Total Least Squares Fitting of k-Spheres in n-D Euclidean Space Using an (n+2)-D Isometric Representation

We fit k-spheres optimally to n-D point data, in a geometrically total least squares sense. A specific practical instance is the optimal fitting of 2D-circles to a 3D point set. Among the optimal fitting methods for 2D-circles based on 2D (!) point data compared in Al-Sharadqah and Chernov (Electron. J. Stat. 3:886–911, 2009), there is one with an algebraic form that permits its extension to optimally fitting k-spheres in n-D. We embed this ‘Pratt 2D circle fit’ into the framework of conformal geometric algebra (CGA), and doing so naturally enables the generalization. The procedure involves a representation of the points in n-D as vectors in an (n+2)-D space with attractive metric properties. The hypersphere fit then becomes an eigenproblem of a specific symmetric linear operator determined by the data. The eigenvectors of this operator form an orthonormal basis representing perpendicular hyperspheres. The intersection of these are the optimal k-spheres; in CGA the intersection is a straightforward outer product of vectors. The resulting optimal fitting procedure can easily be implemented using a standard linear algebra package; we show this for the 3D case of fitting spheres, circles and point pairs. The fits are optimal (in the sense of achieving the KCR lower bound on the variance). We use the framework to show how the hyperaccurate fit hypersphere of Al-Sharadqah and Chernov (Electron. J. Stat. 3:886–911, 2009) is a minor rescaling of the Pratt fit hypersphere.     

A new set distance and its application to shape registration

We propose a new distance measure, called Complement weighted sum of minimal distances, between finite sets in ${\mathbb Z }^n$ and evaluate its usefulness for shape registration and matching. In this set distance the contribution of each point of each set is weighted according to its distance to the complement of the set. In this way, outliers and noise contribute less to the new similarity measure. We evaluate the performance of the new set distance for registration of shapes in binary images and compare it to a number of often used set distances found in the literature. The most extensive evaluation uses a set of synthetic 2D images. We also show three examples of real problems: registering a set of 2D images extracted from synchrotron radiation micro-computed tomography (SR $\upmu $ CT) volumes depicting bone implants; the difficult multi-modal registration task of finding the exact location of a 2D slice of a bone implant, as imaged by a light microscope, within a 3D SR $\upmu $ CT volume of the same implant; and finally recognition of handwritten characters. The evaluation shows that our new set distance performs well for all tasks and outperforms the other observed distance measures in most cases. It is therefore useful in many image registration and shape comparison tasks.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part II: A Posteriori Error Estimation

In this paper new a posteriori error estimates for the local discontinuous Galerkin (LDG) method for one-dimensional fourth-order Euler–Bernoulli partial differential equation are presented and analyzed. These error estimates are computationally simple and are obtained by solving a local steady problem with no boundary condition on each element. We use the optimal error estimates and the superconvergence results proved in Part I to show that the significant parts of the discretization errors for the LDG solution and its spatial derivatives (up to third order) are proportional to \((k+1)\) -degree Radau polynomials, when polynomials of total degree not exceeding \(k\) are used. These results allow us to prove that the \(k\) -degree LDG solution and its derivatives are \(\mathcal O (h^{k+3/2})\) superconvergent at the roots of \((k+1)\) -degree Radau polynomials. We use these results to construct asymptotically exact a posteriori error estimates. We further apply the results proved in Part I to prove that, for smooth solutions, these a posteriori LDG error estimates for the solution and its spatial derivatives at a fixed time \(t\) converge to the true errors at \(\mathcal O (h^{k+5/4})\) rate. We also prove that the global effectivity indices, for the solution and its derivatives up to third order, in the \(L^2\) -norm converge to unity at \(\mathcal O (h^{1/2})\) rate. Our proofs are valid for arbitrary regular meshes and for \(P^k\) polynomials with \(k\ge 1\) , and for periodic and other classical mixed boundary conditions. Our computational results indicate that the observed numerical convergence rates are higher than the theoretical rates. Finally, we present a local adaptive procedure that makes use of our local a posteriori error estimate.     

Distributed transactional memory for general networks

We consider the problem of implementing transactional memory in large-scale distributed networked systems. We present Spiral, a novel distributed directory-based protocol for transactional memory, and theoretically analyze and experimentally evaluate it for the performance boundaries of this approach from the worst-case perspective. Spiral is designed for the data-flow distributed implementation of software transactional memory which supports three basic operations: publish, allowing a shared object to be inserted in the directory so that other nodes can find it; lookup, providing a read-only copy of the object to the requesting node; move, allowing the requesting node to write the object locally after the node gets it. The protocol runs on a hierarchical directory construction based on sparse covers, where clusters at each level are ordered to avoid race conditions while serving concurrent requests. Given a shared object the protocol maintains a directory path pointing to the object. The basic idea is to use “spiral” paths that grow outward to search for the directory path of the object in a bottom-up fashion. For general networks, this protocol guarantees an \(\mathcal{O}(\log ^2 n\cdot \log D)\) approximation in sequential and one-shot concurrent executions of a finite set of move requests, where \(n\) is the number of nodes and \(D\) is the diameter of the network. It also guarantees poly-log approximation for any single lookup request. Our bounds are deterministic and hold in the worst-case. Moreover, this protocol requires only polylogarithmic bits of memory per node. Experimental evaluations in real networks also confirm our theoretical findings. To the best of our knowledge, this is the first deterministic consistency protocol for distributed transactional memory that achieves poly-log approximation in general networks.     

Optimized Parameter Search for Large Datasets of the Regularization Parameter and Feature Selection for Ridge Regression

In this paper we propose mathematical optimizations to select the optimal regularization parameter for ridge regression using cross-validation. The resulting algorithm is suited for large datasets and the computational cost does not depend on the size of the training set. We extend this algorithm to forward or backward feature selection in which the optimal regularization parameter is selected for each possible feature set. These feature selection algorithms yield solutions with a sparse weight matrix using a quadratic cost on the norm of the weights. A naive approach to optimizing the ridge regression parameter has a computational complexity of the order $O(R K N^{2} M)$ with $R$ the number of applied regularization parameters, $K$ the number of folds in the validation set, $N$ the number of input features and $M$ the number of data samples in the training set. Our implementation has a computational complexity of the order $O(KN^3)$ . This computational cost is smaller than that of regression without regularization $O(N^2M)$ for large datasets and is independent of the number of applied regularization parameters and the size of the training set. Combined with a feature selection algorithm the algorithm is of complexity $O(RKNN_s^3)$ and $O(RKN^3N_r)$ for forward and backward feature selection respectively, with $N_s$ the number of selected features and $N_r$ the number of removed features. This is an order $M$ faster than $O(RKNN_s^3M)$ and $O(RKN^3N_rM)$ for the naive implementation, with $N \ll M$ for large datasets. To show the performance and reduction in computational cost, we apply this technique to train recurrent neural networks using the reservoir computing approach, windowed ridge regression, least-squares support vector machines (LS-SVMs) in primal space using the fixed-size LS-SVM approximation and extreme learning machines.     

Better Size Estimation for Sparse Matrix Products

We consider the problem of doing fast and reliable estimation of the number z of non-zero entries in a sparse boolean matrix product. This problem has applications in databases and computer algebra. Let n denote the total number of non-zero entries in the input matrices. We show how to compute a 1±ε approximation of z (with small probability of error) in expected time for any $\varepsilon> 4/\sqrt[4]{z}$ . The previously best estimation algorithm, due to Cohen (J. Comput. Syst. Sci. 53(3):441–453, 1997), uses time . We also present a variant using I/Os in expectation in the cache-oblivious model. In contrast to these results, the currently best algorithms for computing a sparse boolean matrix product use time ω(n ^4/3) (resp. ω(n ^4/3/B) I/Os), even if the result matrix is restricted to nonzero entries. Our algorithm combines the size estimation technique of Bar-Yossef et al. (Proceedings of the 6th International Workshop on Randomization and Approximation Techniques (RANDOM ’02), pp. 1–10, 2002) with a particular class of pairwise independent hash functions that allows the sketch of a set of the form to be computed in expected time and I/Os. We then describe how sampling can be used to maintain (independent) sketches of matrices that allow estimation to be performed in time o(n) if z is sufficiently large. This gives a simpler alternative to the sketching technique of Ganguly et al. (Proceedings of the 24th ACM Symposium on Principles of Database Systems (PODS ’05), pp. 259–270, 2005), and matches a space lower bound shown in that paper. Finally, we present experiments on real-world data sets that show the accuracy of both our methods to be significantly better than the worst-case analysis predicts.     

On surface meshes induced by level set functions

The zero level set of a continuous piecewise-affine function with respect to a consistent tetrahedral subdivision of a domain in ${\mathbb {R}}^3$ is a piecewise-planar hyper-surface. We prove that if a family of consistent tetrahedral subdivions satisfies the minimum angle condition, then after a simple postprocessing this zero level set becomes a consistent surface triangulation which satisfies the maximum angle condition. We treat an application of this result to the numerical solution of PDEs posed on surfaces, using a $P_1$ finite element space on such a surface triangulation. For this finite element space we derive optimal interpolation error bounds. We prove that the diagonally scaled mass matrix is well-conditioned, uniformly with respect to $h$ . Furthermore, the issue of conditioning of the stiffness matrix is addressed.     

Constant Thresholds Can Make Target Set Selection Tractable

Target Set Selection, which is a prominent NP-hard problem occurring in social network analysis and distributed computing, is notoriously hard both in terms of achieving useful polynomial-time approximation as well as fixed-parameter algorithms. Given an undirected graph, the task is to select a minimum number of vertices into a “target set” such that all other vertices will become active in the course of a dynamic process (which may go through several activation rounds). A vertex, equipped with a threshold value t, becomes active once at least t of its neighbors are active; initially, only the target set vertices are active. We contribute further insights into the existence of islands of tractability for Target Set Selection by spotting new parameterizations characterizing some sparse graphs as well as some “cliquish” graphs and developing corresponding fixed-parameter tractability and (parameterized) hardness results. In particular, we demonstrate that upper-bounding the thresholds by a constant may significantly alleviate the search for efficiently solvable, but still meaningful special cases of Target Set Selection.     

Shape preservation of scientific data through rational fractal splines

Fractal interpolation is a modern technique in approximation theory to fit and analyze scientific data. We develop a new class of $\mathcal C ^1$ - rational cubic fractal interpolation functions, where the associated iterated function system uses rational functions of the form $\frac{p_i(x)}{q_i(x)},$ where $p_i(x)$ and $q_i(x)$ are cubic polynomials involving two shape parameters. The rational cubic iterated function system scheme provides an additional freedom over the classical rational cubic interpolants due to the presence of the scaling factors and shape parameters. The classical rational cubic functions are obtained as a special case of the developed fractal interpolants. An upper bound of the uniform error of the rational cubic fractal interpolation function with an original function in $\mathcal C ^2$ is deduced for the convergence results. The rational fractal scheme is computationally economical, very much local, moderately local or global depending on the scaling factors and shape parameters. Appropriate restrictions on the scaling factors and shape parameters give sufficient conditions for a shape preserving rational cubic fractal interpolation function so that it is monotonic, positive, and convex if the data set is monotonic, positive, and convex, respectively. A visual illustration of the shape preserving fractal curves is provided to support our theoretical results.     

The Local Discontinuous Galerkin Method for the Fourth-Order Euler–Bernoulli Partial Differential Equation in One Space Dimension. Part I: Superconvergence Error Analysis

In this paper we develop and analyze a new superconvergent local discontinuous Galerkin (LDG) method for approximating solutions to the fourth-order Euler–Bernoulli beam equation in one space dimension. We prove the $L^2$ stability of the scheme and several optimal $L^2$ error estimates for the solution and for the three auxiliary variables that approximate derivatives of different orders. Our numerical experiments demonstrate optimal rates of convergence. We also prove superconvergence results towards particular projections of the exact solutions. More precisely, we prove that the LDG solution and its spatial derivatives (up to third order) are $\mathcal O (h^{k+3/2})$ super close to particular projections of the exact solutions for $k$ th-degree polynomial spaces while computational results show higher $\mathcal O (h^{k+2})$ convergence rate. Our proofs are valid for arbitrary regular meshes and for $P^k$ polynomials with $k\ge 1$ , and for periodic, Dirichlet, and mixed boundary conditions. These superconvergence results will be used to construct asymptotically exact a posteriori error estimates by solving a local steady problem on each element. This will be reported in Part II of this work, where we will prove that the a posteriori LDG error estimates for the solution and its derivatives converge to the true errors in the $L^2$ -norm under mesh refinement.     

A Simple Conforming Mixed Finite Element for Linear Elasticity on Rectangular Grids in Any Space Dimension

We construct a family of lower-order rectangular conforming mixed finite elements, in any space dimension. In the method, the normal stress is approximated by quadratic polynomials $\{1, x_{i}, x_{i}^{2}\}$ , the shear stress by bilinear polynomials $\{1, x_{i}, x_{j}, x_{i}x_{j}\}$ , and the displacement by linear polynomials $\{1, x_{i} \}$ . The number of total degrees of freedom (dof) per element is 10 plus 4 in 2D, and 21 plus 6 in 3D, while the previous record of least dof for conforming element is 17 plus 4 in 2D, and 72 plus 12 in 3D. The feature of this family of elements is, besides simplicity, that shape function spaces for both stress and displacement are independent of the spatial dimension $n$ . As a result of these choices, the theoretical analysis is independent of the spatial dimension as well. The well-posedness condition and the optimal a priori error estimate are proved. Numerical tests show the stability and effectiveness of these new elements.     

Structuring unreliable radio networks

In this paper we study the problem of building a constant-degree connected dominating set (CCDS), a network structure that can be used as a communication backbone, in the dual graph radio network model (Clementi et al. in J Parallel Distrib Comput 64:89–96, 2004; Kuhn et al. in Proceedings of the international symposium on principles of distributed computing 2009, Distrib Comput 24(3–4):187–206 2011, Proceedings of the international symposium on principles of distributed computing 2010). This model includes two types of links: reliable, which always deliver messages, and unreliable, which sometimes fail to deliver messages. Real networks compensate for this differing quality by deploying low-layer detection protocols to filter unreliable from reliable links. With this in mind, we begin by presenting an algorithm that solves the CCDS problem in the dual graph model under the assumption that every process $u$ is provided with a local link detector set consisting of every neighbor connected to $u$ by a reliable link. The algorithm solves the CCDS problem in $O\left( \frac{\varDelta \log ^2{n}}{b} + \log ^3{n}\right) $ rounds, with high probability, where $\varDelta $ is the maximum degree in the reliable link graph, $n$ is the network size, and $b$ is an upper bound in bits on the message size. The algorithm works by first building a Maximal Independent Set (MIS) in $\log ^3{n}$ time, and then leveraging the local topology knowledge to efficiently connect nearby MIS processes. A natural follow-up question is whether the link detector must be perfectly reliable to solve the CCDS problem. With this in mind, we first describe an algorithm that builds a CCDS in $O(\varDelta $ polylog $(n))$ time under the assumption of $O(1)$ unreliable links included in each link detector set. We then prove this algorithm to be (almost) tight by showing that the possible inclusion of only a single unreliable link in each process’s local link detector set is sufficient to require $\varOmega (\varDelta )$ rounds to solve the CCDS problem, regardless of message size. We conclude by discussing how to apply our algorithm in the setting where the topology of reliable and unreliable links can change over time.     

A Distributed O(1)-Approximation Algorithm for the Uniform Facility Location Problem

We investigate a metric facility location problem in a distributed setting. In this problem, we assume that each point is a client as well as a potential location for a facility and that the opening costs for the facilities and the demands of the clients are uniform. The goal is to open a subset of the input points as facilities such that the accumulated cost for the whole point set, consisting of the opening costs for the facilities and the connection costs for the clients, is minimized. We present a randomized distributed algorithm that computes in expectation an ${\mathcal {O}}(1)$ -approximate solution to the metric facility location problem described above. Our algorithm works in a synchronous message passing model, where each point is an autonomous computational entity that has its own local memory and that communicates with the other entities by message passing. We assume that each entity knows the distance to all the other entities, but does not know any of the other pairwise distances. Our algorithm uses three rounds of all-to-all communication with message sizes bounded to $\mathcal{O}(\log(n))$ bits, where n is the number of input points. We extend our distributed algorithm to constant powers of metric spaces. For a metric exponent ?≥1, we obtain a randomized ${\mathcal {O}}(1)$ -approximation algorithm that uses three rounds of all-to-all communication with message sizes bounded to $\mathcal{O}(\log(n))$ bits.     

Join-Reachability Problems in Directed Graphs

For a given collection \(\mathcal{G}\) of directed graphs we define the join-reachability graph of \(\mathcal{G}\) , denoted by \(\mathcal{J}(\mathcal{G})\) , as the directed graph that, for any pair of vertices u and v, contains a path from u to v if and only if such a path exists in all graphs of  \(\mathcal{G}\) . Our goal is to compute an efficient representation of  \(\mathcal{J}(\mathcal{G})\) . In particular, we consider two versions of this problem. In the explicit version we wish to construct the smallest join-reachability graph for  \(\mathcal{G}\) . In the implicit version we wish to build an efficient data structure, in terms of space and query time, such that we can report fast the set of vertices that reach a query vertex in all graphs of  \(\mathcal{G}\) . This problem is related to the well-studied reachability problem and is motivated by emerging applications of graph-structured databases and graph algorithms. We consider the construction of join-reachability structures for two graphs and develop techniques that can be applied to both the explicit and the implicit problems. First we present optimal and near-optimal structures for paths and trees. Then, based on these results, we provide efficient structures for planar graphs and general directed graphs.