Computing Mahalanobis Distance Between Set of Points and Set of Reference Points

Hello,

I have an n x p matrix - mX which is composed of n points in R^p.

I have another m x p matrix - mY which is composed of m reference points in R^p.

I would like to create an n x m matrix - mD which is the Mahalanobis Distance matrix.

D(i, j) means the Mahalanobis Distance between point j in mX, mX(j, :) and point i in mY, mY(i, :).

Namely, is computes the following:

mD(i, j) = (mX(j, :) - mY(i, :)) * inv(mC) * (mX(j, :) - mY(i, :)).';

Where mC is the given Mahalanobis Distance PSD Matrix.

It is easy to be done in a loop, is there a way to vectorize it?

Namely, is the a function which its inputs are mX, mY and mC and its output is mD and fully vectorized without using any MATLAB toolbox?

Thank You.

[Royi Avital]

Anyone? Any Vectorization wizard? Thank You.

Answer 1

[Royi Avital]

Here is the solution fully vectorized (Though uses much more multiplications than needed):

mA = reshape(bsxfun(@minus, permute(mY, [1, 3, 2]), permute(mX, [3, 1, 2])), [(m * n), p]);
mD = reshape(diag(A* inv(mC) * A.'), [m, n]);

If anyone has faster way (Not necessarily fully vectorized) I'd be happy to see.

Thank You.

Answer 2

[Image Analyst]

Use *mahal(* ) in the Statistics and Machine Learning Toolbox. I haven't used it yet so I don't have any demo for you. Why do you need it?

[Royi Avital]

Hi, First I would like a solution without any toolbox as I noted above. Second, the function you suggested doesn't allow predefined weighting PSD Matrix (It infers it from the data). There's also `pdist`, yet I want to be toolboxes independent. Thank You.

[Image Analyst]

What's m, n, and p? Your code is already at least partially vectorized. If m, n, and p are less than a few million, then it probably won't take much time at all even if it's not vectorized.