Treewidth, crushing, and hyperbolic volume

We prove that there exists a universal constant $c$ such that any closed hyperbolic 3-manifold admits a triangulation of treewidth at most $c$ times its volume. The converse is not true: we show there exists a sequence of hyperbolic 3-manifolds of bounded treewidth but volume approaching infinity. Along the way, we prove that crushing a normal surface in a triangulation does not increase the carving-width, and hence crushing any number of normal surfaces in a triangulation affects treewidth by at most a constant multiple.Comment: 20 pages, 12 figures. V2: Section 4 has been rewritten, as the former argument (in V1) used a construction that relied on a wrong theorem. Section 5.1 has also been adjusted to the new construction. Various other arguments have been clarifie

An Exponential Lower Bound on the Complexity of Regularization Paths

For a variety of regularized optimization problems in machine learning, algorithms computing the entire solution path have been developed recently. Most of these methods are quadratic programs that are parameterized by a single parameter, as for example the Support Vector Machine (SVM). Solution path algorithms do not only compute the solution for one particular value of the regularization parameter but the entire path of solutions, making the selection of an optimal parameter much easier. It has been assumed that these piecewise linear solution paths have only linear complexity, i.e. linearly many bends. We prove that for the support vector machine this complexity can be exponential in the number of training points in the worst case. More strongly, we construct a single instance of n input points in d dimensions for an SVM such that at least \Theta(2^{n/2}) = \Theta(2^d) many distinct subsets of support vectors occur as the regularization parameter changes.Comment: Journal version, 28 Pages, 5 Figure

Introduction to the R package TDA

We present a short tutorial and introduction to using the R package TDA, which provides some tools for Topological Data Analysis. In particular, it includes implementations of functions that, given some data, provide topological information about the underlying space, such as the distance function, the distance to a measure, the kNN density estimator, the kernel density estimator, and the kernel distance. The salient topological features of the sublevel sets (or superlevel sets) of these functions can be quantified with persistent homology. We provide an R interface for the efficient algorithms of the C++ libraries GUDHI, Dionysus and PHAT, including a function for the persistent homology of the Rips filtration, and one for the persistent homology of sublevel sets (or superlevel sets) of arbitrary functions evaluated over a grid of points. The significance of the features in the resulting persistence diagrams can be analyzed with functions that implement recently developed statistical methods. The R package TDA also includes the implementation of an algorithm for density clustering, which allows us to identify the spatial organization of the probability mass associated to a density function and visualize it by means of a dendrogram, the cluster tree

An algorithm for Tambara-Yamagami quantum invariants of 3-manifolds, parameterized by the first Betti number

Quantum topology provides various frameworks for defining and computing invariants of manifolds. One such framework of substantial interest in both mathematics and physics is the Turaev-Viro-Barrett-Westbury state sum construction, which uses the data of a spherical fusion category to define topological invariants of triangulated 3-manifolds via tensor network contractions. In this work we consider a restricted class of state sum invariants of 3-manifolds derived from Tambara-Yamagami categories. These categories are particularly simple, being entirely specified by three pieces of data: a finite abelian group, a bicharacter of that group, and a sign $\pm 1$. Despite being one of the simplest sources of state sum invariants, the computational complexities of Tambara-Yamagami invariants are yet to be fully understood. We make substantial progress on this problem. Our main result is the existence of a general fixed parameter tractable algorithm for all such topological invariants, where the parameter is the first Betti number of the 3-manifold with $\mathbb{Z}/2\mathbb{Z}$ coefficients. We also explain that these invariants are sometimes #P-hard to compute (and we expect that this is almost always the case). Contrary to other domains of computational topology, such as graphs on surfaces, very few hard problems in 3-manifold topology are known to admit FPT algorithms with a topological parameter. However, such algorithms are of particular interest as their complexity depends only polynomially on the combinatorial representation of the input, regardless of size or combinatorial width. Additionally, in the case of Betti numbers, the parameter itself is easily computable in polynomial time.Comment: 24 pages, including 3 appendice