How can I mathematically represent the one-hot encoding?

Question

If we have 5 classes and 3 inputs, let's say [C1, C2, C3, C4, C5] and [X1, X2, X3] then,

• If X1 belongs to the C4 class then the hot encoder for this will be [0, 0, 0, 1, 0].

• If X1 belongs to the C2 class then the hot encoder for this will be [0, 1, 0, 0, 0].

How can I mathematically represent it?

I've also seen math-oriented people refer to a one-hot vector using an indicator function, but I am unable to understand this:

$$ 1_A(x) := \begin{cases} 1 & \text{if}\ x \in A, \\ 0 & \text{if}\ x \notin A. \end{cases} $$

Can anyone explain this above scenario to me?

Answer 1

$A$ is a set. If $x$ is an element of $A$, return $1$. Else return $0$. Thus $A$ is the set of cases that you are assigning a $1$ to in your encoding vector. Thus one-hot encoding is a vector form of this indicator function that applies componentwise.

Answer 2

The mathematical representation of one-hot encoding can indeed be done using indicator functions. An indicator function is a mathematical function that takes a value and returns 1 if the value satisfies a certain condition (belongs to a particular set), and 0 if it doesn't satisfy the condition.

C={C1​,C2​,C3​,C4​,C5​}, and you want to encode an input XX as a one-hot vector. You can define indicator functions for each category:

For category CiCi​, the indicator function 1Ci(X)1Ci​​(X) is defined as: {1if X=Ci0otherwise {10​if X=Ci​otherwise​

In this definition, 1Ci(X)1Ci​​(X) is 1 if XX belongs to category CiCi​, and 0 otherwise. Essentially, it's a function that acts as a "switch" to turn on (1) or off (0) for each category based on whether XX belongs to that category.

So, for your examples:

If X1X1​ belongs to the C4C4​ class, the one-hot encoding can be represented as:
1C1(X1)=0, 1C2(X1)=0, 1C3(X1)=0, 1C4(X1)=1, 1C5(X1)=01C1​​(X1​)=0,1C2​​(X1​)=0,1C3​​(X1​)=0,1C4​​(X1​)=1,1C5​​(X1​)=0

If X1X1​ belongs to the C2C2​ class, the one-hot encoding can be represented as:
1C1(X1)=0, 1C2(X1)=1, 1C3(X1)=0, 1C4(X1)=0, 1C5(X1)=01C1​​(X1​)=0,1C2​​(X1​)=1,1C3​​(X1​)=0,1C4​​(X1​)=0,1C5​​(X1​)=0