# Talks

In this talk, I address two new ideas in sampling geometric objects. The first is a new take on adaptive sampling with respect to the local feature size, i.e., the distance to the medial axis. We recently proved that such samples acn be viewed as uniform samples with respect to an alternative metric on the Euclidean space. The second is a generalization of Voronoi refinement sampling. There, one also achieves an adaptive sample while simultaneously "discovering" the underlying sizing function. We show how to construct such samples that are spaced uniformly with respect to the $k$th nearest neighbor distance function.

This talk addresses some upper and lower bounds techniques for bounding the distortion between mappings between Euclidean metric spaces including circles, spheres, pairs of lines, triples of planes, and the union of a hyperplane and a point.

In their seminal work on homological sensor networks, de Silva and Ghrist showed the surprising fact that its possible to certify the coverage of a coordinate free sensor network even with very minimal knowledge of the space to be covered. We give a new, simpler proof of the de Silva-Ghrist Topological Coverage Criterion that eliminates any assumptions about the smoothness of the boundary of the underlying space, allowing the results to be applied to much more general problems. The new proof factors the geometric, topological, and combinatorial aspects of this approach. This factoring reveals an interesting new connection between the topological coverage condition and the notion of weak feature size in geometric sampling theory. We then apply this connection to the problem of showing that for a given scale, if one knows the number of connected components and the distance to the boundary, one can also infer the higher betti numbers or provide strong evidence that more samples are needed. This is in contrast to previous work which merely assumed a good sample and gives no guarantees if the sampling condition is not met.

Given $n$ points $P$ in a Euclidean space, the Johnson-Linden-strauss lemma guarantees that the distances between pairs of points is preserved up to a small constant factor with high probability by random projection into $O(\log n)$ dimensions. In this paper, we show that the persistent homology of the distance function to $P$ is also preserved up to a comparable constant factor. One could never hope to preserve the distance function to $P$ pointwise, but we show that it is preserved sufficiently at the critical points of the distance function to guarantee similar persistent homology. We prove these results in the more general setting of weighted $k$th nearest neighbor distances, for which $k=1$ and all weights equal to zero gives the usual distance to $P$.

Nested dissection exploits underlying topology to do matrix reductions while persistent homology exploits matrix reductions to reveal underlying topology. It seems natural that one should be able to combine these techniques to beat the currently best bound of matrix multiplication time for computing persistent homology. However, nested dissection works by fixing a reduction order, whereas persistent homology generally constrains the ordering. Despite this obstruction, we show that it is possible to combine these two theories. This shows that one can improve the computation of persistent homology of a filtration by exploiting information about the underlying space. It gives reasonable geometric conditions under which one can beat the matrix multiplication bound for persistent homology. This is a purely theoretical result, but it's a valuable one. We also give a topological view of nested dissection, which should be interesting to the wider theory community. For example, separators are covers, sparsification is simplification, asymmetric reduction is morse reduction, graph separators can be extended to complex separators.

In this talk, I give a gentle introduction to geometric and topological data analysis and then segue into some natural questions that arise when one combines the topological view with the perhaps more well-studied linear algebraic view.

A geometric separator for a set $U$ of $n$ geometric objects (usually balls) is a small (sublinear in $n$) subset whose removal disconnects the intersection graph of $U$ into roughly equal sized parts. These separators provide a natural way to do divide and conquer in geometric settings. A particularly nice geometric separator algorithm originally introduced by Miller and Thurston has three steps: compute a centerpoint in a space of one dimension higher than the input, compute a conformal transformation that ``centers'' the centerpoint, and finally, use the computed transformation to sample a sphere in the original space. The output separator is the subset of $S$ intersecting this sphere. It is both simple and elegant. We show that a change of perspective (literally) can make this algorithm even simpler by eliminating the entire middle step. By computing the centerpoint of the points lifted onto a paraboloid rather than using the stereographic map as in the original method, one can sample the desired sphere directly, without computing the conformal transformation.

We describe a new algorithm for computing the Voronoi diagram of a set of $n$ points in constant-dimensional Euclidean space. The running time of our algorithm is $O(f \log n \log \Delta)$ where $f$ is the output complexity of the Voronoi diagram and $\Delta$ is the spread of the input, the ratio of largest to smallest pairwise distances. Despite the simplicity of the algorithm and its analysis, it improves on the state of the art for all inputs with polynomial spread and near-linear output size. The key idea is to first build the Voronoi diagram of a superset of the input points using ideas from Voronoi refinement mesh generation. Then, the extra points are removed in a straightforward way that allows the total work to be bounded in terms of the output complexity, yielding the output sensitive bound. The removal only involves local flips and is inspired by kinetic data structures.

The word optimal is used in different ways in mesh generation. It could mean that the output is in some sense, "the best mesh" or that the algorithm is, by some measure, "the best algorithm". One might hope that the best algorithm also produces the best mesh, but maybe some tradeoffs are necessary. In this talk, I will survey several different notions of optimality in mesh generation and explore the different tradeoffs between them. The bias will be towards Delaunay/Voronoi methods.

Note that I have made some corrections from the version presented at SoCG.
Specifically, I had presented the analyssi for why the feature size intergral counts the vertices size in a quality mesh.
This was meant to be simple, but it ignores the fact that one must first prove the algorithm produces verties that are spaced according to the feature size.
As this may have been confusing, I updated the slides and added a note.

Voronoi diagrams and their duals, Delaunay triangulations, are used in many areas of computing and the sciences. Starting in 3-dimensions, there is a substantial (i.e. polynomial) difference between the best case and the worst case complexity of these objects when starting with $n$ points. This motivates the search for algorithms that are output-senstiive rather than relying only on worst-case guarantees. In this talk, I will describe a simple, new algorithm for computing Voronoi diagrams in $d$-dimensions that runs in $O(f \log n \log \Delta)$ time, where $f$ is the output size and the spread $\Delta$ of the input points is the ratio of the diameter to the closest pair distance. For a wide range of inputs, this is the best known algorithm. The algorithm is novel in the that it turns the classic algorithm of Delaunay refinement for mesh generation on its head, working backwards from a quality mesh to the Delaunay triangulation of the input. Along the way, we will see instances of several other classic problems for which no higher-dimensional results are known, including kinetic convex hulls and splitting Delaunay triangulations.

In this talk, I show how mesh generation and topological data analysis are a natural fit. (Watch the video here)

We show that filtering the barycentric decomposition of a \v Cech complex by the cardinality of the vertices captures precisely the topology of $k$-covered regions among a collection of balls for all values of $k$. Moreover, we relate this result to the Vietoris-Rips complex to get an approximation in terms of the persistent homology.

The theory of optimal size meshes gives a method for analyzing the output size (number of simplices) of a Delaunay refinement mesh in terms of the integral of a sizing function over the input domain. The input points define a maximal such sizing function called the feature size. This paper presents a way to bound the feature size integral in terms of an easy to compute property of a suitable ordering of the point set. The key idea is to consider the pacing of an ordered point set, a measure of the rate of change in the feature size as points are added one at a time. In previous work, Miller et al.\ showed that if an ordered point set has pacing $\phi$, then the number of vertices in an optimal mesh will be $O(\phi^dn)$, where $d$ is the input dimension. We give a new analysis of this integral showing that the output size is only $\Theta(n + n\log \phi)$. The new analysis tightens bounds from several previous results and provides matching lower bounds. Moreover, it precisely characterizes inputs that yield outputs of size $O(n)$.

This talk explores upper and lower bounds on sampling rates for determining the homology of a maifold from noisy random samples.

The Vietoris-Rips filtration is a versatile tool in topological data analysis.
Unfortunately, it is often too large to construct in full.
We show how to construct an $O(n)$-size filtered simplicial complex on an $n$-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration.
The filtration can be constructed in $O(n\log n)$ time.
The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation.
For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets.

Our approach uses a hierarchical net-tree to sparsify the filtration.
We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration,
or sparsify the underlying graph by throwing out edges at larger scales to give a standard filtration.
Both methods yield the same guarantees.

The Vietoris-Rips filtration is a versatile tool in topological data analysis.
Unfortunately, it is often too large to construct in full.
We show how to construct an $O(n)$-size filtered simplicial complex on an $n$-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration.
The filtration can be constructed in $O(n\log n)$ time.
The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation.
For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets.

Our approach uses a hierarchical net-tree to sparsify the filtration.
We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration,
or sparsify the underlying graph by throwing out edges at larger scales to give a standard filtration.
Both methods yield the same guarantees.

The Vietoris-Rips filtration is a versatile tool in topological data analysis.
Unfortunately, it is often too large to construct in full.
We show how to construct an $O(n)$-size filtered simplicial complex on an $n$-point metric space such that the persistence diagram is a good approximation to that of the Vietoris-Rips filtration.
The filtration can be constructed in $O(n\log n)$ time.
The constants depend only on the doubling dimension of the metric space and the desired tightness of the approximation.
For the first time, this makes it computationally tractable to approximate the persistence diagram of the Vietoris-Rips filtration across all scales for large data sets.

Our approach uses a hierarchical net-tree to sparsify the filtration.
We can either sparsify the data by throwing out points at larger scales to give a zigzag filtration,
or sparsify the underlying graph by throwing out edges at larger scales to give a standard filtration.
Both methods yield the same guarantees.

We present NetMesh, a new algorithm that produces a conforming Delaunay mesh for point sets in any fixed dimension with guaranteed optimal mesh size and quality.
Our comparison based algorithm runs in time $O(n\log n + m)$, where $n$ is the input size and $m$ is the output size, and with constants depending only on the dimension and the desired element quality bounds.
It can terminate early in $O(n\log n)$ time returning a $O(n)$ size Voronoi diagram of a superset of $P$ with a relaxed quality bound, which again matches the known lower bounds.

The previous best results in the comparison model depended on the log of the **spread** of the input, the ratio of the largest to smallest pairwise distance among input points.
We reduce this dependence to $O(\log n)$ by using a sequence of $\epsilon$-nets to determine input insertion order in an incremental Voronoi diagram.
We generate a hierarchy of well-spaced meshes and use these to show that the complexity of the Voronoi diagram stays linear in the number of points throughout the construction.

In this talk, I argue that mesh generation is good tool to preprocess point clouds for geometric inference and discuss how to compute such meshes.

What is the difference between a mesh and a net?
What is the difference between a metric space epsilon-net and a range
space epsilon-net?
What is the difference between geometric divide-and-conquer and
combinatorial divide-and-conquer?

In this talk, I will answer these questions and discuss how these
different ideas come together to finally settle the question of how to
compute conforming point set meshes in optimal time. The meshing problem
is to discretize space into as few pieces as possible and yet still
capture the underlying density of the input points. Meshes are fundamental
in scientific computing, graphics, and more recently, topological data
analysis.

This is joint work with Gary Miller and Todd Phillips

Here's a toy problem: What is the SMALLEST number of unit balls you can fit in a box such that no more will fit?
In this talk, I will show how just thinking about a naive greedy approach to this problem leads to a simple derivation of several of the most important theoretical results in the field of mesh generation.
We'll prove classic upper and lower bounds on both the number of balls and the complexity of their interrelationships.

Then, we'll relate this problem to a similar one called the Fat Voronoi Problem, in which we try to find point sets such that every Voronoi cell is fat
(the ratio of the radii of the largest contained to smallest containing ball is bounded).
This problem has tremendous promise in the future of mesh generation as it can circumvent the classic lowerbounds presented in the first half of the talk.
Unfortunately the simple approach no longer works.
In the end we will show that the number of neighbors of any cell in a Fat Voronoi Diagram in the plane is bounded by a constant
(if you think that's obvious, spend a minute to try to prove it).
We'll also talk a little about the higher dimensional version of the problem and its wide range of applications.

In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales. To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation. This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.

In topological inference, the goal is to extract information about a shape, given only a sample of points from it. There are many approaches to this problem, but the one we focus on is persistent homology. We get a view of the data at different scales by imagining the points are balls and consider different radii. The shape information we want comes in the form of a persistence diagram, which describes the components, cycles, bubbles, etc in the space that persist over a range of different scales. To actually compute a persistence diagram in the geometric setting, previous work required complexes of size n^O(d). We reduce this complexity to O(n) (hiding some large constants depending on d) by using ideas from mesh generation. This talk will not assume any knowledge of topology. This is joint work with Gary Miller, Benoit Hudson, and Steve Oudot.

Geometry, Topology and All of You Wildest Dreams Will Come True.

In this light talk, I give a high level view of some of my recent research in using ideas from mesh generation to lower the complexity of computing persistent homology in geomemtric settings. Because this talk is for a general audience, I will focus on three related applications (where related is interpreted loosely) that I think have the widest appeal. The applications are: (1) Winning Nobel Peace Prizes, (2)Winning Olympic Gold Medals, and (3) Finding True Love

There are many depth measures on point sets that yield centerpoint theorems. These theorems guarantee the existence of points of a specified depth, a kind of geometric median. However, the deep point guaranteed to exist is not guaranteed to be among the input, and often, it is not. The alpha-wedge depth of a point with respect to a point set is a natural generalization of halfspace depth that replaces halfspaces with wedges (cones or cocones) of angle alpha. We introduce the notion of a centervertex, a point with depth at least n/(d+1) among the set S. We prove that for any finite subset S of R^d, a centervertex exists. We also present a simple algorithm for computing an approximate centervertex.

We show how to derandomize the Iterated-Radon algorithm to achieve the first deterministic algorithm for computing approximate centerpoints that runs in time subexponential in the dimension. The algorithm also returns a polynomial-time checkable proof of the quality of the returned centerpoint.

In 1864, James Clerk Maxwell published his now famous correspondence between equilibrium stresses and liftings of planar graphs into 3D "terrains." We will explore this classic theorem and how it has influenced many areas of modern theoretical computer science. This survey will touch on results in graph drawing, regular triangulations, network routing, rigidity theory, and spectral graph theory (time permitting).

We show how to adapt Delaunay refinement meshing technology to produce linear size Delaunay meshes of any input point set. The new technique that makes this possible is the use of well-paced points, or those that can be ordered so that consecutive prefixes have a bounded difference in local feature size.

We give an algorithm for computing approximate nearest neighbors that also achieves spatial adaptivity, a kind of geometric finger search. The algorithm is general dimensional.

We generalize the Tukey depth to use cones instead of halfspaces. We prove a generalization of the Center Point Theorem that for $S\subset \reals^d$, there is a vertex $s\in S$, with depth at least $\frac{n}{d+1}$ for cones of half-angle $45^\circ$. This gives a notion of data depth for which an approximate median can always be found among the original set. We present a simple algorithm for computing an approximate center vertex.

Convexity is a powerful tool for extracting combinatorial information from geometric objects. In this talk, I will survey several classical convexity theorems going back to the 1920s and show how they are useful in computational geometry today. In particular, I will focus on the problem of computing center points. A center point p for a set S is a point such that any line through p divides S evenly (at worst its a 1/3-2/3 cut). The existence of a center point follows independently from both Helly's Theorem (1923) and Tverberg's Theorem (1966). By borrowing intuition from both theorems, we arrive at a new deterministic algorithm for computing approximate center points in higher dimensions.

In this talk, I will present Overlay Stitch Meshing, a novel new algorithm that simultaneously solves two different problems in the field of triangulating planar straight-line graphs in the plane for scientific computing applications: It is the first Delaunay Refinement-type triangulation algorithm that terminates with a full complement of guarantees even on degenerate inputs and with no dependence on the size of the smallest input angle. It is the first algorithm that gives a log competitive output size for the problem of no-large-angle triangulation with Steiner points. The workings of the algorithm are very easy to state and understand. The brunt of the talk will focus on new analytic techniques that allow us to prove strong guarantees.

In this talk, we will be looking at a basic primitive in computational geometry, the flip. Also known as bistellar flips, edge-flips, rotations, and Pachner moves, this local change operation has been discovered and rediscovered in a variety of fields (thus the many names) and has proven useful both as an algorithmic tool as well as a proof technology. For algorithm designers working outside of computational geometry, one can consider the flip move as a higher dimensional analog of the tree rotations used in binary trees. I will survey some of the most important results about flips with an emphasis on developing a general geometric intuition that has led to many advances.