Given three vectors in $\mathbb{R}^{512}$, my task is to compute a Minimum Volume Enclosing Ellipsoid (MVEE). I have tried Kachiyan's algorithm, but it requires at least as many vectors as there are dimensions, which I do not have at my disposal (attempts to create synthetic vectors resulted in useless ellipsoids).
Thus far, the most accurate and useful method I have found is to formulate the problem as a semidefinite program.
$$\min\; -\operatorname{logdet}(A) \\ \text{s.t. } A \geq 0 \\ (x_{i} - c)^{T}A(x_{i} - c) \leq 1$$
In the above formulation, $c$ is the specified center of the ellipsoid, $A\geq 0$ is the positive semi-definiteness constraint, and $x_{i}$ is the $i$th vector in $\mathbb{R}^{512}$.
This method works very well but is too slow for my application (10+ seconds using SCS with very few iterations, all other SDP methods I've tried are glacial by comparison). Is there a faster method for finding an MVEE for a small number of points in a high number of dimensions? I understand that said MVEE will not be unique due to the small number of points, as long as the volume is minimal or close to minimal and the points are enclosed).
