Convex hull in higher dimensions, finding the vertices of a polytope

Question

Suppose I have a point cloud given in 6-dimensional space, which I can make as dense as needed. These points turn out to lie on the surface of a lower-dimensional polytope (i.e.

Answer 1

In this answer, I will assume you have already used PCA to near-losslessly compress the data to 4-dimensional data, where the reduced data lies in a 4-dimensional polytope with conceptually few faces. I will describe an approach to solve for the faces of this polytope, which will in turn give you the vertices.

Let x_i in R⁴, i = 1, ..., m, be the PCA-reduced data points.

Let F = (a, b) be a face, where a is in R⁴ with a • a = 1 and b is in R.

We define the face loss function L as follows, where λ⁺, λ^- > 0 are parameters you choose. λ⁺ should be a very small positive number. λ^- should be a very large positive number.

L(F) = sum_i(λ⁺ • max(0, a • x_i + b) - λ^- • min(0, a • x_i + b))

We want to find minimal faces F with respect to the loss function L. The small set of all minimal faces will describe your polytope. You can solve for minimal faces by randomly initializing F and then performing gradient descent using the partial derivatives ∂L / ∂a_j, j = 1, 2, 3, 4, and ∂L / ∂b. At each step of gradient descent, constrain a • a to be 1 by normalizing.

∂L / ∂a_j = sum_i(λ⁺ • x_j • [a • x_i + b > 0] - λ^- • x_j • [a • x_i + b < 0]) for j = 1, 2, 3, 4

∂L / ∂b = sum_i(λ⁺ • [a • x_i + b > 0] - λ^- • [a • x_i + b < 0])

Note Iverson brackets: [P] = 1 if P is true and [P] = 0 if P is false.