Search for: All records

We give several ways to derive and express classic summation problems in terms of polycubes. We visualize them with 3D printed models. The video is here: http://go.ncsu.edu/sum_of_squares. 

Over the last 50 years, there have been many data structures proposed to perform proximity search problems on metric data. Perhaps the simplest of these is the ball tree, which was independently discovered multiple times over the years. However, there is a lack of strong theoretical guarantees for standard ball trees, often leading to more complicated structures when guarantees are required. In this paper, we present the greedy tree, a simple ball tree construction for which we can prove strong theoretical guarantees for proximity search queries, matching the state of the art under reasonable assumptions. To our knowledge, this is the first ball tree construction providing such guarantees. Like a standard ball tree, it is a binary tree with the points stored in the leaves. Only a point, a radius, and an integer are stored for each node. The asymptotic running times of search algorithms in the greedy tree match those of more complicated structures regularly used in practice. 

We illustrate the computation of a greedy permutation using finite Voronoi diagrams. We describe the neighbor graph, which is a sparse graph data structure that facilitates efficient point location to insert a new Voronoi cell. This data structure is not dependent on a Euclidean metric space. The greedy permutation is computed in O(n log Delta) time for low-dimensional data using this method. 

The space of persistence diagrams under bottleneck distance is known to have infinite doubling dimension. Because many metric search algorithms and data structures have bounds that depend on the dimension of the search space, the high-dimensionality makes it difficult to analyze and compare asymptotic running times of metric search algorithms on this space. We introduce the notion of nearly-doubling metrics, those that are Gromov-Hausdorff close to metric spaces of bounded doubling dimension and prove that bounded $$k$$-point persistence diagrams are nearly-doubling. This allows us to prove that in some ways, persistence diagrams can be expected to behave like a doubling metric space. We prove our results in great generality, studying a large class of quotient metrics (of which the persistence plane is just one example). We also prove bounds on the dimension of the $$k$$-point bottleneck space over such metrics. The notion of being nearly-doubling in this Gromov-Hausdorff sense is likely of more general interest. Some algorithms that have a dependence on the dimension can be analyzed in terms of the dimension of the nearby metric rather than that of the metric itself. We give a specific example of this phenomenon by analyzing an algorithm to compute metric nets, a useful operation on persistence diagrams. 

We show how a filtration of Delaunay complexes can be used to approximate the persistence diagram of the distance to a point set in ℝ^d. Whereas the full Delaunay complex can be used to compute this persistence diagram exactly, it may have size O(n^⌈d/2⌉). In contrast, our construction uses only O(n) simplices. The central idea is to connect Delaunay complexes on progressively denser subsamples by considering the flips in an incremental construction as simplices in d+1 dimensions. This approach leads to a very simple and straightforward proof of correctness in geometric terms, because the final filtration is dual to a (d+1)-dimensional Voronoi construction similar to the standard Delaunay filtration. We also, show how this complex can be efficiently constructed. 

Given a persistence diagram with n points, we give an algorithm that produces a sequence of n persistence diagrams converging in bottleneck distance to the input diagram, the ith of which has i distinct (weighted) points and is a 2-approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the ith and the (i+1)st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in O(n) space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams - a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation. 

We generalize the local-feature size definition of adaptive sampling used in surface reconstruction to relate it to an alternative metric on Euclidean space. In the new metric, adaptive samples become uniform samples, making it simpler both to give adaptive sampling versions of homological inference results and to prove topological guarantees using the critical points theory of distance functions. This ultimately leads to an algorithm for homology inference from samples whose spacing depends on their distance to a discrete representation of the complement space. 

We adapt and generalize a heuristic for k-center clustering to the permutation case, where every pre x of the ordering is a guaranteed approximate solution. The one-hop greedy permutations work by choosing at each step the farthest unchosen point and then looking in its local neighborhood for a point that covers the most points at a certain scale. This balances the competing demands of reducing the coverage radius and also covering as many points as possible. This idea first appeared in the work of Garcia-Diaz et al. and their algorithm required O(n2 log n) time for a fi xed k (i.e. not the whole permutation). We show how to use geometric data structures to approximate the entire permutation in O(n log D
) time for metrics sets with spread D. Notably, this running time is asymptotically the same as the running time for computing the ordinary greedy permutation. 

Finding the kth nearest neighbor to a query point is a ubiquitous operation in many types of metric computations, especially those in unsupervised machine learning. In many such cases, the distance to k sample points is used as an estimate of the local density of the sample. In this paper, we give an algorithm that takes a finite metric (P,d) and an integer k and produces a subset S ⊆ P with the property that for any q ∈ P, the distance to the second nearest point of S to q is a constant factor approximation to the distance to the kth nearest point of P to q. Thus, the sample S may be used in lieu of P. In addition to being much smaller than P, the distance queries on S only require finding the second nearest neighbor instead of the kth nearest neighbor. This is a significant improvement, especially because theoretical guarantees on kth nearest neighbor methods often require k to grow as a function of the input size n. 

DOI: 10.1137/1.9781611975994.25