I've been looking for a computationally tractable transformation $T$ which can be used to embed a given set of vectors $X\in\mathbb{R}^n$ into some space, say $\mathcal{H}$, such that orthogonality within it is redefined, i.e. $\langle Tx,Ty\rangle_{X}\neq 0$, but $\langle Tx,Ty\rangle_{\mathcal{H}}= 0$ with any $x\perp y\in X$. I can imagine such a embedding in a 3-dimensional representation where $n$ evenly distributed rays projected onto a sphere surface are the new (othogonal) coordinates in $\mathcal{H}$.
I've rencetly observed that Spherical Harmonics transform can be useful, however, I'm not sure how to use it and to obtain a kind of Singular Value Decomposition (SVD) from $X$ into $\mathcal{H}$. Furthermore, for computer applications, probably it is a good approach to ensure that $\mathcal{H}$ is a Hilbert space. Also I'm not sure.
Thanks in advance...