Projecting points on a Pareto-front onto a (hyper)plane?

Question

I'm trying to implement the Biased Crowding Distance in NSGA-II as described in the paper Integrating User Preferences into Evolutionary Multi-Objective Optimization by Branke and Deb.

Basically, from what I understand, given a set of points on a Pareto-front, we need to project the points onto a hyperplane with a direction vector $\eta$. Doing this will give the new crowding distance $d'$ based on the locations of the individualson the hyperplane with direction $\eta$.

To illustrate, see the figure below.

My question is simple: how do we obtain the hyperplane given a user-specified direction vector $\nu$? Is it simply by getting the perpendicular of the direction vector?

And a follow up question would be: how would I project the points onto the hyperplane? If $p$ is the point and $v$ is the vector representing the hyperplane, is it correct that the new location of $p$, call it $p'$ is $\frac{v.x(p.x) + v.y(p.y)}{||v||^2} (v.x, v.y)$ based on this?

Thanks and any information would be very helpful.

Answer 1

I'm not sure if this should be considered a computer science question or not, it seems more an algebra question than anything else.

A vector is a direction vector of a hyperplane if it is parallel to that hyperplane (source).

For any vector $\vec{v} = (v_1, . . . , v_n)^T$ you can find a plane parallel to it by taking the dot product of $v$ and your variable vector $\vec{x} = (x_1, . . . , x_n)^T$ and setting the result equal to some constant $c$.

Consider $\eta$ as described in the paper, then the hyperplane he is referring to will be $$\eta_1 f_1 + . . . + \eta_n f_n = c $$ for a constant $c$. Because you are only interested in the distance between the points after projection, and not the location of the projected points, you can select $c = 0$, as shifting around the hyperplane won't affect the distances between the projected points.

Since the origin is contained in this hyperplane, the hyperplane is a subspace, call it $H$. We can then represent each $p_i$ as a vector $\vec{p_i}$ and project the vector onto the subspace.

To project a vector onto $H$ we need to find a basis for $H$. This can be done by making $n - 1$ similar vectors to $\eta$, where the difference is a single value is set to $0$ (distinct from previous selections). E.g., the plane $3x_1 + 4x_2 + 5x_3 = 0$ has a basis of $$\Bigg\{\begin{bmatrix}3 \\ 4 \\ 0\end{bmatrix}, \begin{bmatrix}3 \\ 0 \\ 5\end{bmatrix}\Bigg\},\Bigg\{\begin{bmatrix}3 \\ 4 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 4 \\ 5\end{bmatrix}\Bigg\}, \text{ or } \Bigg\{\begin{bmatrix}0 \\ 4 \\ 5\end{bmatrix}, \begin{bmatrix}3 \\ 0 \\ 5\end{bmatrix}\Bigg\}$$

Once you have a basis of $H$ you can use this guide guide to perform the projection. Note, since you have to do this multiple times, coming up with $Q$ as described in the guide and saving it will allow you to perform the process on many points.