We study algorithms for matching user tracks, consisting of time-ordered location points, to paths in the road network. Previous work has focused on the scenario where the location data is linearly ordered and consists of fairly dense and regular samples. In this work, we consider the , where the location data comes from different trips on the same route, each with very sparse samples. This captures the realistic scenario where users repeatedly travel on regular routes and samples are sparsely collected, either due to energy consumption constraints or because samples are only collected when the user actively uses a service. In the multi-track problem, the total set of combined locations is only partially ordered, rather than globally ordered as required by previous map-matching algorithms. We propose two methods, the iterative projection scheme and the graph Laplacian scheme, to solve the multi-track problem by using a single-track map-matching subroutine. We also propose a boosting technique which may be applied to either approach to improve the accuracy of the estimated paths. In addition, in order to deal with variable sampling rates in single-track map matching, we propose a method based on a particular regularized cost function that can be adapted for different sampling rates and measurement errors. We evaluate the effectiveness of our techniques for reconstructing tracks under several different configurations of sampling error and sampling rate.
Another line of research does not use any statistical methods to address the problem. Authors in @cite_4 use curve simplification for approximating the Fr 'echet distance of curves. Given a polygonal curve @math and an embedded graph @math with edges embedded as straight line segments, they attack the problem of finding the closest path in @math to the curve @math with respect to the Fr 'echet distance. @cite_9 , the average Fr 'echet distance is used to reduce the effect of outliers.
@cite_9 @cite_4
1544411027 2403500509
Two central limit theorems for sample Fr 'echet means are derived, both significant for nonparametric inference on non-Euclidean spaces. The first one, Theorem 2.2, encompasses and improves upon most earlier CLTs on Fr 'echet means and broadens the scope of the methodology beyond manifolds to diverse new non-Euclidean data including those on certain stratified spaces which are important in the study of phylogenetic trees. It does not require that the underlying distribution @math have a density, and applies to both intrinsic and extrinsic analysis. The second theorem, Theorem 3.3, focuses on intrinsic means on Riemannian manifolds of dimensions @math and breaks new ground by providing a broad CLT without any of the earlier restrictive support assumptions. It makes the statistically reasonable assumption of a somewhat smooth density of @math . The excluded case of dimension @math proves to be an enigma, although the first theorem does provide a CLT in this case as well under a support restriction. Theorem 3.3 immediately applies to spheres @math , @math , which are also of considerable importance in applications to axial spaces and to landmarks based image analysis, as these spaces are quotients of spheres under a Lie group @math of isometries of @math . A straight-line grid embedding of a planar graph is a drawing of the graph on a plane where the vertices are located at grid points and the edges are represented by nonintersecting segments of straight, lines joining their incident vertices. Given an n-vertex embedded planar graph with @math , a straight-line embedding on a grid of size @math can be computed deterministically in @math time with @math processors. If randomization is used, the complexity is improved to @math expected time with the same optimal linear work. These algorithms run on a parallel random access machine that allows concurrent reads and concurrent writes of the shared memory and permits an arbitrary processor to succeed in case of a write conflict. What is the densest packing of points in an infinite strip of widthw, where any two of the points must be separated by distance at least I? This question was raised by Fejes-Toth a number of years ago. The answer is trivial for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai abgsMiJoaakaaabaGaaG4maaWcbeaakiaac+cacaaIYaaaaa!3AF6! @math and, surprisingly, it is not difficult to prove [M2] for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83Dai aa-1dacaWGUbWaaOaaaeaacaaIZaaaleqaaOGaai4laiaaikdaaaa!3AF2! @math , wheren is a positive integer, that the regular triangular lattice gives the optimal packing. Kertesz [K] solved the case MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbiGaa83DaG qaaiaa+bcacaGF8aWaaOaaaeaacaaIYaaaleqaaaaa!392C! @math . Here we fill the first gap, i.e., the maximal density is determined for MathType!MTEF!2!1!+- feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9 vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaie aacaWFZaaaleqaaOGaai4laiaaikdacqGH8aapcaWG3bGaeyizIm6a aOaaaeaacaaIZaaaleqaaaaa!3CCB! @math . A new piecewise linear method is presented for the approximation of digitized curves. This method produces a sequence of consecutive line segments and has the following characteristics: (i) it approximates the digitized curve with the minimum number of line segments, (ii) the Euclidean distance between each point of the digitized curve and the line segment that approximates it, does not exceed a boundary value spl epsiv and (iii) the vertices of the produced line are not (necessarily) points of the input curve. Sliced Wasserstein metrics between probability measures solve the optimal transport (OT) problem on univariate projections, and average such maps across projections. The recent interest for the SW distance shows that much can be gained by looking at optimal maps between measures in smaller subspaces, as opposed to the curse-of-dimensionality price one has to pay in higher dimensions. Any transport estimated in a subspace remains, however, an object that can only be used in that subspace. We propose in this work two methods to extrapolate, from an transport map that is optimal on a subspace, one that is nearly optimal in the entire space. We prove that the best optimal transport plan that takes such "subspace detours" is a generalization of the Knothe-Rosenblatt transport. We show that these plans can be explicitly formulated when comparing Gaussians measures (between which the Wasserstein distance is usually referred to as the Bures or Fr 'echet distance). Building from there, we provide an algorithm to select optimal subspaces given pairs of Gaussian measures, and study scenarios in which that mediating subspace can be selected using prior information. We consider applications to NLP and evaluation of image quality (FID scores). This paper addresses the basic question of how well a tree can approximate distances of a metric space or a graph. Given a graph, the problem of constructing a spanning tree in a graph which strongly preserves distances in the graph is a fundamental problem in network design. We present scaling distortion embeddings where the distortion scales as a function of @math , with the guarantee that for each @math simultaneously, the distortion of a fraction @math of all pairs is bounded accordingly. Quantitatively, we prove that any finite metric space embeds into an ultrametric with scaling distortion @math . For the graph setting, we prove that any weighted graph contains a spanning tree with scaling distortion @math . These bounds are tight even for embedding into arbitrary trees. These results imply that the average distortion of the embedding is constant and that the @math distortion is @math . For probabilistic embedding into spanning trees we prove a scaling distortion of @math , which implies constant @math -distortion for every fixed @math . This paper addresses deep face recognition (FR) problem under open-set protocol, where ideal face features are expected to have smaller maximal intra-class distance than minimal inter-class distance under a suitably chosen metric space. However, few existing algorithms can effectively achieve this criterion. To this end, we propose the angular softmax (A-Softmax) loss that enables convolutional neural networks (CNNs) to learn angularly discriminative features. Geometrically, A-Softmax loss can be viewed as imposing discriminative constraints on a hypersphere manifold, which intrinsically matches the prior that faces also lie on a manifold. Moreover, the size of angular margin can be quantitatively adjusted by a parameter @math . We further derive specific @math to approximate the ideal feature criterion. Extensive analysis and experiments on Labeled Face in the Wild (LFW), Youtube Faces (YTF) and MegaFace Challenge show the superiority of A-Softmax loss in FR tasks. The code has also been made publicly available. We extend recent results using curve simplification for approximating the Frechet distance of realistic curves in near linear time to map matching: the problem of matching a curve in an embedded graph. We show that the theoretical bounds on the running time of the previous result still hold if only one of the curves is simplified during the course of the approximation algorithm. This enables our extension to the case of map matching under the assumption that the graph is p-low density for a constant p. We present experimental evidence for this assumption and implement the extended approximate matching algorithm. We show that it performs well on real world data, such as GPS traces and road networks of urban areas. In particular, it is able to perform matching tasks that took several hours with the exact matching algorithm in under a second. Let @math XźA2r be a real curve embedded into an even-dimensional affine space. We characterise when the @math rth secant variety to @math X is an irreducible component of the algebraic boundary of the convex hull of the real points @math X(R) of @math X. This fact is then applied to @math 4-dimensional @math SO(2)-orbitopes and to the so called Barvinok---Novik orbitopes to study when they are basic closed semi-algebraic sets. In the case of @math 4-dimensional @math SO(2)-orbitopes, we find all irreducible components of their algebraic boundary. This paper proposes a novel closed-form solution to complete line-segment extraction. Given a voting angle in image space, the voting distribution is analyzed and two functional relationships are deduced. Regarding the corresponding column in Hough space, voting along the distance axis is considered as being a random variable, and voting values in cells are considered as forming a probability distribution. Statistical characteristics of this distribution are used to fit a quadratic polynomial curve and a linear curve. Direction, length, and width of a line segment are simultaneously computed in a closed form based on coefficients of fitted quadratic polynomial curves. The midpoint of a line segment is determined based on the fitted linear curve. The method is tested on simulated and real-world images; results show that the proposed closed-form solution is feasible in the presence of quantization errors or image noise. HighlightsWe proposed a method for obtaining the complete set of line-segment parameters.The direction, length and width of a line-segment are extracted simul- taneously in a closed form.We provided a complete theoretical derivation of voting variance with respect to voting angle.The midpoint of a line-segment is determined by the coefficients of the fitted linear curve.Parallel, crossing, and aligned line-segments are discussed by analysing events in image space and Hough space. Given a set S of points in the plane, there is a triangulation of S such that a path found within this triangulation has length bounded by a constant times the straight-line distance between the endpoints of the path. Specifically, for any two points a and b of S there is a path along edges of the triangulation with length less than √10 times |ab|, where |ab| is the straight-line Euclidean distance between a and b. Thus, a shortest path in this planar graph is less than about 3 times longer than the corresponding straight-line distance. The triangulation that has this property is the L 1 metric Delaunay triangulation for the set 5. This result can be applied to motion planning in the plane. Given a source, a destination, and a set of polygonal obstacles of size n, an O(n) size data structure can be used to find a reasonable approximation to the shortest path between the source and the destination in O(n log n) time. Finding the so-called characteristic numbers of the complex projective plane @math is a classical problem of enumerative geometry posed by Zeuthen more than a century ago. For a given @math and @math one has to find the number of degree @math genus @math curves that pass through a certain generic configuration of points and in the same time are tangent to certain generic configuration of lines. The total number of points and lines in these two configurations is @math so that the answer is a finite integer number. In this paper we translate this classical problem to the corresponding enumerative problem of tropical geometry in the case when @math . Namely, we show that the tropical problem is well-posed and establish a special case of the correspondence theorem that ensures that the corresponding tropical and classical numbers coincide. Then we use the floor diagram calculus to reduce the problem to pure combinatorics. As a consequence, we compute genus 0 characteristic numbers of @math in terms of open Hurwitz numbers. We analyze the problem of approximating a smooth quantum waveguide with a quantum graph. We consider a planar curve with compactly supported curvature and a strip of constant width around the curve. We rescale the curvature and the width in such a way that the strip can be approximated by a singular limit curve, consisting of one vertex and two infinite, straight edges, i.e., a broken line. We discuss the convergence of the Laplacian, with Dirichlet boundary conditions on the strip, in a suitable sense and we obtain two possible limits: the Laplacian on the line with Dirichlet boundary conditions in the origin and a nontrivial family of point perturbations of the Laplacian on the line. The first case generically occurs and corresponds to the decoupling of the two components of the limit curve, while in the second case a coupling takes place. We present also two families of curves which give rise to coupling. The page migration problem is to manage a globally addressed shared memory in a multiprocessor system. Each physical page of memory is located at a given processor, and memory references to that page by other processors incur a cost proportional to the network distance. At times the page may migrate between processors at cost proportional to the distance times @math , a page size factor. The problem is to schedule movements on-line so that the total cost of memory references is within a constant factor @math of the best off-line schedule. An algorithm that does so is called c-competitive. Black and Sleator gave 3-competitive deterministic on-line algorithms for uniform networks (complete graphs with unit edge lengths) and for trees with arbitrary edge lengths. No good deterministic algorithm is known for general networks with arbitrary edge lengths. Randomized algorithms are presented for the migration problem that are both simple and better than 3-competitive against an oblivious adversary. An algorithm for uniform graphs is given. It is approximately 2.28-competitive as @math grows large. A second, more powerful algorithm that works on graphs with arbitrary edge distances is also given. This algorithm is approximately 2.62-competitive (or, 1 plus the golden ratio) for large @math . Both these algorithms use random bits only during an initialization phase, and from then on run deterministically. The competitiveness of a very simple coin-flipping algorithm is also examined. Given a graph embedded in a metric space, its dilation is the maximum over all distinct pairs of vertices of the ratio between their distance in the graph and the metric distance between them. Given such a graph G with n vertices and m edges and consisting of at most two connected components, we consider the problem of augmenting G with an edge such that the resulting graph has minimum dilation. We show that we can find such an edge in @math time using linear space which solves an open problem of whether a linear-space algorithm with o(n 4) running time exists. We show that O(n 2logn) time is achievable if G is a simple path or the union of two vertex-disjoint simple paths. Finally, we show how to find an edge that maximizes the dilation of the resulting graph in O(n 3) time with O(n 2) space and in O(n 3logn) time with linear space. Here we study the kth symmetric trigonometric moment curve and its convex hull, the Barvinok–Novik orbitope. In 2008, Barvinok and Novik introduced these objects and showed that there is some threshold so that for two points on @math with arclength below this threshold the line segment between their lifts to the curve forms an edge on the Barvinok–Novik orbitope, and for points with arclength above this threshold their lifts do not form an edge. They also gave a lower bound for this threshold and conjectured that this bound is tight. Results of Smilansky prove tightness for k=2. Here we prove this conjecture for all k. Finite convex geometries are combinatorial structures. It follows from a recent result of M. Richter and L.G. Rogers that there is an infinite set @math of planar convex polygons such that @math with respect to geometric convex hulls is a locally convex geometry and every finite convex geometry can be represented by restricting the structure of @math to a finite subset in a natural way. An @math is a differentiable convex simple closed curve @math in the plane having an inscribed circle of radius @math and a circumscribed circle of radius @math such that the ratio @math is at least @math . Motivated by Richter and Rogers' result, we construct a set @math such that (1) @math contains all points of the plane as degenerate singleton circles and all of its non-singleton members are differentiable convex simple closed planar curves; (2) @math with respect to the geometric convex hull operator is a locally convex geometry; (3) as opposed to @math , @math is closed with respect to non-degenerate affine transformations; and (4) for every (small) positive @math and for every finite convex geometry, there are continuum many pairwise affine-disjoint finite subsets @math of @math such that each @math consists of almost-circles of accuracy @math and the convex geometry in question is represented by restricting the convex hull operator to @math . The affine-disjointness of @math and @math means that, in addition to @math , even @math is disjoint from @math for every non-degenerate affine transformation @math . Hand-drawn schematized maps traditionally make extensive use of curves. However, there are few automated approaches for curved schematization; most previous work focuses on straight lines. We present a new algorithm for area-preserving curved schematization of territorial outlines. Our algorithm converts a simple polygon into a schematic crossing-free representation using circular arcs. We use two basic operations to iteratively replace consecutive arcs until the desired complexity is reached. Our results are not restricted to arcs ending at input vertices. The method can be steered towards different degrees of “curviness”: we can encourage or discourage the use of arcs with a large central angle via a single parameter. Our method creates visually pleasing results even for very low output complexities. To evaluate the effectiveness of our design choices, we present a geometric evaluation of the resulting schematizations. Besides the geometric qualities of our algorithm, we also investigate the potential of curved schematization as a concept. We conducted an online user study investigating the effectiveness of curved schematizations compared to straight-line schematizations. While the visual complexity of curved shapes was judged higher than that of straight-line shapes, users generally preferred curved schematizations. We observed that curves significantly improved the ability of users to match schematized shapes of moderate complexity to their unschematized equivalents. Right Angle Crossing (RAC) drawings are polyline drawings where each crossing forms four right angles. RAC drawings have been introduced because cognitive experiments provided evidence that increasing the number of crossings does not decrease the readability of the drawing if the edges cross at right angles. We investigate to what extent RAC drawings can help in overcoming the limitations of widely adopted planar graph drawing conventions, providing both positive and negative results. First, we prove that there exist acyclic planar digraphs not admitting any straight-line upward RAC drawing and that the corresponding decision problem is NP-hard. Also, we show digraphs whose straight-line upward RAC drawings require exponential area. Second, we study if RAC drawings allow us to draw bounded-degree graphs with lower curve complexity than the one required by more constrained drawing conventions. We prove that every graph with vertex-degree at most 6 (at most 3) admits a RAC drawing with curve complexity 2 (resp. 1) and with quadratic area. Third, we consider a natural non-planar generalization of planar embedded graphs. Here we give bounds for curve complexity and area different from the ones known for planar embeddings. Given a polygon @math , for two points @math and @math contained in the polygon, their is the length of the shortest @math -path within @math . A of radius @math centered at a point @math is the set of points in @math whose geodesic distance to @math is at most @math . We present a polynomial time @math -approximation algorithm for finding a densest geodesic unit disk packing in @math . Allowing arbitrary radii but constraining the number of disks to be @math , we present a @math -approximation algorithm for finding a packing in @math with @math geodesic disks whose minimum radius is maximized. We then turn our focus on of @math and present a @math -approximation algorithm for covering @math with @math geodesic disks whose maximal radius is minimized. Furthermore, we show that all these problems are @math -hard in polygons with holes. Lastly, we present a polynomial time exact algorithm which covers a polygon with two geodesic disks of minimum maximal radius.