An Extended Sparse Linear System for Exact Marginal Inference on Factor Graphs

Computer Solution of Large Linear Systems

We introduce a new framework for feature grouping based on factor graphs, which are graphical models that encode interactions among arbitrary numbers of random variables. The ability of factor graphs to express interactions higher than pairwise order (the highest order encountered in most graphical models used in computer vision) is useful for modeling a variety of pattern recognition problems. In particular, we show how this property makes factor graphs a natural framework for performing grouping and segmentation, which we apply to the problem of finding text in natural scenes. We demonstrate an implementation of our factor graph-based algorithm for finding text on a Nokia camera phone, which is intended for eventual use in a camera phone system that finds and reads text (such as street signs) in natural environments for blind users.

The simulation of several physical phenomena, arising from a wide class of scientific fields, requires solving large sparse linear systems effectively. In order to exploit appropriately the available supercomputing infrastructures, designing efficient and scalable solvers for large sparse linear systems is necessary. A hybrid parallel algebraic iterative method based on the Schur complement method and multiprojection techniques that utilizes semiaggregated subdomains is proposed. The graph corresponding to the coefficient matrix of the sparse linear system is partitioned by a graph partitioning algorithm and the Schur complement problem is solved by a preconditioned Krylov subspace method. The preconditioning scheme is based on domain decomposition and the resulting subdomains consist of fine and aggregated components from the inner boundaries. Moreover, two variants for approximating the local Schur complements, derived from the semiaggregation procedure, are proposed in order to improve memory requirements and preprocessing time. The proposed method is suitable for distributed memory systems with multicore nodes and has improved convergence behavior for a large number of workstations. Numerical and comparative results concerning the applicability, the convergence behavior, and the parallel performance of the proposed scheme are given.

Semipositive matrices (matrices that map at least one nonnegative vector to a positive vector) and minimally semipositive matrices (semipositive matrices whose no column-deleted submatrix is semipositive) are well studied in matrix theory. In this article, this notion is revisited and new results are presented. It is shown that the set of all $m \times n$ minimally semipositive matrices contains a basis for the linear space of all $m \times n$ matrices. Apart from considerations involving principal pivot transforms and the Schur complement, results on semipositivity and/or minimal semipositivity for the following classes of matrices are presented: intervals of rectangular matrices, skew-symmetric and almost skew-symmetric matrices, copositive matrices, $N$-matrices, almost $N$-matrices and almost $P$-matrices.

Given a linear system Ax = b , one can construct a related “normal equations” system AA T y = b, x = A T y . Björck and Elfving have shown that the SSOR algorithm, applied to the normal equations, can be accelerated by the conjugate gradient algorithm (CG). The resulting algorithm, called CGMN, is error-reducing and in theory it always converges even when the equation system is inconsistent and/or nonsquare. SSOR on the normal equations is equivalent to the Kaczmarz algorithm (KACZ), with a fixed relaxation parameter, run in a double (forward and backward) sweep on the original equations. CGMN was tested on nine well-known large and sparse linear systems obtained by central-difference discretization of elliptic convection-diffusion partial differential equations (PDEs). Eight of the PDEs were strongly convection-dominated, and these are known to produce very stiff systems with large off-diagonal elements. CGMN was compared with some of the foremost state-of-the art Krylov subspace methods: restarted GMRES, Bi-CGSTAB, and CGS. These methods were tested both with and without various preconditioners. CGMN converged in all the cases, while none of the preceding algorithm/preconditioner combinations achieved this level of robustness. Furthermore, on varying grid sizes, there was only a gradual increase in the number of iterations as the grid was refined. On the eight convection-dominated cases, the initial convergence rate of CGMN was better than all the other combinations of algorithms and preconditioners, and the residual decreased monotonically. The CGNR algorithm was also tested, and it was as robust as CGMN, but slower.

Evolutionary credibility risk premium

Spectral methods for mesh processing and analysis rely on the eigenvalues, eigenvectors, or eigenspace projections derived from appropriately defined mesh operators to carry out desired tasks. Early work in this area can be traced back to the seminal paper by Taubin in 1995, where spectral analysis of mesh geometry based on a combinatorial Laplacian aids our understanding of the low‐pass filtering approach to mesh smoothing. Over the past 15 years, the list of applications in the area of geometry processing which utilize the eigenstructures of a variety of mesh operators in different manners have been growing steadily. Many works presented so far draw parallels from developments in fields such as graph theory, computer vision, machine learning, graph drawing, numerical linear algebra, and high‐performance computing. This paper aims to provide a comprehensive survey on the spectral approach, focusing on its power and versatility in solving geometry processing problems and attempting to bridge the gap between relevant research in computer graphics and other fields. Necessary theoretical background is provided. Existing works covered are classified according to different criteria: the operators or eigenstructures employed, application domains, or the dimensionality of the spectral embeddings used. Despite much empirical success, there still remain many open questions pertaining to the spectral approach. These are discussed as we conclude the survey and provide our perspective on possible future research.

Domain adaptation (DA) and Transfer Learning (TL) are terms coined by the machine learning community, particular computer vision. However, the chemometrics community has been working on similar problems (with chemical and spectroscopic contexts) for much longer and these techniques go under the moniker of calibration transfer and maintenance (CTM). Both the machine learning and chemometrics communities often encounter the same problem: their prediction models have a tendency to rely too much on the distribution of the data on which they have been trained. In practice, we are constantly updating a predictive model on data that evolves over time. Ramin Nikzad-Langerodi received his MSc in Biochemistry and Biophysics from the University of Zurich, Switzerland and his PhD in Pharmaceutical Sciences from the University of Vienna, Austria. He currently leads a Data Science research group at the Software Competence Center Hagenberg (SCCH), Austria. His research interests are in analytical chemistry, chemometrics and process analytical technology (PAT). He has published several scientific papers at the intersection between chemometrics and artificial intelligence. Erik Andries resides in New Mexico and currently wears many hats: mathematics instructor at Central New Mexico Community College, visiting research scientist at the Center for Advanced Research Computing (CARC) at the University of New Mexico (UNM), and an active consultant for biomedical start-ups in the Albuquerque and San Francisco Bay areas. His interests revolve around the intersection of chemometrics and numerical linear algebra, particularly the incorporation of chemical and spectroscopic domain knowledge into analyte prediction models. Unsupervised domain adaptation (i.e., calibration transfer and maintenance using secondary samples without reference measurements) is his current obsession. Previously, Erik Andries was a research scientist at InLight Solutions working on data analysis algorithms for non-invasive glucose sensing. Before that, he had a joint postdoctoral fellowship at the UNM Department of Pathology (spatio-kinetic Monte-Carlo methods and stochastic differential equations) and Sandia National Laboratories (biomolecular imaging). In 2004, he received a Ph.D. in applied mathematics from UNM, where he was a research assistant at the UNM Cancer Research Center. He is originally from New Orleans, Louisiana, USA.

Well-quasi-ordering of matrices under Schur complement and applications to directed graphs

Symmetry is one of apparent features of objects in both the man-made world and nature. In computer vision, finding axes of skewed symmetry is an important task. This paper presents a new algorithm for detecting the skewed symmetry axes of closed contours according to their geometric properties obtained before. The algorithm runs in linear time in the number of contour points without knowing the number of axes of skewed symmetry in advance. The experimental results show that the method we propose is reliable, accurate and quick. The contribution of the paper has many great meanings in recognizing the symmetry of 3-D objects and recovering their structure.

We present and release in open source format a sparse linear solver which efficiently exploits heterogeneous parallel computers. The solver can be easily integrated into scientific applications that need to solve large and sparse linear systems on modern parallel computers made of hybrid nodes hosting Nvidia Graphics Processing Unit (GPU) accelerators. The work extends previous efforts of some of the authors in the exploitation of a single GPU accelerator and proposes an implementation, based on the hybrid MPI-CUDA software environment, of a Krylov-type linear solver relying on an efficient Algebraic MultiGrid (AMG) preconditioner already available in the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">BootCMatchG</i> library. Our design for the hybrid implementation has been driven by the best practices for minimizing data communication overhead when multiple GPUs are employed, yet preserving the efficiency of the GPU kernels. Strong and weak scalability results of the new version of the library on well-known benchmark test cases are discussed. Comparisons with the Nvidia AmgX solution show a speedup, in the solve phase, up to 2.0x.

Frequency-domain elastic wave modeling relies on an efficient linear solver for the large, sparse, ill-conditioned linear system derived from the discretization of the elastic wave equation. Direct solvers are mostly based on LU decomposition. These methods are efficient for multiple right-hand sides problems, but they require significant memory resources. Conversely, iterative solvers benefit from the sparsity of the system, but they require sophisticated preconditioners to converge due to the system ill-conditioning. In this study, we investigate the performance of an iterative method named CARP-CG for frequency-domain elastic wave modeling. The CARP-CG method transforms the original system into a symmetric positive semi-definite system by cyclic row-projections. This system is efficiently solved with the conjugate gradient (CG) method. The cyclic row-projection transformation can be seen as a purely algebraic preconditioning technique which is easy to implement. The algorithm can be parallelized through a row-block decomposition combined with component-averaging operations. Numerical experiments on the 2D frequency-domain elastic problem with high Poisson's ratio exhibit a good scalability of CARP-CG. Comparisons for different frequencies between CARP-CG and standard Krylov iterative solvers (GMRES and CG on the normal equations) emphasize the robustness and the fast convergence of the method.

A spatial domain decomposition method for parabolic optimal control problems

Parallel two level block ILU preconditioning techniques for solving large sparse linear systems

Hessian matrix is the square matrix of second partial derivatives of a scalar-valued function and is well known for object recognition in computer vision and medical shape analysis. Previous curvature based polyp detection algorithms generate myriad of false positives. Hessian-matrix based method, however, is more sensitive to local shape features, so easily reduce false positives. Calculation of Hessian matrix on 3D CT data and Eigen decomposition of the matrix gives three Eigen values and vectors at each voxel. Using these Eigen values, we can figure out which type of intensity structures (blob, line, and sheet-like) is on the given voxel. We focus on detecting blob-like object automatically. In the inner colonic wall structures, blob-like, line-like, and sheet-like objects represent polyps, folds and wall, respectively. In addition, to improve the performance of the algorithm, Gaussian blurring factor and shape threshold parameters are optimized. Before Hessian matrix calculation, smoothing the given region using Gaussian kernel with small deviation is necessary to enhance local intensity structures. To optimize the parameters and validate this method, we have produced anthropomorphic pig phantoms. Fourteen phantoms with 103 polyps (16 polyps <6mm, 87 >= 6mm) were used. CT scan was performed with 1mm slice thickness. Our detection algorithm found 84 polyps (81.6%) correctly. Average number of false positives is 7.9 at each CT scan. This results show that our algorithm is clinically applicable for polyp detection, because of high sensitivity and relatively low false positive detections.