What is the relationship between pdist2(..., 'mahalanobis') and mahal?

[kj]

This question is about the relationship between two different ways to
compute Mahalanobis distances with MATLAB: `pdist2(..., 'mahalanobis')`
and `mahal(...)`. (Both functions are part of MATLAB's stats
toolbox.)

To be concrete, suppose we have the following:

>> rng(0)
>> X = rand(5, 3)
X =
0.8147 0.0975 0.1576
0.9058 0.2785 0.9706
0.1270 0.5469 0.9572
0.9134 0.9575 0.4854
0.6324 0.9649 0.8003
>> Y = rand(4, 3)
Y =
0.1419 0.9595 0.9340
0.4218 0.6557 0.6787
0.9157 0.0357 0.7577
0.7922 0.8491 0.7431
>> PD2 = pdist2(Y, X, 'mahalanobis')
PD2 =
7.6812 2.9122 1.4405 4.5954 1.9977
5.2445 4.4568 2.8991 2.9615 2.4820
6.9355 2.4993 2.3643 4.3252 2.4828
6.9843 3.2129 3.5415 2.7514 0.5562
>> MAH = mahal(Y, X)
MAH =
3.4400
0.8204
3.1139
0.7368

Is it possible to get the same result `PD2` (ignoring round-off
errors) using only `mahal`? Conversely, can one obtain `MAH` using
`pdist2(..., 'mahalanobis')` alone?

More generally, what exactly is the relationship between `PD2` and
`MAH`?

(Just to be clear: I'm asking about the mathematical/conceptual
relationship between these two functions. I know that `pdist2(...,
'mahalanobis')` does not invoke `mahal`, and I suspect that the
converse is also true.)

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(!!!IMPORTANT!!!: All the post's essential information is given above.
What follows may be skipped.)

The documentation for the function `pdist2` begins as follows:

D = pdist2(X,Y) returns a matrix D containing the Euclidean distances
between each pair of observations in the MX-by-N data matrix X and
MY-by-N data matrix Y. Rows of X and Y correspond to observations,
and columns correspond to variables. D is an MX-by-MY matrix, with the
(I,J) entry equal to distance between observation I in X and
observation J in Y.

D = pdist2(X,Y,DISTANCE) computes D using DISTANCE.

One of the possible values for the `DISTANCE` parameter is
`mahalanobis`; the relevant part of the `pdist2` documentation begins
as follows:

'mahalanobis' - Mahalanobis distance, using the sample covariance
of X as computed by NANCOV.

AFAICT, in the definition of `pdist2`, `X` and `Y` are the same "sort
of thing". IOW, `pdist2(X, Y, 'mahalanobis')` and `pdist2(Y, X,
'mahalanobis')` both make sense, even if they are not equal.

In contrast, the main arguments to the built-in `mahal` function are
neither semantically nor mathematically symmetric:


D2 = mahal(Y,X) returns the Mahalanobis distance (in squared units) of
each observation (point) in Y from the sample data in X, i.e.,

D2(I) = (Y(I,:)-MU) * SIGMA^(-1) * (Y(I,:)-MU)',

where MU and SIGMA are the sample mean and covariance of the data in X.
Rows of Y and X correspond to observations, and columns to variables. X
and Y must have the same number of columns, but can have different numbers
of rows. X must have more rows than columns.

In particular, for some pairs of `Y` and `X`, `mahal(Y, X)` may make
sense, while `mahal(X, Y)` will result in an error. For example:

>> rng(0)
>> Y = rand(1, 3); X = rand(5, 3);
>> mahal(Y, X)
ans =
7.3939
>> mahal(X, Y)
Error using mahal (line 38)
The number of rows of X must exceed the number of columns.

IOW, the *domain* of `pdist2(..., ..., 'mahalanobis')` is a symmetric
Cartesian product A x A, whereas the domain of `mahal`
is an asymmetric one, A x B with A ~= B.