How much smaller is the Čech complex than the Vietoris-Rips complex?

Question

The Čech complex is a subcomplex of the Vietoris-Rips complex.

The V-R complex includes as a simplex a set of points with pairwise distances at most $\epsilon$, whereas the Č complex includes as a simplex a set of points with non-empty intersection of diameter $\epsilon$-balls centered on the points.

One advantage of the Č complex is it can be (and generally is) smaller than the V-R complex. My question is essentially: How much smaller?

Q. What results are known for the relative sizes of the two complexes for random point clouds?

By size I mean some measure of combinatorial complexity, such as the total number of simplicies. I am open to any definition of what constitutes a "random point cloud": uniformly distributed within a sphere, multidimensional Gaussian distribution, benchmark data sets, ... I'm primarily interested in points in $\mathbb{R}^3$ but higher-dimensional results would be equally welcomed.

Answer 1

I'm going to offer an answer mainly to get an idea off my brain and maybe someone will point out why this is incorrect. However, in my view, a lot of discussions about Čech Vs Vietoris-Rips seem to overlook a key connection: The Vietoris-Rips Complex is a particular type of Čech complex.

Given a topological space $X$ and a cover $\mathcal{U} = \{U_i\}$ of $X$, the Čech complex is like a free vector space generated by higher intersections of elements of the cover.

• If the $U_i$'s are $\epsilon$-balls centered at finitely many chosen points then this is still a Čech complex and it is commonly referred to as "the Čech complex" when the finitely many chosen points are exactly the point cloud in question.
• If the $U_i$'s are $\epsilon'$-cubes centered at finitely many chosen points then this is still a Čech complex and it is commonly referred to as "the Vietoris-Rips complex" when the finitely many chosen points are exactly the point cloud in question.

So most of the relationships and discussions around Čech vs V-R complexes to me are really overly-complicated conversations about cubes and balls. Once we let $\epsilon$ vary, the discussions are not really important anymore since we are supposedly imagining our data points are sitting on some unknown manifold and so the nerve theorem says that cubes and balls will eventually compute (co)homology if epsilon is small enough.

Looking forward to seeing if this is a valuable thought to others. (I didn't realize any of this until I was working with some undergrads on this stuff.)