I'm trying to deal with the following problem (working in metric learning).
For a classic discriminant analysis, given a set of data $X\in \mathbb{R}^{n\times p}$, in the case of a two classes problem. We are trying to find the best axe which can separate the two classes. To do so, we're working with the euclidean distance and we solve the following problem : $$\underset{u}{max} \dfrac{uBu^t}{uVu^t},$$
where u is the discriminant axe (in other words, an eigenvector), B et V are respectively the between-classe and the variance matrices.
But now I'm trying to learn a metric M, which will give me a new representation of my data that this new metric will increase the largest value of the previous problem. Here is my question :
How to learn such a metric ? Is it possible to do so without constraints (but keeping the fast M should be PSD and symmetric) ? Can we directly solve the problem using the fact we want to maximize the largest eigenvalue.
Thank you for your help,
Guillaume.