Mahalanobis distance in a LDA classifier

Question

I've read that Mahalanobis distance is as effective as the Euclidean distance when comparing 2 projected feature vectors in classification using a LDA classifier.

I was wondering if this statement were true?

It would be nice if someone could comment on this.

I'm working on projecting a 36 dimensional feature vector to a 1d feature vector using a 2 class LDA classifier and comparing projected feature vectors.

Answer 1

Given that the covariance matrix S = I, the identity matrix, the Mahalanobis distance is equal to the normalised euclidean distance - which is a scale invariant Euclidean distance.

Answer 2

For a multivariate normal distribution the contours of constant probability fall on an ellipsoid. LDA is most often used in the context of multivariate normal features. A contour of constant probability is also the set of points in the sapce of the multivariate normal distribution that have the same Mahalanobis distance. So the greater the Mahalanobis distance is the lower the probability density is. The interior of an ellipsoid with boundary given by a fixed Mahalanobis distance. is the central region that best describes where most of the data will lie and outside it is like being in the tail of a univariate distribution. This is the sense that effective can be used in the description you gave.