straight line are called *non-linear problems*. These require more complex shapes, like curves, to solve.

<br/>

## Solving Non-Linear Problems with a Multi-Layer Perceptron

A `Multi-Layer Perceptron (MLP)` is an artificial neural network composed of multiple perceptrons arranged in layers.

Unlike a single-layer perceptron, which classifies data with a single line, a multi-layer perceptron transforms input data through several layers, enabling it to learn more complex patterns.

Between the input layer and the output layer in a multi-layer perceptron, there exists a `hidden layer`.

In the hidden layer, `weights` are applied to the input data, and an `activation function` is used to generate new features.

Through this transformation process, data that couldn't be linearly separated in its original form is **transformed into a new dimension**.

In the XOR problem, for example, single-layer separation in the original 2D space is not possible, but after going through the hidden layer, the data can be converted to a `higher-dimensional space` (e.g., a three-dimensional space) where it is possible to separate linearly.

Thus, a multi-layer perceptron utilizes hidden layers to transform input data and solve non-linear problems.

The XOR problem outputs 1 only when the two input values are different, meaning (0,0) and (1,1) are in the same group, while (0,1) and (1,0) are in different groups. A single-layer perceptron cannot separate these groups with a single line, so it cannot solve the XOR problem.

### A single-layer perceptron can solve the XOR problem.

Input 1 ( $x_1$ )	Input 2 ( $x_2$ )	Output ( $y$ )
0	0	0
0	1	1
1	0	1
1	1	0