213 research outputs found
Complexity Analysis and Efficient Measurement Selection Primitives for High-Rate Graph SLAM
Sparsity has been widely recognized as crucial for efficient optimization in
graph-based SLAM. Because the sparsity and structure of the SLAM graph reflect
the set of incorporated measurements, many methods for sparsification have been
proposed in hopes of reducing computation. These methods often focus narrowly
on reducing edge count without regard for structure at a global level. Such
structurally-naive techniques can fail to produce significant computational
savings, even after aggressive pruning. In contrast, simple heuristics such as
measurement decimation and keyframing are known empirically to produce
significant computation reductions. To demonstrate why, we propose a
quantitative metric called elimination complexity (EC) that bridges the
existing analytic gap between graph structure and computation. EC quantifies
the complexity of the primary computational bottleneck: the factorization step
of a Gauss-Newton iteration. Using this metric, we show rigorously that
decimation and keyframing impose favorable global structures and therefore
achieve computation reductions on the order of and , respectively,
where is the pruning rate. We additionally present numerical results
showing EC provides a good approximation of computation in both batch and
incremental (iSAM2) optimization and demonstrate that pruning methods promoting
globally-efficient structure outperform those that do not.Comment: Pre-print accepted to ICRA 201
An Object-Oriented Algorithmic Laboratory for Ordering Sparse Matrices
We focus on two known NP-hard problems that have applications in sparse matrix computations: the envelope/wavefront reduction problem and the fill reduction problem. Envelope/wavefront reducing orderings have a wide range of applications including profile and frontal solvers, incomplete factorization preconditioning, graph reordering for cache performance, gene sequencing, and spatial databases. Fill reducing orderings are generally limited toâbut an inextricable part ofâsparse matrix factorization.
Our major contribution to this field is the design of new and improved heuristics for these NP-hard problems and their efficient implementation in a robust, cross-platform, object-oriented software package. In this body of research, we (1) examine current ordering algorithms, analyze their asymptotic complexity, and characterize their behavior in model problems, (2) introduce new and improved algorithms that address deficiencies found in previous heuristics, (3) implement an object-oriented library of these algorithms in a robust, modular fashion without significant loss of efficiency, and (4) extend our algorithms and software to address both generalized and constrained problems. We stress that the major contribution is the algorithms and the implementation; the whole being greater than the sum of its parts.
The initial motivation for implementing our algorithms in object-oriented software was to manage the inherent complexity. During our research came the realization that the object-oriented implementation enabled new possibilities for augmented algorithms that would not have been as natural to generalize from a procedural implementation. Some extensions are constructed from a family of related algorithmic components, thereby creating a poly-algorithm that can adapt its strategy to the properties of the specific problem instance dynamically. Other algorithms are tailored for special constraints by aggregating algorithmic components and having them collaboratively generate the global ordering.
Our software laboratory, âSpindle,â implements state-of-the-art ordering algorithms for sparse matrices and graphs. We have used it to examine and augment the behavior of existing algorithms and test new ones. Its 40,000+ lines of C++ code includes a base library test drivers, sample applications, and interfaces to C, C++, Matlab, and PETSc. Spindle is freely available and can be built on a variety of UNIX platforms as well as WindowsNT
Sparse Cholesky factorization by greedy conditional selection
Dense kernel matrices resulting from pairwise evaluations of a kernel
function arise naturally in machine learning and statistics. Previous work in
constructing sparse approximate inverse Cholesky factors of such matrices by
minimizing Kullback-Leibler divergence recovers the Vecchia approximation for
Gaussian processes. These methods rely only on the geometry of the evaluation
points to construct the sparsity pattern. In this work, we instead construct
the sparsity pattern by leveraging a greedy selection algorithm that maximizes
mutual information with target points, conditional on all points previously
selected. For selecting points out of , the naive time complexity is
, but by maintaining a partial Cholesky factor we reduce
this to . Furthermore, for multiple () targets we
achieve a time complexity of , which is
maintained in the setting of aggregated Cholesky factorization where a selected
point need not condition every target. We apply the selection algorithm to
image classification and recovery of sparse Cholesky factors. By minimizing
Kullback-Leibler divergence, we apply the algorithm to Cholesky factorization,
Gaussian process regression, and preconditioning with the conjugate gradient,
improving over -nearest neighbors selection
A Fast Minimum Degree Algorithm and Matching Lower Bound
The minimum degree algorithm is one of the most widely-used heuristics for
reducing the cost of solving large sparse systems of linear equations. It has
been studied for nearly half a century and has a rich history of bridging
techniques from data structures, graph algorithms, and scientific computing. In
this paper, we present a simple but novel combinatorial algorithm for computing
an exact minimum degree elimination ordering in time, which improves on
the best known time complexity of and offers practical improvements
for sparse systems with small values of . Our approach leverages a careful
amortized analysis, which also allows us to derive output-sensitive bounds for
the running time of , where is
the number of unique fill edges and original edges that the algorithm
encounters and is the maximum degree of the input graph.
Furthermore, we show there cannot exist an exact minimum degree algorithm
that runs in time, for any , assuming
the strong exponential time hypothesis. This fine-grained reduction goes
through the orthogonal vectors problem and uses a new low-degree graph
construction called -fillers, which act as pathological inputs and cause any
minimum degree algorithm to exhibit nearly worst-case performance. With these
two results, we nearly characterize the time complexity of computing an exact
minimum degree ordering.Comment: 17 page
Book of Abstracts of the Sixth SIAM Workshop on Combinatorial Scientific Computing
Book of Abstracts of CSC14 edited by Bora UçarInternational audienceThe Sixth SIAM Workshop on Combinatorial Scientific Computing, CSC14, was organized at the Ecole Normale Supérieure de Lyon, France on 21st to 23rd July, 2014. This two and a half day event marked the sixth in a series that started ten years ago in San Francisco, USA. The CSC14 Workshop's focus was on combinatorial mathematics and algorithms in high performance computing, broadly interpreted. The workshop featured three invited talks, 27 contributed talks and eight poster presentations. All three invited talks were focused on two interesting fields of research specifically: randomized algorithms for numerical linear algebra and network analysis. The contributed talks and the posters targeted modeling, analysis, bisection, clustering, and partitioning of graphs, applied in the context of networks, sparse matrix factorizations, iterative solvers, fast multi-pole methods, automatic differentiation, high-performance computing, and linear programming. The workshop was held at the premises of the LIP laboratory of ENS Lyon and was generously supported by the LABEX MILYON (ANR-10-LABX-0070, Université de Lyon, within the program ''Investissements d'Avenir'' ANR-11-IDEX-0007 operated by the French National Research Agency), and by SIAM
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