Broadcasted subtraction notation in mathematics

Question

Consider the calculation of the (uncorrected) sample covariance as an example. Given a set of $N$ vector observations $\mathbf { X } = \{ \mathbf { x } _ 1, \dots, \mathbf { x } _ N \}$, the sample covariance is calculated as follows:

• Step 1: $ ~~~~~~~ \displaystyle \boldsymbol { \mu } = \frac{1}{N} \sum _ {n = 1} ^ N \mathbf { x } _ n $

• Step 2. $ ~~~~~~~ \displaystyle \boldsymbol { \Sigma } = \frac{1}{N} \sum _ {n = 1} ^ N ~( \mathbf { x } _ n - \boldsymbol { \mu } ) ( \mathbf { x } _ n - \boldsymbol { \mu } )^ { \mathrm T } $

There exists in many scientific programming languages the idea of "broadcasting", i.e. expanding singleton dimensions of a matrix or array, to perform elementwise operations with another array of higher dimensionality.

Let's notate such a "broadcasted subtraction" operation between a matrix and a vector as "$\circleddash$", such that: $$ \mathbf { X } \circleddash \boldsymbol { \mu } ~~\equiv~~ \Bigl (~ \mathbf { x } _ 1 \! - \! \boldsymbol { \mu } ~~~~~~ \mathbf { x } _ 2 \! - \! \boldsymbol { \mu } ~~~~~~\dots ~~~~~~ \mathbf { x } _ n \! - \! \boldsymbol { \mu } ~ \Bigr ) $$

With this notation, we could then rewrite step 2 above simply as:

• Step 2. $ ~~~~~~~ \displaystyle \boldsymbol { \Sigma } = \frac{1}{N} ~( \mathbf { X } \circleddash \boldsymbol { \mu } ) ( \mathbf { X } \circleddash \boldsymbol { \mu } )^ { \mathrm T } $

Does such notation for broadcasting exist? Or would people simply:

• use the normal subtraction "$-$" symbol and infer broadcasting from context
• rely on collecting into intermediate variables via some sort of "generator syntax" to bypass the need for such an operator altogether, e.g. $$ \mathbf { D } \equiv \Bigl ( \mathbf { d }_1 ~~ \mathbf { d }_2 ~~ \dots ~~ \mathbf { d }_N \Bigr ) ~~:~~ \mathbf { d } _ n = \mathbf { x } _ n - \boldsymbol { \mu }, ~~~n \in \{1, \dots, N\} $$ (and therefore $\boldsymbol { \Sigma } = \frac{1}{N} D D ^ { \mathrm T } $ )
• define their own broadcasted operators as (I've done above)

etc?

Answer 1

I would avoid introducing extra symbols, and simply write something like:

Slightly abusing notation, let $\mathbf{X} - \boldsymbol{\mu}$ denote the matrix $(\mathbf{x}_1 - \boldsymbol{\mu}, \dotsm, \mathbf{x}_n - \boldsymbol{\mu})$. Then the covariance can be written as $\displaystyle \boldsymbol{\Sigma} = \frac{1}{N} ~(\mathbf{X} - \boldsymbol{\mu}) (\mathbf{X} - \boldsymbol{\mu})^{ \mathrm T}$.