Sparse Gaussian elimination with partial pivoting computes the factorization PAQ = LU of a sparse matrix A, where the row ordering P is selected during factorization using standard partial pivoting with row interchanges. The goal is to select a column preordering, Q, based solely on the nonzero pattern of A such that the factorization remains as sparse as possible, regardless of the subsequent choice of P. The choice of Q can have a dramatic impact on the number of nonzeros in L and U. One scheme for determining a good column ordering for A is to compute a symmetric ordering that reduces fill-in in the Cholesky factorization of ATA. This approach, which requires the sparsity structure of ATA to be computed, can be expensive both in

Solving the SLAM problem is one way to enable a robot to explore, map, and navigate in a previously unknown environment. We investigate smoothing approaches as a viable alternative to extended Kalman filter-based solutions to the problem. In particular, we look at approaches that factorize either the associated information matrix or the measurement Jacobian into square root form. Such techniques have several significant advantages over the EKF: they are faster yet exact, they can be used in either batch or incremental mode, are better equipped to deal with non-linear process and measurement models, and yield the entire robot trajectory, at lower cost for a large class of SLAM problems. In addition, in an indirect but dramatic way, column ordering heuristics automatically exploit the locality inherent in the geographic nature of the SLAM problem. In this paper we present the theory underlying these methods, along with an interpretation of factorization in terms of the graphical model associated with the SLAM problem. We present both simulation results and actual SLAM experiments in large-scale environments that underscore the potential of these methods as an alternative to EKF-based approaches. 1

An approximate minimum degree column ordering algorithm (COLAMD) for preordering an unsymmetric sparse matrix A prior to...

This article presents an algebraic analysis of a mesh-compression technique called high-pass quantization [Sorkine et al. 2003]. In high-pass quantization, a rectangular matrix based on the mesh topological Laplacian is applied to the vectors of the Cartesian coordinates of a polygonal mesh. The resulting vectors, called δ-coordinates, are then quantized. The applied matrix is a function of the topology of the mesh and the indices of a small set of mesh vertices (anchors) but not of the location of the vertices. An approximation of the geometry can be reconstructed from the quantized δ-coordinates and the spatial locations of the anchors. In this article, we show how to algebraically bound the reconstruction error that this method generates. We show that the small singular value of the transformation matrix can be used to bound both the quantization error and the rounding error which is due to the use of floating-point arithmetic. Furthermore, we prove a bound on this singular value. The bound is a function of the topology of the mesh and of the selected anchors. We also propose a new anchor-selection algorithm, inspired by this bound. We show experimentally that the method is effective and that the computed upper bound on the error is not too pessimistic.

In this report a new version of the multifrontal sparse QR factorization routine sqr, originally by Matstoms, for general sparse matrices is described and evaluated. In the previous version the orthogonal factor Q is discarded due to storage considerations. The new version provides Q and uses the multifrontal structure to store this orthogonal factor in a compact way. A new data class with overloaded operators is implemented in Matlab to provide an easy usage of the compact orthogonal factors. This implicit way of storing the orthogonal factor also results in faster computation and application of Q and Q T . Examples are given, where the new version is up to four times faster when computing only R and up to 1000 times faster when computing both Q and R, than the built-in function qr in Matlab. The sqr package is available at URL: http://www.mai.liu.se/~milun/sls/. Key words: QR factorization, sparse problems, multifrontal method, orthogonal factorization. 1 Introduction. Let A 2 IR...

In this paper we present a novel data structure, the Bayes tree, which exploits the connections between graphical model inference and sparse linear algebra. The proposed data structure provides a new perspective on an entire class of simultaneous localization and mapping (SLAM) algorithms. Similar to a junction tree, a Bayes tree encodes a factored probability density, but unlike the junction tree it is directed and maps more naturally to the square root information matrix of the SLAM problem. This makes it eminently suited to encode the sparse nature of the problem, especially in a smoothing and mapping (SAM) context. The inherent sparsity of SAM has already been exploited in the literature to produce efficient solutions in both batch and online mapping. The graphical model perspective allows us to develop a novel incremental algorithm that seamlessly incorporates reordering and relinearization. This obviates the need for expensive periodic batch operations from previous approaches, which negatively affect the performance and detract from the intended online nature of the algorithm. The new method is evaluated using simulated and real-world datasets in both landmark and pose SLAM settings. I.

Abstract We present a novel data structure, the Bayes tree, that provides an algorithmic foundation enabling a better understanding of existing graphical model inference algorithms and their connection to sparse matrix factorization methods. Similar to a clique tree, a Bayes tree encodes a factored probability density, but unlike the clique tree it is directed and maps more naturally to the square root information matrix of the simultaneous localization and mapping (SLAM) problem. In this paper, we highlight three insights provided by our new data structure. First, the Bayes tree provides a better understanding of batch matrix factorization in terms of probability densities. Second, we show how the fairly abstract updates to a matrix factorization translate to a simple editing of the Bayes tree and its conditional densities. Third, we apply the Bayes tree to obtain a completely novel algorithm for sparse nonlinear incremental optimization, that combines incremental updates with fluid relinearization of a reduced set of variables for efficiency, combined with fast convergence to the exact solution. We also present a novel strategy for incremental variable reordering to retain sparsity. We evaluate our algorithm on standard datasets in both landmark and pose SLAM settings.

We present a novel data structure, the Bayes tree, that provides an algorithmic foundation enabling a better understanding of existing graphical model inference algorithms and their connection to sparse matrix factorization methods. Similar to a clique tree, a Bayes tree encodes a factored probability density, but unlike the clique tree it is directed and maps more naturally to the square root information matrix of the simultaneous localization and mapping (SLAM) problem. In this paper, we highlight three insights provided by our new data structure. First, the Bayes tree provides a better understanding of the matrix factorization in terms of probability densities. Second, we show how the fairly abstract updates to a matrix factorization translate to a simple editing of the Bayes tree and its conditional densities. Third, we apply the Bayes tree to obtain a completely novel algorithm for sparse nonlinear incremental optimization, named iSAM2, which achieves improvements in efficiency through incremental variable re-ordering and fluid relinearization, eliminating the need for periodic batch steps. We analyze various properties of iSAM2 in detail, and show on a range of real and simulated datasets that our algorithm compares favorably with other recent mapping algorithms in both quality and efficiency.