I'm attempting to implement this algorithm https://arxiv.org/pdf/1711.00748.pdf. Part of this algorithm requires determining whether a point $p \in \mathbb{R}^d$ is located within the volume of a rotated ellipsoid centered at some arbitrary point $x \in \mathbb{R}^d$. See sections A and B on pages 3 and 4 of the paper. I think I might know how to go about doing this but want to confirm.
The direction of the ellipsoid radii is determined by the SVD of a set of points which forms a matrix, $Y \in \mathbb{R}^{m \times d}$ where $m$ is the number of data points and $d$ is the dimensionality of the data points. (Just to note, $Y$ has been centered or demeaned. The mean of each column is $0$). Then $Y = U\Sigma V^T$. $V$ is a $d\times d$ unitary, orthogonal matrix. The directions of the radii come from $V$.
To calculate if a point $p$ is located within the volume of an ellipsoid centered at $x$ with radii in the direction of $V$, you can map the point $p$ to the axes of the ellipsoid determined by $V$. This is where I'm unclear. To do this, I imagine I can do the mapping by first centering $p$ relative to $x$. $$p_{centered} = p - x$$ Then apply the mapping to $p_{centered}$. To do this, I imagine I can use $V^T$ that came from the SVD. $$p_{mapped} = V^{T}p_{centered}$$
Once I have $p_{mapped}$ which is mapped to the same axes as the ellipse, I can just use the equation for an ellipsoid $$\sum\limits_{i=1}^{d}\left( \frac{x_i}{r_i} \right)^{2}= 1$$ Using radii magnitudes of my choosing, I plug in the the values for $p_{mapped}$ into the left hand side of the equation above and if the resulting quantity is less than 1, it is contained in the ellipse.
Does this look correct?
Thanks for the help!