arXiv subjects

Clément Maria

Publications and source records attributed to Clément Maria.

23 records · Page 2Linked to original sources

arXiv2016

A polynomial time algorithm to compute quantum invariants of 3-manifolds with bounded first Betti number

In this article, we introduce a fixed parameter tractable algorithm for computing the Turaev-Viro invariants TV(4,q), using the dimension of the first homology group of the manifold as parameter. This is, to our knowledge, the first parameterised algorithm in computational 3-manifold topology using a topological parameter. The computation of TV(4,q) is known to be #P-hard in general; using a topological parameter provides an algorithm polynomial in the size of the input triangulation for the extremely large family of 3-manifolds with first homology group of bounded rank. Our algorithm is easy to implement and running times are comparable with running times to compute integral homology groups for standard libraries of triangulated 3-manifolds. The invariants we can compute this way are powerful: in combination with integral homology and using standard data sets we are able to roughly double the pairs of 3-manifolds we can distinguish. We hope this qualifies TV(4,q) to be added to the short list of standard properties (such as orientability, connectedness, Betti numbers, etc.) that can be computed ad-hoc when first investigating an unknown triangulation.

math.GT↗

arXiv2015

Admissible colourings of 3-manifold triangulations for Turaev-Viro type invariants

Turaev Viro invariants are amongst the most powerful tools to distinguish 3-manifolds: They are implemented in mathematical software, and allow practical computations. The invariants can be computed purely combinatorially by enumerating colourings on the edges of a triangulation T. These edge colourings can be interpreted as embeddings of surfaces in T. We give a characterisation of how these embedded surfaces intersect with the tetrahedra of T. This is done by characterising isotopy classes of simple closed loops in the 3-punctured disk. As a direct result we obtain a new system of coordinates for edge colourings which allows for simpler definitions of the tetrahedron weights incorporated in the Turaev-Viro invariants. Moreover, building on a detailed analysis of the colourings, as well as classical work due to Kirby and Melvin, Matveev, and others, we show that considering a much smaller set of colourings suffices to compute Turaev-Viro invariants in certain significant cases. This results in a substantial improvement of running times to compute the invariants, reducing the number of colourings to consider by a factor of $2^n$. In addition, we present an algorithm to compute Turaev-Viro invariants of degree four -- a problem known to be #P-hard -- which capitalises on the combinatorial structure of the input. The improved algorithms are shown to be optimal in the following sense: There exist triangulations admitting all colourings the algorithms consider. Furthermore, we demonstrate that our new algorithms to compute Turaev-Viro invariants are able to distinguish the majority of $\mathbb{Z}$-homology spheres with complexity up to $11$ in $O(2^n)$ operations in $\mathbb{Q}$.

cs.CG↗

arXiv2015

Algorithms and complexity for Turaev-Viro invariants

The Turaev-Viro invariants are a powerful family of topological invariants for distinguishing between different 3-manifolds. They are invaluable for mathematical software, but current algorithms to compute them require exponential time. The invariants are parameterised by an integer $r \geq 3$. We resolve the question of complexity for $r=3$ and $r=4$, giving simple proofs that computing Turaev-Viro invariants for $r=3$ is polynomial time, but for $r=4$ is \#P-hard. Moreover, we give an explicit fixed-parameter tractable algorithm for arbitrary $r$, and show through concrete implementation and experimentation that this algorithm is practical---and indeed preferable---to the prior state of the art for real computation.

math.GT↗

arXiv2015

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.

cs.MS↗

arXiv2012

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 Θ(2^{n/2}) = Θ(2^d) many distinct subsets of support vectors occur as the regularization parameter changes.

cs.LG↗