Suppose $A$ is a $d\times d$ matrix. My application requires finding the following decomposition. Does it have a name?
$$A=BB^T$$
Matrix $B$ is not required to be symmetric/triangular/skinny.
Edit the application is iteratively reweighted least squared. For logistic regression/scalar output, I can reweigh each example $x$ by by $\sqrt{g''(x)}$ and treat it as regular least squares. When moving to multiclass case, the Hessian $g''$ is now a matrix, and I need to replace $\sqrt{\cdot}$ by this decomposition. I would call it "square root", but this name is already taken. For the most common loss (softmax), $B$ can be chosen as a sum of a diagonal and a rank-1 matrix.
Of course, Cholesky's decomposition gives you something of the form $A = LL^T$, where $L$ is lower triangular; every other such decomposition is of the form $A = BB^T$ with $B = LU$ for some orthogonal matrix $U$.
I do not believe that freedom in the choice of $B$ leads to any computational improvements over the usual Cholesky method, which is hard to beat. I do not know of a name for this more general form of decomposition.