# 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;

$$v \in [0, 1]^m; \sum_{i=1}^m v_i = 1$$

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,