# Compact notation for one-hot indicator vectors?

Many machine learning approaches use one-hot vectors to represent categorical data. This is sometimes called using indicator features, indicator vectors, regular categorical encoding, dummy coding, or one-hot encoding (among other names).

I'm searching for a compact way to denote a one-hot vector within a model.

Say we have a categorical variable with $$m$$ categories. First, apply some arbitrary sorting to the categories. A one-hot vector $$v$$ is then a binary vector of length $$m$$ where only a single entry can be one, all others must be zero. We set the $$i^\text{th}$$ entry to 1, and all others to 0, to indicate that this $$v$$ represents the categorical variable taking on the $$i^\text{th}$$ possible value.

One clunky attempt based on misguided set notation;

$$v \in \{0, 1\}^m \qquad\qquad \sum_{i=1}^m v_i = 1$$

I've also seen math-oriented people refer to a one-hot vector using the notation

$$\mathbf{e}_i$$

But I don't understand where this notation comes from or what it is called.

Can anyone help me out? Is there a paper that does a good job of this?

Thank you,

There are several ways to note dummy variables (or one-hot encoded), one of them is the indicator function :

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

For $$e_i$$ it is a vector of the standard base, where $$e_i$$ denotes the vector with a $$1$$ in the $$i$$ ith coordinate and $$0$$'s elsewhere. For example, in $$\mathbb{R}^5$$, $$e_3 = (0, 0, 1, 0, 0)$$

• This is a good answer! Thanks @Fisher. Mar 19 '20 at 0:17