Linear-Size Approximations to the Vietoris-Rips Filtration

The Vietoris-Rips filtration is a versatile tool in topological data analysis. It is a sequence of simplicial complexes built on a metric space to add topological structure to an otherwise disconnected set of points. It is widely used because it encodes useful information about the topology of the underlying metric space. This information is often extracted from its so-called persistence diagram. Unfortunately, this filtration 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 its persistence diagram is a good approximation to that of the Vietoris-Rips filtration. This new filtration can be constructed in $O(n\log n)$ time. The constant factors in both the size and the running time 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. We describe two different sparse filtrations. The first is a zigzag filtration that removes points as the scale increases. The second is a (non-zigzag) filtration that yields the same persistence diagram. Both methods are based on a hierarchical net-tree and yield the same guarantees

Efficient Classification for Metric Data

Recent advances in large-margin classification of data residing in general metric spaces (rather than Hilbert spaces) enable classification under various natural metrics, such as string edit and earthmover distance. A general framework developed for this purpose by von Luxburg and Bousquet [JMLR, 2004] left open the questions of computational efficiency and of providing direct bounds on generalization error. We design a new algorithm for classification in general metric spaces, whose runtime and accuracy depend on the doubling dimension of the data points, and can thus achieve superior classification performance in many common scenarios. The algorithmic core of our approach is an approximate (rather than exact) solution to the classical problems of Lipschitz extension and of Nearest Neighbor Search. The algorithm's generalization performance is guaranteed via the fat-shattering dimension of Lipschitz classifiers, and we present experimental evidence of its superiority to some common kernel methods. As a by-product, we offer a new perspective on the nearest neighbor classifier, which yields significantly sharper risk asymptotics than the classic analysis of Cover and Hart [IEEE Trans. Info. Theory, 1967].Comment: This is the full version of an extended abstract that appeared in Proceedings of the 23rd COLT, 201

Fast Construction of Nets in Low Dimensional Metrics, and Their Applications

We present a near linear time algorithm for constructing hierarchical nets in finite metric spaces with constant doubling dimension. This data-structure is then applied to obtain improved algorithms for the following problems: Approximate nearest neighbor search, well-separated pair decomposition, compact representation scheme, doubling measure, and computation of the (approximate) Lipschitz constant of a function. In all cases, the running (preprocessing) time is near-linear and the space being used is linear.Comment: 41 pages. Extensive clean-up of minor English error

The Traveling Salesman Problem: Low-Dimensionality Implies a Polynomial Time Approximation Scheme

The Traveling Salesman Problem (TSP) is among the most famous NP-hard optimization problems. We design for this problem a randomized polynomial-time algorithm that computes a (1+eps)-approximation to the optimal tour, for any fixed eps>0, in TSP instances that form an arbitrary metric space with bounded intrinsic dimension. The celebrated results of Arora (A-98) and Mitchell (M-99) prove that the above result holds in the special case of TSP in a fixed-dimensional Euclidean space. Thus, our algorithm demonstrates that the algorithmic tractability of metric TSP depends on the dimensionality of the space and not on its specific geometry. This result resolves a problem that has been open since the quasi-polynomial time algorithm of Talwar (T-04)

MapReduce and Streaming Algorithms for Diversity Maximization in Metric Spaces of Bounded Doubling Dimension

Given a dataset of points in a metric space and an integer $k$, a diversity maximization problem requires determining a subset of $k$ points maximizing some diversity objective measure, e.g., the minimum or the average distance between two points in the subset. Diversity maximization is computationally hard, hence only approximate solutions can be hoped for. Although its applications are mainly in massive data analysis, most of the past research on diversity maximization focused on the sequential setting. In this work we present space and pass/round-efficient diversity maximization algorithms for the Streaming and MapReduce models and analyze their approximation guarantees for the relevant class of metric spaces of bounded doubling dimension. Like other approaches in the literature, our algorithms rely on the determination of high-quality core-sets, i.e., (much) smaller subsets of the input which contain good approximations to the optimal solution for the whole input. For a variety of diversity objective functions, our algorithms attain an $(\alpha+\epsilon)$-approximation ratio, for any constant $\epsilon>0$, where $\alpha$ is the best approximation ratio achieved by a polynomial-time, linear-space sequential algorithm for the same diversity objective. This improves substantially over the approximation ratios attainable in Streaming and MapReduce by state-of-the-art algorithms for general metric spaces. We provide extensive experimental evidence of the effectiveness of our algorithms on both real world and synthetic datasets, scaling up to over a billion points.Comment: Extended version of http://www.vldb.org/pvldb/vol10/p469-ceccarello.pdf, PVLDB Volume 10, No. 5, January 201