{"title": "Random Projections and Sampling Algorithms for Clustering of High-Dimensional Polygonal Curves", "book": "Advances in Neural Information Processing Systems", "page_first": 12827, "page_last": 12837, "abstract": "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 Fr\\'echet 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 Fr\\'echet 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 Fr\\'echet distance and use it to evaluate our results empirically.", "full_text": "Random Projections and Sampling Algorithms for\nClustering of High-Dimensional Polygonal Curves\n\nStefan Meintrup\n\nFaculty of Computer Science\n\nTU Dortmund University\n\nDortmund, Germany\n\nAlexander Munteanu\n\nDortmund Data Science Center\n\nTU Dortmund University\n\nDortmund, Germany\n\nstefan.meintrup@tu-dortmund.de\n\nalexander.munteanu@tu-dortmund.de\n\nChair of Ef\ufb01cient Algorithms and Complexity Theory\n\nDennis Rohde\n\nTU Dortmund University\n\nDortmund, Germany\n\ndennis.rohde@tu-dortmund.de\n\nAbstract\n\nWe study the k-median clustering problem for high-dimensional polygonal curves\nwith \ufb01nite but unbounded number of vertices. We tackle the computational issue\nthat arises from the high number of dimensions by de\ufb01ning a Johnson-Lindenstrauss\nprojection for polygonal curves. We analyze the resulting error in terms of the\nFr\u00e9chet distance, which is a tractable and natural dissimilarity measure for curves.\nOur clustering algorithms achieve sublinear dependency on the number of input\ncurves via subsampling. Also, we show that the Fr\u00e9chet distance can not be\napproximated within any factor of less than\n2 by probabilistically reducing the\ndependency on the number of vertices of the curves. As a consequence we provide\na fast, CUDA-parallelized version of the Alt and Godau algorithm for computing\nthe Fr\u00e9chet distance and use it to evaluate our results empirically.\n\n\u221a\n\n1\n\nIntroduction\n\nTime-series are sequences of measurements taken at certain instants of time. They arise in numerous\napplications, e.g., in the physical, geo-spatial, technical or \ufb01nancial domains (Zhang et al., 2007;\nChapados and Bengio, 2008; Zimmer et al., 2018). Often there are multiple measurements per time\ninstant, e.g., when there are numerous synchronized sensors. While the analysis of time-series is\na well-studied topic, cf. Hamilton (1994); Liao (2005); Aghabozorgi et al. (2015), there are only\nfew approaches that take high-dimensional multivariate time-series into account. In this work we\nbuild upon Driemel et al. (2016), who developed the \ufb01rst (1 + \u03b5)-approximation algorithms for\nclustering univariate time-series under the Fr\u00e9chet distance. Their idea is that \u2013due to environmental\ncircumstances\u2013 time-series often have heterogeneous lengths and their measurements are taken with\ndifferent time-intervals in between. Thus, common approaches, where univariate time-series are\nrepresented by a point in a high-dimensional space, each dimension corresponding to one instant\nof time, become hard or even impossible to apply. Additionally, when these time-intervals differ\nsubstantially, depending on the sampling rates, continuous distance measures perform much better\nthan discrete ones. This is due to the fact that they are inherently independent of the sampling rates:\nby interpreting a sequence of measurements as the vertices of a polygonal curve, those induce a\nlinear interpolation between every two consecutive measurements. We extend this further to the\nmultivariate case. When synchronized sensors are available, multiple univariate time-series are\n\n33rd Conference on Neural Information Processing Systems (NeurIPS 2019), Vancouver, Canada.\n\n\f\u221a\n\ninterpreted as a high-dimensional polygonal curve, i.e., the number of dimensions equals the number\nof simultaneously measured attributes and the number of vertices equals the number of measurements.\nWe focus on big data with large number of curves n and speci\ufb01cally on a large number of dimensions,\nsay d \u2208 \u2126(n) as well as a high complexity of the curves, i.e., the number of their vertices is bounded\nby, say m \u2208 \u2126(n) each. This setting rules out the possibility of using sum-based continuous similarity\nmeasures like the continuous dynamic time warping distance, cf. Efrat et al. (2007). For this measure\nthere is only one tractable algorithm, which is strongly related to paths on a two-dimensional manifold.\nUnfortunately, it also restricts to polygonal curves in R2. In contrast, the Alt and Godau algorithm\n(Alt and Godau (1995)) for computing the Fr\u00e9chet distance works for any number of dimensions. The\nFr\u00e9chet distance intuitively measures the maximum distance one must traverse, when continuously\nand monotonously walking along two curves under an optimal speed adjustment, which is a suitable\nsetting for comparing time-series in most cases. The Alt and Godau algorithm has running-time\nO(d \u00b7 m2 log(m)), so we still end up with a worst-case running-time super-cubic in the number of\ninput curves, in our setting. Unfortunately, it is impossible to reduce the complexity of the curves\ndeterministically such that the Fr\u00e9chet distance is preserved up to any multiplicative, which we\nprove in Theorem 14. Also it is not possible to reduce the complexity of the curves probabilistically\nsuch that the Fr\u00e9chet distance is preserved up to any multiplicative less than\n2, which we prove in\nTheorem 15. We tackle this issue by parallelizing the Alt and Godau algorithm via CUDA-enabled\nGPUs and thus preserve the original distance.\nThe main part of our work focuses on dimension reduction. SVD-based feature-selection approaches\nare common in practice, cf. Billsus and Pazzani (1998); Hong (1991). Unfortunately, these work\npoorly for polygonal curves, which we assess experimentally. Instead, we focus on Gaussian random\nprojections via the seminal Johnson-Lindenstrauss Lemma (Johnson and Lindenstrauss, 1984) which\nperform much better. Explicit error-guarantees for discrete dissimilarity measures, like dynamic time\nwarping or the discrete Fr\u00e9chet distance, are immediate from the approximation of a \ufb01nite number\nof Euclidean distances. But if we restrict to these measures, we loose the aforementioned linear\ninterpolation which is not desirable in practice.\nWe thus study how the error of the Johnson-Lindenstrauss embedding propagates in the continuous\ncase. In our theoretical analysis we show the \ufb01rst explicit error bound for the continuous Fr\u00e9chet\ndistance by extending the Johnson-Lindenstrauss embedding to polygonal curves. We project the\nvertices of the curve down from d to O(\u03b5\u22122log(nm)) dimensions and re-connect their images in the\nlow-dimensional space in the given order. The error is bounded by an \u03b5-fraction relative to the Fr\u00e9chet\ndistance and to the length of the largest edge of the input curves, which we prove in Theorem 8. This\ngives a combined multiplicative and additive approximation guarantee, similar to the lightweight\ncoresets of Bachem et al. (2018). All in all, we reduce the running-time of one Fr\u00e9chet distance\ncomputation to O(\u03b5\u22122 m2\n#cc log(m) log(nm)), where #cc is the number of CUDA cores available.\nWe analyze various data sets. Our experiments show promising results concerning the approximation\nof the Fr\u00e9chet distance under the Johnson-Lindenstrauss embedding and a massive improvement in\nterms of running-time.\nJust as Driemel et al. (2016), we study median clustering. Since the median is a measure of central\ntendency that is robust when up to half of the data is arbitrarily corrupted, it is particularly useful\nfor providing a summary of the massive data set. To the best of our knowledge, there is no tractable\nalgorithm to compute an exact median polygonal curve. Thus, we restrict the search space of feasible\nsolutions to the input. This problem, known as the discrete median, has a polynomial-time exhaustive-\nsearch algorithm: calculate the cost of each possible curve by summing over all other input curves. In\nour setting, this is prohibitive since it takes O(n2) distance computations. Therefore, we propose and\nanalyze a sampling-scheme for the discrete 1-median under the Fr\u00e9chet distance when the number\nof input curves is also high. In Theorem 11 we show that a sample of constant size already yields a\n(2 + \u03b5)-approximation in the worst case. Under reasonable assumptions on the distribution of the data,\nthe same algorithm yields a (1 + \u03b5)-approximation, which we prove in Theorem 12. To this end we\nintroduce a natural parameter that quanti\ufb01es the fraction of outliers as a function of the input, setting\nthis approach in the light of beyond worst-case analysis, cf. Roughgarden (2019). The number of\nsamples needed depends on this parameter and is almost always constant unless the fraction of outliers\ntends to 1/2 at a high rate, depending on n. If those assumptions hold, we meet the requirements to\napply Theorem 1.1 from Ackermann et al. (2010) and thus obtain a k-median (1 + \u03b5)-approximation\nalgorithm for the Fr\u00e9chet distance that uses n \u00b7 2\n\n\u03b53 )) distance computations.\n\nO( k\n\n\u03b52 +log( k\n\n2\n\n\fFinally, we note that our techniques do not only apply to multivariate time-series, but to high-\ndimensional polygonal curves in general and thus may be valuable to the communities of computa-\ntional geometry as well as the \ufb01eld of machine learning.\nOur contributions We advance the study of clustering high-dimensional polygonal curves under the\nFr\u00e9chet distance both, in theory and in practice. Speci\ufb01cally,\n1) we show an extension of the Gaussian random projections of Johnson-Lindenstrauss to polygonal\ncurves and provide rigorous bounds on the distortion of their continuous Fr\u00e9chet distance,\n2) we provide sublinear sampling algorithms for the 1-median clustering of time series resp. polygonal\ncurves under the Fr\u00e9chet distance that can be extended (under natural assumptions) to a k-median\n(1 + \u03b5)-approximation,\n3) we prove lower bounds for reducing the curves complexity,\n4) we provide a highly ef\ufb01cient CUDA-parallelized implementation of the algorithm by Alt and\nGodau (1995) for computing the Fr\u00e9chet distance,\n5) and we evaluate the proposed methods on benchmark and real-world data.\n\n1.1 Related work\n\nClustering under the Fr\u00e9chet distance Driemel et al. (2016) developed the \ufb01rst k-center and\nk-median clustering algorithms for one-dimensional polygonal curves under the Fr\u00e9chet distance,\nwhich provably achieve an approximation factor of (1 + \u03b5). The resulting centers are curves from a\ndiscretized family of simpli\ufb01ed curves, whose complexity is parameterized by a parameter (cid:96). Their\nalgorithms have near-linear running-time in the input size for constant \u03b5, k and (cid:96) but are exponential in\nthe latter quantities. The \ufb01rst extension of k-center to higher dimensional curves was done in Buchin\net al. (2019a). In that paper, however it was shown that there is no polynomial-time approximation\nscheme unless P=NP. In the case of the discrete Fr\u00e9chet distance on two-dimensional curves, the\nhardness of approximation within a factor close to 2.598 was established even for k = 1. Finally,\nGonzalez\u2019 algorithm yields a 3-approximation in any number of dimensions. Even more recently\nBuchin et al. (2019b) showed that the k-median problem is also NP-hard for k = 1 and improved upon\nthe aforementioned (1 + \u03b5)-approximations. Open problems thus include dimensionality reduction for\nhigh-dimensional curves and practical algorithms that do not depend exponentially on the parameters.\nAlgorithm engineering for the Fr\u00e9chet distance Bringmann et al. (2019) describe an improved\nversion of one of the best algorithms that was developed by the participants of the GIS Cup 2017.\nThe goal of the cup was to answer Fr\u00e9chet queries as fast as possible, i.e., given a set of curves T , a\nquery curve q and a positive real r, return all curves from T that are within distance r to q. Roughly\nspeaking, all top algorithms (see also Baldus and Bringmann (2018); Buchin et al. (2017); D\u00fctsch\nand Vahrenhold (2017)) utilized heuristics to \ufb01lter out all \u03c4 \u2208 T , that are certainly within distance\nr to q or certainly not. In the best case, the common algorithm by Alt and Godau only served as a\nrelapse option when no clear decision could be found in advance. Since the heuristics mostly have\nsublinear running-time, the Fr\u00e9chet distance computation is speed up massively in the average case.\nThe Alt and Godau algorithm is also improved by simplifying the resulting free-space diagram.\nRandom projections for problems in computational geometry Random projections have several\napplications as embedding techniques in computational geometry. One of the most in\ufb02uential work\nwas Agarwal et al. (2013) who applied the Johnson-Lindenstrauss embedding, among others, to\nsurfaces and curves for the sake of tracking moving points. Only recently Driemel and Krivosija\n(2018) studied the \ufb01rst probabilistic embeddings of the Fr\u00e9chet distance by projecting the curves on a\nrandom line. Another work that inspired our dimensionality reduction approach is due to Sheehy\n(2014). He noticed that a Johnson-Lindenstrauss embedding of points yields an embedding for their\nentire convex hull with additive error. Our results are in line with a recent lower bound of \u2126(n) for\nsketching, i.e., compressing the strongly related Dynamic Time Warping distance of sequences via\nlinear embeddings, due to Braverman et al. (2019).\nBeyond-worst-case and relaxations A common assumption is that \u201cClustering is dif\ufb01cult only when\nit does not matter\u201d (Daniely et al., 2012). Similarly, it has been noted for many other problems that\nwhile being particularly hard to solve in the worst-case, they are relatively simple to solve for typical\nor slightly perturbed inputs. Beyond-worst-case-analysis tries to parametrize the notion of typical\n\n3\n\n\fand to derive better bounds in terms of this parameter assuming its value is small. See Munteanu\net al. (2018) for a recent contribution in machine learning. These assumptions are usually weaker\nthan the norm in statistical machine learning which is closer to average-case analysis, for example\nwhen data points are modeled as i.i.d. samples from some distribution. See (Roughgarden, 2019) for\nan extensive overview and more details. Another complementary recent approach is weakening the\nusual multiplicative error guarantees by an additional additive error term in favor of a computational\nspeedup. Those relaxations still perform competitively well in practice, cf. (Bachem et al., 2018).\n\n2 Dimension Reduction for Polygonal Curves\n\nWe begin with the basic de\ufb01nitions, all proofs can be found in Appendix A in the supplement.\nPolygonal curves are composed of line segments, which we de\ufb01ne as follows.\nDe\ufb01nition 1 (line segment). A line segment between two points p1, p2 \u2208 Rd, denoted by p1p2, is\nthe set of points {(1 \u2212 \u03bb)p1 + \u03bbp2 | \u03bb \u2208 [0, 1]}. For \u03bb \u2208 [0, 1] we denote by lp (p1p2, \u03bb) the point\n(1 \u2212 \u03bb)p1 + \u03bbp2, lying on p1p2.\nWe next de\ufb01ne polygonal curves. Thereby we need an exact parametrization of the points on the\nindividual line segments to express any point on the curve in terms of its segments vertices. This\nunusually complicates the de\ufb01nition but simpli\ufb01es the notation and will later be needed in the context\nof Johnson-Lindenstrauss embeddings.\nDe\ufb01nition 2 (polygonal curve). A parameterized curve is a continuous mapping \u03c4 : [0, 1] \u2192 Rd.\nLet H be the set of all continuous, injective and non-decreasing functions h : [0, 1] \u2192 [0, 1] with\nh(0) = 0 and h(1) = 1, which we call reparameterizations.\nA curve \u03c4 is polygonal, if there exist h \u2208 H, v1, . . . , vm \u2208 Rd, no three consecutive on a line, called\n\u03c4\u2019s vertices and t1, . . . , tm \u2208 [0, 1] with t1 < \u00b7\u00b7\u00b7 < tm, t1 = 0 and tm = 1, called \u03c4\u2019s instants, such\nthat\n\n(cid:16)\n(cid:16)\n\n\uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3\n\nlp\n...\nlp\n\n\u03c4 (h(t)) =\n\n(cid:17)\n\n,\n\nv1v2, h(t)\u2212t1\nt2\u2212t1\n\nvm\u22121vm, h(t)\u2212tm\u22121\ntm\u2212tm\u22121\n\n(cid:17)\n\nif h(t) \u2208 [0, t2)\n\n.\n\nif h(t) \u2208 [tm\u22121, 1]\n\n,\n\nIn the following we will assume that h is the identity function, because the Fr\u00e9chet distance, which is\nsubsequently de\ufb01ned, is invariant under reparameterizations. We only need h to keep our de\ufb01nition\ngeneral. Further, we call m the complexity of \u03c4, denoted by |\u03c4|. We are now ready to de\ufb01ne the\n(continuous) Fr\u00e9chet distance.\nDe\ufb01nition 3 (continuous Fr\u00e9chet distance). The Fr\u00e9chet distance between polygonal curves \u03c4 and \u03c3\nis de\ufb01ned as dF (\u03c4, \u03c3) := inf h\u2208H maxt\u2208[0,1](cid:107)\u03c4 (t) \u2212 \u03c3(h(t))(cid:107), where (cid:107)\u00b7(cid:107) is the Euclidean norm.\nWe next give a basic de\ufb01ntion of the seminal Johnson-Lindenstrauss embedding result, cf. Johnson\nand Lindenstrauss (1984). Speci\ufb01cally, they showed that a properly rescaled Gaussian matrix mapping\nfrom d to d(cid:48) \u2208 O(\u03b5\u22122 log n) dimensions satis\ufb01es the following de\ufb01nition with positive constant\nprobability.\nDe\ufb01nition 4 ((1 \u00b1 \u03b5)-Johnson-Lindenstrauss embedding). Given a set P \u2282 Rd of points, a function\nf : Rd \u2192 Rd(cid:48)\n\nis a (1 \u00b1 \u03b5)-Johnson-Lindenstrauss embedding for P , if it holds that\n\u2200p, q \u2208 P : (1 \u2212 \u03b5)(cid:107)p \u2212 q(cid:107) \u2264 (cid:107)f (p) \u2212 f (q)(cid:107) \u2264 (1 + \u03b5)(cid:107)p \u2212 q(cid:107),\n\nwith constant probability at least \u03c1 \u2208 (0, 1] over the random construction of f.\n\nIn De\ufb01nition 5 we extend the mapping f from De\ufb01nition 4 to polygonal curves by applying it to the\nvertices of the curves and re-connecting their images in the given order.\nDe\ufb01nition 5 ((1 \u00b1 \u03b5)-Johnson-Lindenstrauss embedding for polygonal curves). Let \u03c4 be a polyg-\nonal curve, t1, . . . , tm be its instants and v1, . . . , vm be its vertices. Let f be a (1 \u00b1 \u03b5)-Johnson-\nLindenstrauss embedding for {v1, . . . , vm}. By F (\u03c4 ) we de\ufb01ne the (1 \u00b1 \u03b5)-Johnson-Lindenstrauss\n\n4\n\n\fembedding of \u03c4 as follows:\n\nF (\u03c4 )(t) :=\n\n(cid:16)\n(cid:16)\n\n\uf8f1\uf8f4\uf8f4\uf8f4\uf8f2\uf8f4\uf8f4\uf8f4\uf8f3\n\nlp\n...\nlp\n\n(cid:17)\n\n,\n\nf (v1)f (v2), t\u2212t1\nt2\u2212t1\n\nf (vm\u22121)f (vm),\n\nt\u2212tm\u22121\ntm\u2212tm\u22121\n\n(cid:17)\n\nif t \u2208 [0, t2)\n\n.\n\nif t \u2208 [tm\u22121, 1]\n\n,\n\nFor a set T := {\u03c41, . . . , \u03c4n} of polygonal curves we de\ufb01ne F (T ) := {F (\u03c4 ) | \u03c4 \u2208 T} and require the\nfunction f to be a (1 \u00b1 \u03b5)-Johnson-Lindenstrauss embedding for the set of all vertices of all \u03c4 \u2208 T .\nWe next give an explicit bound on the distortion of the Fr\u00e9chet distance when the map of De\ufb01nition 5\nis applied to the input curves. Note that the previously mentioned approach by Sheehy (2014) for the\nconvex hull of points is not directly applicable since two curves might be drawn apart from each other\nmaking the error arbitrary large. Our additive error will depend only on the length of line segments\nbetween consecutive points of a curve, which is usually bounded.\nWe \ufb01rst express the distance between two points on two distinct line segments using their relative\npositions on the respective line segment.\nProposition 6. Let s1\n:= q1q2 be line segments between two points\np1 := (p1,1, . . . , p1,d), p2 := (p2,1, . . . , p2,d) \u2208 Rd, respective q1 := (q1,1, . . . , q1,d), q2 :=\n(q2,1, . . . , q2,d) \u2208 Rd. For any \u03bbp, \u03bbq \u2208 [0, 1] and p := lp (p1p2, \u03bbp) lying on s1, as well as\nq := lp (q1q2, \u03bbq) lying on s2, it holds that\n(cid:107)p \u2212 q(cid:107)2 = \u2212 (\u03bbp \u2212 \u03bb2\n\nq)(cid:107)q1 \u2212 q2(cid:107)2 + (1 \u2212 \u03bbp \u2212 \u03bbq + \u03bbp\u03bbq)(cid:107)p1 \u2212 q1(cid:107)2\n\np)(cid:107)p1 \u2212 p2(cid:107)2 \u2212 (\u03bbq \u2212 \u03bb2\n\n:= p1p2 and s2\n\n+ (\u03bbq \u2212 \u03bbp\u03bbq)(cid:107)p1 \u2212 q2(cid:107)2 + (\u03bbp \u2212 \u03bbp\u03bbq)(cid:107)p2 \u2212 q1(cid:107)2 + \u03bbp\u03bbq(cid:107)p2 \u2212 q2(cid:107)2.\n\n(cid:16)\n\n(cid:17)\n\n(cid:17)\n\n(cid:16)\n\nf (p1)f (p2), \u03bbp\n\n, as well as q := lp (q1q2, \u03bbq), q(cid:48) := lp\n\nProposition 6 can be proven using the law of cosines, the geometric and algebraic de\ufb01nition of the\ndot product and tedious algebraic manipulations.\nUsing Proposition 6, our calculation yields an explicit error-bound when applying De\ufb01nition 5 to\nboth line-segments. This is formalized in Lemma 7.\nLemma 7. Let P := {p1, . . . , pn} \u2282 Rd be a set of points and f be a (1\u00b1 \u03b5)-Johnson-Lindenstrauss\nembedding for P . Let p1, p2, q1, q2 \u2208 P , for arbitrary \u03bbp, \u03bbq \u2208 [0, 1] and p := lp (p1p2, \u03bbp),\np(cid:48) := lp\n(1\u2212\u03b5)2(cid:107)p\u2212q(cid:107)2\u2212\u03b5((cid:107)p1\u2212p2(cid:107)2+(cid:107)q1\u2212q2(cid:107)2) \u2264 (cid:107)p(cid:48)\u2212q(cid:48)(cid:107)2 \u2264 (1+\u03b5)2(cid:107)p\u2212q(cid:107)2+\u03b5((cid:107)p1\u2212p2(cid:107)2+(cid:107)q1\u2212q2(cid:107)2)\nis satis\ufb01ed with probability at least \u03c1 \u2208 (0, 1] over the random construction of f.\nThis \ufb01nally yields our main theorem which states the desired error guarantee for the Fr\u00e9chet distance\nof a set of polygonal curves.\nTheorem 8. Let T := {\u03c41, . . . , \u03c4n} be a set of polygonal curves and for \u03c4 \u2208 T let \u03b1(\u03c4 ) denote\nthe maximum distance of two consecutive vertices of \u03c4. Furher, for \u03c4, \u03c3 \u2208 T let \u03b1(\u03c4, \u03c3) :=\nmax{\u03b1(\u03c4 ), \u03b1(\u03c3)}. Now, let F be a (1 \u00b1 \u03b5)-Johnson-Lindenstrauss embedding for T . With constant\nprobability at least \u03c1 \u2208 (0, 1] it holds for all \u03c4, \u03c3 \u2208 T that\n\nf (q1)f (q2), \u03bbq\n\nit holds that\n\n(cid:113)\n\nF (\u03c4, \u03c3) \u2212 2\u03b5\u03b1(\u03c4, \u03c3)2 \u2264 dF (F (\u03c4 ), F (\u03c3)) \u2264(cid:113)\n\n(1 \u2212 \u03b5)2d2\n\n(1 + \u03b5)2d2\n\nF (\u03c4, \u03c3) + 2\u03b5\u03b1(\u03c4, \u03c3)2,\n\nwhere the exact value for \u03c1 stems from the technique used for obtaining f.\n\nLet us \ufb01rst note that these bounds tend to dF (\u03c4, \u03c3) as \u03b5 tends to 0. The multiplicative error bounds\nare similar to \u03b5-coresets which are popular data reduction techniques in clustering, cf. (Feldman et al.,\n2010, 2013; Sohler and Woodruff, 2018). The additional additive error is in line with the relaxation\ngiven by lightweight coresets (Bachem et al., 2018).\nWe believe that the additive error is necessary. Consider the following two polygonal curves in\nRd, for d \u2265 3. Let \u03b1 \u2208 R>0 be arbitrary. The \ufb01rst curve is p := p1p2 with p1 := (0, . . . , 0) and\np2 := (\u03b1, 0, . . . , 0). The second curve q has vertices q1 := (0, 1, 0, . . . , 0), q2 := ( \u03b1\n2 , 2, 1, 0, . . . , 0)\nand q3 := (\u03b1, 1, 0, . . . , 0). It\u2019s edges are q1q2 and q2q3. Clearly, we have (cid:107)p1\u2212p2(cid:107) = \u03b1 = (cid:107)q1\u2212q3(cid:107),\n\n5\n\n\fFigure 1: Empirical relative er-\nror in terms of the distortion\nof the Fr\u00e9chet distance between\nthe curves p and q.\nIt can be\nobserved that the distortion de-\npends on the value \u03b1, which de-\ntermines the curves lengths, but\nnot their Fr\u00e9chet distance. Note\nthat an empirical relative error\nabove 1 means that not even a\n2-approximation of the distance\nwas achieved. However, this\nonly happens for line segments\nof length larger than 1015.\n\n\u221a\n\n(cid:107)p1 \u2212 q1(cid:107) = 1 = (cid:107)p2 \u2212 q3(cid:107) and (cid:107)p1 \u2212 q2(cid:107) = ( \u03b12\n4 + 5)1/2 = (cid:107)p2 \u2212 q2(cid:107), as well as (cid:107)q1 \u2212 q2(cid:107) =\n4 + 2)1/2 = (cid:107)q2 \u2212 q3(cid:107). Also note that dF (p, q) =\n( \u03b12\n5, a constant that does not depend on \u03b1. The\npairwise distances among the points will be distorted by at most (1 \u00b1 \u03b5). Now the embedding has its\nmass concentrated in the interval (1 \u00b1 \u03b5) but inspecting the concentration inequalities most of this\nmass is between (1 \u00b1 \u03b5\nc ) and (1 \u00b1 \u03b5) for large c. Thus, with reasonably large probability the error on\nq2 will depend on \u03b5\u03b1\nWe assess the distortion of the Fr\u00e9chet distance between p and q experimentally. We use the target\ndimension of the proof in (Dasgupta and Gupta, 2003) and all combinations of \ufb01ve choices for \u03b5, as\nwell as sixteen choices for \u03b1, we conduct one experiment with one hundred repetitions. The results\nare depicted in Fig. 1.\n\nc which is additive since \u03b1 is unrelated to the original Fr\u00e9chet distance.\n\n3 Median Clustering under the Fr\u00e9chet Distance\n\nWe study the k-median problem. As discussed before, we restrict the centers to subsets of the input.\nDe\ufb01nition 9 (discrete median clustering). Given a set of T of polygonal curves, the k-median\nclustering problem is to \ufb01nd a set C \u2286 T of k centers such that the sum of the distances from the\ncurves in T to the closest center in C is minimized.\n\nAt \ufb01rst, we restrict to k = 1. Instead of exhaustively trying out all curves as possible median, thus\ncomputing all pairwise distances among the input curves, we aim to \ufb01nd a small candidate set of\npossible medians and another small witness set which serves as a proxy to sum over. We will use\nthe following theorem of Indyk (2000) to bound the number of required witnesses, given a set of\ncandidates of certain size.\nTheorem 10. (Indyk, 2000, Theorem 31) Let \u03b5 \u2208 (0, 1] be a constant and T be a set of\npolygonal curves. Further let W be a non-empty uniform sample from T . For \u03c4, \u03c3 \u2208\n\u03c4(cid:48)\u2208W dF (\u03c4, \u03c4(cid:48)) \u2264\n\n\u03c4(cid:48)\u2208T dF (\u03c3, \u03c4(cid:48)) it holds that Pr[(cid:80)\n\nT with (cid:80)\n\u03c4(cid:48)\u2208T dF (\u03c4, \u03c4(cid:48)) > (1 + \u03b5)(cid:80)\n\u03c4(cid:48)\u2208W dF (\u03c3, \u03c4(cid:48))] < exp(cid:0)\u2212\u03b52|W|/64(cid:1).\n(cid:80)\n\nuniform sample W := {w1, . . . , w(cid:96)W } of cardinality O(cid:0)ln((cid:96)S/\u03b4)/\u03b52(cid:1) of witnesses, to obtain a\n\nUsing only this theorem, we still have to cope with all n input curves as candidates. In what follows,\nwe reduce this to a constant size sample of the input. Without assumptions on the input, by standard\nprobabilistic arguments and the triangle-inequality, we obtain a (2 + \u03b5)-approximation.\nTheorem 11. Given constants \u03b5, \u03b4 \u2208 (0, 1/2) and a non-empty set T of polygonal curves, we\ncan use a uniform sample S := {s1, . . . , s(cid:96)S} of cardinality O (ln(1/\u03b4)/\u03b5) of candidates and a\n(2 + \u03b5)-approximate 1-median cS \u2208 S with probability at least 1 \u2212 \u03b4.\nUnder natural assumptions, setting our analysis in the Beyond-Worst-Case regime (Roughgarden,\n2019), we can even get a (1 + \u03b5)-approximation on a subsample of sublinear size. The high-level idea\nbehind is that the curves are usually not equidistant to an optimal median. Relative to the average\ncost, there will be some outliers, some curves at medium distance and also some curves very close to\nan optimal median. Now if there are quite a good number of outliers, but also not too many, they\n\n6\n\n\f(cid:17)\n\n1/2\u2212\u03b3(T )\n\nmake up a good share of the total cost. This implies that the number of curves at medium distance is\nbounded by a constant fraction of the curves. Finally this implies that the number of curves that are\nclose to an optimal median, is not too small such that a small sample will include at least one of them\nwith constant probability.\nTheorem 12. Let \u03b5, \u03b4 \u2208 (0, 1/2) be constants, and T be a non-empty set of polygonal curves with at\nleast (1 \u2212 \u03b5)\u03b3(T )n outliers, for 0 < \u03b3(T ) < 1/2. We can use a uniform sample S := {s1, . . . , s(cid:96)S}\nof candidates and a uniform sample W := {w1, . . . , w(cid:96)W } of\n\nof cardinality (cid:96)S = O(cid:16) ln(1/\u03b4)\ncardinality O(cid:0)ln((cid:96)S/\u03b4)/\u03b52(cid:1) of witnesses, to obtain a (1 + \u03b5)-approximate 1-median cS \u2208 S with\n\nprobability at least 1 \u2212 \u03b4.\nNote in particular, that the samples in Theorem 12 still have constant size unless the fraction of\noutliers \u03b3(T ) tends arbitrarily close to 1/2 depending on |T| = n. In this case the usual notion of an\noutlier is not met for two reasons: \ufb01rst, more than a quarter of the curves would be considered outliers,\nand second their distance to an optimal median is not much larger than the medium curves implying\nthat basically all curves are in a narrow annulus around the average distance. Both observations\nmake the notion of outliers highly questionable. The details are in the proof and Fig. 5, which can be\nfound in the supplement. Note that in practice it is neither necessary nor desirable to compute \u03b3(T ).\nInstead, one should set \u03b3(T ) = 1/2 \u2212 1/c, for a large enough constant c. Now, if our assumptions on\nthe input hold, dF has the [\u03b5, \u03b4]-sampling property from Ackermann et al. (2010) and we can apply\ntheir Theorem 1.1, yielding the following corollary:\nCorollary 13. Under the assumptions of Theorem 12, there exists an algorithm for the discrete k-\nmedian under the Fr\u00e9chet distance that, given a set of n polygonal curves and \u03b5 \u2208 (0, 1), returns with\npositive constant probability a (1 + \u03b5)-approximation using only n \u00b7 2O(k\u00b7(|S|+|W|) log( k\n\u03b5 \u00b7(|S|+|W|))\ndistance computations, where S is the candidate sample and W is the witness sample.\n\n4 Complexity Reduction for Polygonal Curves\n\nWe study the space complexity of compressing polygonal curves such that their complexity, i.e., their\nnumber of vertices, is reduced while their Fr\u00e9chet distance is preserved. Recall that m dominates\nthe running-time of the Alt and Godau algorithm. Now, for reducing this dependence, the goal is to\nde\ufb01ne a randomized function S together with an estimation procedure E so that for any polygonal\ncurves \u03c4, \u03c3, we take the compressed representations S(\u03c4 ) and the estimation procedure satis\ufb01es\nwith constant probability dF (\u03c4, \u03c3) \u2264 E(S(\u03c4 ), \u03c3) \u2264 \u03b7 \u00b7 d(\u03c4, \u03c3) for some approximation factor \u03b7,\ncf. Braverman et al. (2019). The challenge is to bound the size of S(\u03c4 ) depending on the complexity\nof \u03c4 in order to obtain an approximation factor of \u03b7.\nWe prove that the Fr\u00e9chet distance can not be approximated up to any factor by reducing the\ncomplexity of the curves deterministically, even in one dimension. We achieve this result by reducing\nfrom the equality test communication problem, which requires a linear number of bits, cf. Wegener\n(2005).\nTheorem 14. Let \u03c4, \u03c3 be polygonal curves in Rd, for d \u2265 1, with m vertices each. Any deterministic\ndata oblivious sketching function S for which there exists a deterministic estimation function E\nsatisfying dF (\u03c4, \u03c3) \u2264 E(S(\u03c4 ), \u03c3) \u2264 \u03b7 \u00b7 dF (\u03c4, \u03c3), for an arbitrary \u03b7 \u2208 [1,\u221e), uses \u2126(m) bits to\nrepresent S(\u03c4 ).\n\u221a\nAlso, we prove that the Fr\u00e9chet distance can not be approximated within any factor less than\n2\nby reducing the complexity of the curves probabilistically. We show this by reducing from the set\ndisjointness communication problem, which also requires a linear number of bits for any randomized\nprotocol succeeding with constant probability, cf. H\u00e5stad and Wigderson (2007).\nTheorem 15. Let \u03c4, \u03c3 be polygonal curves in Rd, for d \u2265 2, with m vertices each. Any randomized\ndata oblivious sketching function S for which there exists a randomized estimation function E\nsatisfying dF (\u03c4, \u03c3) \u2264 E(S(\u03c4 ), \u03c3) \u2264 \u03b7 \u00b7 dF (\u03c4, \u03c3), for \u03b7 \u2208 [1,\n2], uses \u2126(m) bits to represent S(\u03c4 ).\n\n\u221a\n\n5 Experiments\n\nThe main practical motivation for our work is that any application utilizing the Fr\u00e9chet distance\nsuffers from its computational cost. In general, there are three parameters on which the running-time\n\n7\n\n\fFigure 2: (a): Distortion under the (1 \u00b1 \u03b5)-Johnson-Lindenstrauss embedding. The lateral axis shows\nthe values for \u03b5 plugged into the embedding (and the corresponding number of dimensions). The\nlongitutdinal axis shows the empirical relative error. (b): Running-times of the algorithms, where\nsequential is an na\u00efve implementation of the Alt and Godau algorithm, parallel is our CUDA-enabled\nvariant and the suf\ufb01x \u201c_rp\u201d means that the data was randomly projected before.\n\n(a) Distortion\n\n(b) Running-times\n\n\u221a\n\ndepends: the dimension of the ambient space d, the number of curves n and their complexity m.\nWe tackle the \ufb01rst utilizing our results from Section 2, i.e., the dimension reduction and the second\nby utilizing our results from Section 3, i.e., the sampling schemes. For the last, by Section 4 we\nwould loose a factor of at least\n2 and can not hope to design a (1 + \u03b5)-approximation algorithm\nwith subquadratic running-time in m. We thus decide to tackle the dependence of the Alt and Godau\nalgorithm on m by parallelization.1 We now seek to answer:\nQ1 Does the random projection induce a reasonably small distortion on the Fr\u00e9chet distance?\nQ2 What is the impact of our techniques on the running-time of the Fr\u00e9chet distance computation?\nQ3 Do we obtain reasonable results combining the sampling scheme and the random projection?\nQ4 Does PCA lead to better results than random projections?\nBefore we answer these questions based on our experimental results, we describe our data sets, the\nmodi\ufb01cations we applied to the Alt and Godau algorithm, and our setup. Also, note that we used the\nempirical constant of 2 for the experiments in this section, cf. Venkatasubramanian and Wang (2011).\nTherefore, we projected from d to d(cid:48) = 2\u03b5\u22122 ln(nm) dimensions.\nData sets Our \ufb01rst data set was taken by monitoring a hydraulic test rig via multiple sensors\n(cf. Helwig et al. (2015)), including six pressure sensors PS1, . . . , PS6. In a total of 2205 test-cycles,\neach sensor measured 6000 values in each cycle. We chose to build six polygonal curves with 2205\nvertices each in the 6000-dimensional Euclidean space. Also, for comparison we generate curves\nof equal complexity and ambient dimension by picking their vertices uniformly at random from\na d + 1-simplex scaled by a large number, thus obtaining curves of high intrinsic dimension. For\n1-median clustering we use weather simulation data (Lucas et al., 2015) from which we construct\n2922 curves with 15 vertices each in 327-dimensional Euclidean space.\nAlgorithm modi\ufb01cations We decided to parallelize the Alt and Godau algorithm utilizing\nCUDA-enabled graphic cards. We improve the worst-case running-time of the algorithm from\nO(dm2 log(m)) to O(d m2\nSetup We ran our experiments on a high perfomance linux cluster, which has twenty GPU nodes\nwith two Intel Xeon E5-2640v4 CPUs, 64 GB of RAM and two Nvidia K40 GPUs each. This makes\n2880 CUDA cores per card. To minimize interference, each experiment was run on an exclusive\ncore and both GPUs, with 30 GB of RAM guaranteed. Each experiment was run ten times for each\nparametrization. Every experiment concerning the curves sampled from the simplex was even run\none hundred times for each parameterization.\n\n#cc log(m)), where #cc is the number of available CUDA cores.\n\n1Code available at https://www.dennisrohde.work/rp4frechet-code.\n\n8\n\n\fFigure 3: (a): Running-times and (b): deviations for the Fr\u00e9chet 1-median sampling scheme. The\nlateral axis shows the values for \u03b5 plugged into the sampling algorithm. The deviations are with\nrespect to the optimal objective value. epsilon rp is the value for \u03b5 that is plugged into the embedding.\n\n(a) Running-times\n\n(b) Deviations\n\nFigure 4: Comparison of Johnson Linden-\nstrauss embedding vs. embedding via PCA\nshowing the trade-off between the method\u2019s\nrunning-time and its achieved quality.\n\nQ1 Concerning all data sets we can say that the distortion of the Fr\u00e9chet distance after applying\nthe Johnson-Lindenstrauss embedding is reasonably small. In Fig. 2(a) we depict the results of the\nFr\u00e9chet distance computations vs. the chosen values for \u03b5. It can be observed that even for larger\nvalues of \u03b5, the effective error never exceeds the given margin.\nQ2 In Fig. 2(b) we depict the running-times of the Alt and Godau algorithm under our measures. The\nresults stem from the same experiments that lead to the values depicted in Fig. 2(a). The random\nprojection and the parallelization speed up the computations by a factor of 10 each independently.\nBoth together yield a speedup of factor 100. While the na\u00efve implementation of the algorithm took\nabout roughly three hours, we were able to lower the running-time to about 30 seconds on average.\nQ3 We conducted experiments on the weather simulation dataset. Fig. 3 shows that employing\nthe subsampling schemes yields substantial improvements in terms of running-times while the\napproximation error remains robust to the choices of the approximation parameters \u201cepsilon sampling\u201d\nand \u201cepsilon rp\u201d plugged into the subsampling scheme and the embedding, respectively. This indicates\nthat the approximation is indeed dependent on the data paramater \u03b3(T ).\nQ4 In Fig. 4 we compare the Johnson-Lindenstrauss embedding for polygonal curves to PCA applied\nto the vertices in a similar fashion. Here, we only used the curves whose vertices were sampled from\na d + 1-simplex to emphasize the impact of hard inputs on the distortion. We depict the methods\nrunning-time vs. distortion. It can be observed that for all choices of \u03b5, the Johnson Lindenstrauss\nembedding performs much better in terms of distortion as well as running-time.\n\nAcknowledgments\n\nWe thank the anonymous reviewers for their valuable comments. This work was supported by the German Science\nFoundation (DFG), Collaborative Research Center SFB 876 \"Providing Information by Resource-Constrained\nAnalysis\", project C4 and by the Dortmund Data Science Center (DoDSc).\n\n9\n\n\fReferences\nAckermann, M. R., Bl\u00f6mer, J., and Sohler, C. (2010). Clustering for metric and nonmetric distance measures.\n\nACM Transactions on Algorithms, 6(4):59:1\u201359:26.\n\nAgarwal, P. K., Har-Peled, S., and Yu, H. (2013). Embeddings of surfaces, curves, and moving points in\n\nEuclidean space. 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