arXiv subjects

Dennis Rohde

Publications and source records attributed to Dennis Rohde.

9 recordsLinked to original sources

arXiv2024

Fast Approximations and Coresets for (k, l)-Median under Dynamic Time Warping

We present algorithms for the computation of $\varepsilon$-coresets for $k$-median clustering of point sequences in $\mathbb{R}^d$ under the $p$-dynamic time warping (DTW) distance. Coresets under DTW have not been investigated before, and the analysis is not directly accessible to existing methods as DTW is not a metric. The three main ingredients that allow our construction of coresets are the adaptation of the $\varepsilon$-coreset framework of sensitivity sampling, bounds on the VC dimension of approximations to the range spaces of balls under DTW, and new approximation algorithms for the $k$-median problem under DTW. We achieve our results by investigating approximations of DTW that provide a trade-off between the provided accuracy and amenability to known techniques. In particular, we observe that given $n$ curves under DTW, one can directly construct a metric that approximates DTW on this set, permitting the use of the wealth of results on metric spaces for clustering purposes. The resulting approximations are the first with polynomial running time and achieve a very similar approximation factor as state-of-the-art techniques. We apply our results to produce a practical algorithm approximating $(k,\ell)$-median clustering under DTW.

cs.CG↗

arXiv2023

Random projections for curves in high dimensions

Modern time series analysis requires the ability to handle datasets that are inherently high-dimensional; examples include applications in climatology, where measurements from numerous sensors must be taken into account, or inventory tracking of large shops, where the dimension is defined by the number of tracked items. The standard way to mitigate computational issues arising from the high dimensionality of the data is by applying some dimension reduction technique that preserves the structural properties of the ambient space. The dissimilarity between two time series is often measured by ``discrete'' notions of distance, e.g. the dynamic time warping or the discrete Fréchet distance. Since all these distance functions are computed directly on the points of a time series, they are sensitive to different sampling rates or gaps. The continuous Fréchet distance offers a popular alternative which aims to alleviate this by taking into account all points on the polygonal curve obtained by linearly interpolating between any two consecutive points in a sequence. We study the ability of random projections à la Johnson and Lindenstrauss to preserve the continuous Fréchet distance of polygonal curves by effectively reducing the dimension. In particular, we show that one can reduce the dimension to $O(ε^{-2} \log N)$, where $N$ is the total number of input points while preserving the continuous Fréchet distance between any two determined polygonal curves within a factor of $1\pm ε$. We conclude with applications on clustering.

cs.CG↗

arXiv2023

Stabbing balls with line segments and polygonal paths

We study the problem of ordered stabbing of $n$ balls (of arbitrary and possibly different radii, no ball contained in another) in $\mathbb{R}^d$, $d \geq 3$, with either a directed line segment or a (directed) polygonal curve. Here, the line segment, respectively polygonal curve, shall visit (intersect) the given sequence of balls in the order of the sequence. We present a deterministic algorithm that decides whether there exists a line segment stabbing the given sequence of balls in order, in time $O(n^{4d-2} \log n)$. Due to the descriptional complexity of the region containing these line segments, we can not extend this algorithm to actually compute one. We circumvent this hurdle by devising a randomized algorithm for a relaxed variant of the ordered line segment stabbing problem, which is built upon the central insights from the aforementioned decision algorithm. We further show that this algorithm can be plugged into an algorithmic scheme by Guibas et al., yielding an algorithm for a relaxed variant of the minimum-link ordered stabbing path problem that achieves approximation factor 2 with respect to the number of links. We conclude with experimental evaluations of the latter two algorithms, showing practical applicability.

cs.CG↗

arXiv2022

Approximating Length-Restricted Means under Dynamic Time Warping

We study variants of the mean problem under the $p$-Dynamic Time Warping ($p$-DTW) distance, a popular and robust distance measure for sequential data. In our setting we are given a set of finite point sequences over an arbitrary metric space and we want to compute a mean point sequence of given length that minimizes the sum of $p$-DTW distances, each raised to the $q$\textsuperscript{th} power, between the input sequences and the mean sequence. In general, the problem is $\mathrm{NP}$-hard and known not to be fixed-parameter tractable in the number of sequences. On the positive side, we show that restricting the length of the mean sequence significantly reduces the hardness of the problem. We give an exact algorithm running in polynomial time for constant-length means. We explore various approximation algorithms that provide a trade-off between the approximation factor and the running time. Our approximation algorithms have a running time with only linear dependency on the number of input sequences. In addition, we use our mean algorithms to obtain clustering algorithms with theoretical guarantees.

cs.CG↗

arXiv2021

Coresets for $(k, \ell)$-Median Clustering under the Fréchet Distance

We present an algorithm for computing $ε$-coresets for $(k, \ell)$-median clustering of polygonal curves in $\mathbb{R}^d$ under the Fréchet distance. This type of clustering is an adaption of Euclidean $k$-median clustering: we are given a set of $n$ polygonal curves in $\mathbb{R}^d$, each of complexity (number of vertices) at most $m$, and want to compute $k$ median curves such that the sum of distances from the given curves to their closest median curve is minimal. Additionally, we restrict the complexity of the median curves to be at most $\ell$ each, to suppress overfitting, a problem specific for sequential data. Our algorithm has running time linear in $n$, sub-quartic in $m$ and quadratic in $ε^{-1}$. With high probability it returns $ε$-coresets of size quadratic in $ε^{-1}$ and logarithmic in $n$ and $m$. We achieve this result by applying the improved $ε$-coreset framework by Langberg and Feldman to a generalized $k$-median problem over an arbitrary metric space. Later we combine this result with the recent result by Driemel et al. on the VC dimension of metric balls under the Fréchet distance. Furthermore, our framework yields $ε$-coresets for any generalized $k$-median problem where the range space induced by the open metric balls of the underlying space has bounded VC dimension, which is of independent interest. Finally, we show that our $ε$-coresets can be used to improve the running time of an existing approximation algorithm for $(1,\ell)$-median clustering.

cs.CG↗

arXiv2020

Approximating $(k,\ell)$-Median Clustering for Polygonal Curves

In 2015, Driemel, Krivošija and Sohler introduced the $(k,\ell)$-median problem for clustering polygonal curves under the Fréchet distance. Given a set of input curves, the problem asks to find $k$ median curves of at most $\ell$ vertices each that minimize the sum of Fréchet distances over all input curves to their closest median curve. A major shortcoming of their algorithm is that the input curves are restricted to lie on the real line. In this paper, we present a randomized bicriteria-approximation algorithm that works for polygonal curves in $\mathbb{R}^d$ and achieves approximation factor $(1+ε)$ with respect to the clustering costs. The algorithm has worst-case running-time linear in the number of curves, polynomial in the maximum number of vertices per curve, i.e. their complexity, and exponential in $d$, $\ell$, $ε$ and $δ$, i.e., the failure probability. We achieve this result through a shortcutting lemma, which guarantees the existence of a polygonal curve with similar cost as an optimal median curve of complexity $\ell$, but of complexity at most $2\ell-2$, and whose vertices can be computed efficiently. We combine this lemma with the superset-sampling technique by Kumar et al. to derive our clustering result. In doing so, we describe and analyze a generalization of the algorithm by Ackermann et al., which may be of independent interest.

cs.CG↗

arXiv2020

Random Projections and Sampling Algorithms for Clustering of High-Dimensional Polygonal Curves

We study the $k$-median clustering problem for high-dimensional polygonal curves with finite but unbounded number of vertices. We tackle the computational issue that arises from the high number of dimensions by defining a Johnson-Lindenstrauss projection for polygonal curves. We analyze the resulting error in terms of the Fréchet distance, which is a tractable and natural dissimilarity measure for curves. Our clustering algorithms achieve sublinear dependency on the number of input curves via subsampling. Also, we show that the Fréchet distance can not be approximated within any factor of less than $\sqrt{2}$ by probabilistically reducing the dependency on the number of vertices of the curves. As a consequence we provide a fast, CUDA-parallelized version of the Alt and Godau algorithm for computing the Fréchet distance and use it to evaluate our results empirically.

cs.LG↗

arXiv2019

Coresets for $(k,l)$-Clustering under the Fréchet Distance

Clustering is the task of partitioning a given set of geometric objects. This is thoroughly studied when the objects are points in the euclidean space. There are also several approaches for points in general metric spaces. In this thesis we consider clustering polygonal curves, i.e., curves composed of line segments, under the Fréchet distance. We obtain clusterings by minimizing an objective function, which yields a set of centers that induces a partition of the input. The objective functions we consider is the so called $(k,l)$-\textsc{center}, where we are to find the $k$ center-curves that minimize the maximum distance between any input-curve and a nearest center-curve and the so called $k$-\textsc{median}, where we are to find the $k$ center-curves that minimize the sum of the distances between the input-curves and a nearest center-curve. Given a set of $n$ polygonal curves, we are interested in reducing this set to an $ε$-coreset, i.e., a notably smaller set of curves that has a very similar clustering-behavior. We develop a construction method for such $ε$-coresets for the $(k,l)$-\textsc{center}, that yields $ε$-coresets of size of a polynomial of $\frac{1}ε$, in time linear in $n$ and a polynomial of $\frac{1}ε$, for line segments. Also, we develop a construction technique for the $(k,l)$-\textsc{center} that yields $ε$-coresets of size exponential in $m$ with basis $\frac{1}ε$, in time sub-quadratic in $n$ and exponential in $m$ with basis $\frac{1}ε$, for general polygonal curves. Finally, we develop a construction method for the $k$-\textsc{median}, that yields $ε$-coresets of size polylogarithmic in $n$ and a polynomial of $\frac{1}ε$, in time linear in $n$ and a polynomial of $\frac{1}ε$.

cs.CG↗

arXiv2018

A Theory-Based Evaluation of Nearest Neighbor Models Put Into Practice

In the $k$-nearest neighborhood model ($k$-NN), we are given a set of points $P$, and we shall answer queries $q$ by returning the $k$ nearest neighbors of $q$ in $P$ according to some metric. This concept is crucial in many areas of data analysis and data processing, e.g., computer vision, document retrieval and machine learning. Many $k$-NN algorithms have been published and implemented, but often the relation between parameters and accuracy of the computed $k$-NN is not explicit. We study property testing of $k$-NN graphs in theory and evaluate it empirically: given a point set $P \subset \mathbb{R}^δ$ and a directed graph $G=(P,E)$, is $G$ a $k$-NN graph, i.e., every point $p \in P$ has outgoing edges to its $k$ nearest neighbors, or is it $ε$-far from being a $k$-NN graph? Here, $ε$-far means that one has to change more than an $ε$-fraction of the edges in order to make $G$ a $k$-NN graph. We develop a randomized algorithm with one-sided error that decides this question, i.e., a property tester for the $k$-NN property, with complexity $O(\sqrt{n} k^2 / ε^2)$ measured in terms of the number of vertices and edges it inspects, and we prove a lower bound of $Ω(\sqrt{n / εk})$. We evaluate our tester empirically on the $k$-NN models computed by various algorithms and show that it can be used to detect $k$-NN models with bad accuracy in significantly less time than the building time of the $k$-NN model.

cs.LG↗