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Register a new userFitting a Graph to One-Dimensional Data
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Added by
Otfried Cheong
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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Given n data points in R^d, an appropriate edge-weighted graph connecting the data points finds application in solving clustering, classification, and regresssion problems. The graph proposed by Daitch, Kelner and Spielman (ICML~2009) can be computed by quadratic programming and hence in polynomial time. While a more efficient algorithm would be preferable, replacing quadratic programming is challenging even for the special case of points in one dimension. We develop a dynamic programming algorithm for this case that runs in O(n^2) time.
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Geometric graphs form an important family of hidden structures behind data. In this paper, we develop an efficient and robust algorithm to infer a graph skeleton behind a point cloud data (PCD)embedded in high dimensional space. Previously, there has been much work to recover a hidden graph from a low-dimensional density field, or from a relatively clean high-dimensional PCD (in the sense that the input points are within a small bounded distance to a true hidden graph). Our proposed approach builds upon the recent line of work on using a persistence-guided discrete Morse (DM) theory based approach to reconstruct a geometric graph from a density field defined over a triangulation of low-dimensional Euclidean domain. In particular, we first give a very simple generalization of this DM-based algorithm from a density-function perspective to a general filtration perspective. On the theoretical front, we show that the output of the generalized algorithm contains a so-called lexicographic-optimal persistent cycle basis w.r.t the input filtration, justifying that the output is indeed meaningful. On the algorithmic front, this generalization allows us to use the idea of sparsified weighted Rips filtration (developed by Buchet etal) to develop a new graph reconstruction algorithm for noisy point cloud data (PCD) (which do not need to be embedded). The new algorithm is robust to background noise and non-uniform distribution of input points. We provide various experimental results to show the efficiency and effectiveness of our new graph reconstruction algorithm for PCDs.
The Hausdorff distance, the Gromov-Hausdorff, the Frechet and the natural pseudo-distances are instances of dissimilarity measures widely used in shape comparison. We show that they share the property of being defined as $inf_rho F(rho)$ where $F$ is a suitable functional and $rho$ varies in a set of correspondences containing the set of homeomorphisms. Our main result states that the set of homeomorphisms cannot be enlarged to a metric space $mathcal{K}$, in such a way that the composition in $mathcal{K}$ (extending the composition of homeomorphisms) passes to the limit and, at the same time, $mathcal{K}$ is compact.
Given a tesselation of the plane, defined by a planar straight-line graph $G$, we want to find a minimal set $S$ of points in the plane, such that the Voronoi diagram associated with $S$ fits $G$. This is the Generalized Inverse Voronoi Problem (GIVP), defined in cite{Trin07} and rediscovered recently in cite{Baner12}. Here we give an algorithm that solves this problem with a number of points that is linear in the size of $G$, assuming that the smallest angle in $G$ is constant.
The efficiency of extracting topological information from point data depends largely on the complex that is built on top of the data points. From a computational viewpoint, the most favored complexes for this purpose have so far been Vietoris-Rips and witness complexes. While the Vietoris-Rips complex is simple to compute and is a good vehicle for extracting topology of sampled spaces, its size is huge--particularly in high dimensions. The witness complex on the other hand enjoys a smaller size because of a subsampling, but fails to capture the topology in high dimensions unless imposed with extra structures. We investigate a complex called the {em graph induced complex} that, to some extent, enjoys the advantages of both. It works on a subsample but still retains the power of capturing the topology as the Vietoris-Rips complex. It only needs a graph connecting the original sample points from which it builds a complex on the subsample thus taming the size considerably. We show that, using the graph induced complex one can (i) infer the one dimensional homology of a manifold from a very lean subsample, (ii) reconstruct a surface in three dimension from a sparse subsample without computing Delaunay triangulations, (iii) infer the persistent homology groups of compact sets from a sufficiently dense sample. We provide experimental evidences in support of our theory.
It is shown that there exists a dihedral acute triangulation of the three-dimensional cube. The method of constructing the acute triangulation is described, and symmetries of the triangulation are discussed.
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