Bhattacharya Distance on Distributions (Matrices) with Different Number of Variables (Dimensions)

Question

We have two matrices, $A$ and $B$, representing two different probability distributions, with dimensions, $m*n$ and $k*n$, respectively.

How can we calculate the Bhattacharya distance or another measure of similarity or dissimilarity between $A$ and $B$?

Here, $m$ and $k$ denote the number of variables captured by the two matrices $A$ and $B$. In general, $m$ and $k$ are not equal.

$n$ is the number of observations, which is the same across the two distributions.

Related Broader Question:

Bhattacharya Distance (or A Measure of Similarity) --- On Matrices with Different Dimensions

Answer 1

Using the Johnson Lindenstrauss transformation, we can reduce the dimensions of the larger dataset to be the same dimension as the smaller dataset and then compute the Bhattacharya Distance.

Link: "https://en.wikipedia.org/wiki/Johnson–Lindenstrauss_lemma"