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Uniqueness of lowrank matrix completion by rigidity theory
, 2009
"... Abstract. The problem of completing a lowrank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and control. Most recent work had been focused on constr ..."
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Cited by 20 (1 self)
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Abstract. The problem of completing a lowrank matrix from a subset of its entries is often encountered in the analysis of incomplete data sets exhibiting an underlying factor model with applications in collaborative filtering, computer vision and control. Most recent work had been focused on constructing efficient algorithms for exact or approximate recovery of the missing matrix entries and proving lower bounds for the number of known entries that guarantee a successful recovery with high probability. A related problem from both the mathematical and algorithmic point of view is the distance geometry problem of realizing points in a Euclidean space from a given subset of their pairwise distances. Rigidity theory answers basic questions regarding the uniqueness of the realization satisfying a given partial set of distances. We observe that basic ideas and tools of rigidity theory can be adapted to determine uniqueness of lowrank matrix completion, where inner products play the role that distances play in rigidity theory. This observation leads to an efficient randomized algorithm for testing both local and global unique completion. Crucial to our analysis is a new matrix, which we call the completion matrix, that serves as the analogue of the rigidity matrix. Key words. Low rank matrices, missing values, rigidity theory, rigid graphs, iterative methods.
Composing parallel software efficiently with Lithe
 In Proc. of the SIGPLAN 2010 Conference on Programming Language Design and Implementation (PLDI
, 2010
"... Applications composed of multiple parallel libraries perform poorly when those libraries interfere with one another by obliviously using the same physical cores, leading to destructive resource oversubscription. This paper presents the design and implementation of Lithe, a lowlevel substrate that p ..."
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Cited by 16 (3 self)
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Applications composed of multiple parallel libraries perform poorly when those libraries interfere with one another by obliviously using the same physical cores, leading to destructive resource oversubscription. This paper presents the design and implementation of Lithe, a lowlevel substrate that provides the basic primitives and a standard interface for composing parallel codes efficiently. Lithe can be inserted underneath the runtimes of legacy parallel libraries to provide bolton composability without needing to change existing application code. Lithe can also serve as the foundation for building new parallel abstractions and libraries that automatically interoperate with one another. In this paper, we show versions of Threading Building Blocks (TBB) and OpenMP perform competitively with their original implementations when ported to Lithe. Furthermore, for two applications composed of multiple parallel libraries, we show that leveraging our substrate outperforms their original, even expertly tuned, implementations.
Multilevel submap based slam using nested dissection
 in IEEE/RSJ Intl. Conf. on Intelligent Robots and Systems (IROS
, 2010
"... Fig. 1. Our algorithm recursively partitions the SLAM graph into a submap tree, and the optimization runs from the leaves to the root. Following the treemap visualization [1], each rectangle represents a submap, and the subrectangles represent the submaps in the child level. The red and green dots ..."
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Cited by 11 (7 self)
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Fig. 1. Our algorithm recursively partitions the SLAM graph into a submap tree, and the optimization runs from the leaves to the root. Following the treemap visualization [1], each rectangle represents a submap, and the subrectangles represent the submaps in the child level. The red and green dots are robot poses and landmarks respectively. From left to right: 1). the finest level of submaps; 2). the coarsest level of submaps; 3). the optimized full map. Abstract — We propose a novel batch algorithm for SLAM problems that distributes the workload in a hierarchical way. We show that the original SLAM graph can be recursively partitioned into multiplelevel submaps using the nested dissection algorithm, which leads to the cluster tree, a powerful graph representation. By employing the nested dissection algorithm, our algorithm greatly minimizes the dependencies between two subtrees, and the optimization of the original SLAM graph can be done using a bottomup inference along the corresponding cluster tree. To speed up the computation, we also introduce a base node for each submap and use it to represent the rigid transformation of the submap in the global coordinate frame. As a result, the optimization moves the base nodes rather than the actual submap variables. We demonstrate that our algorithm is not only exact but also much faster than alternative approaches in both simulations and realworld experiments. I.
Probabilistic Simultaneous Pose and NonRigid Shape Recovery
"... We present an algorithm to simultaneously recover nonrigid shape and camera poses from point correspondences between a reference shape and a sequence of input images. The key novel contribution of our approach is in bringing the tools of the probabilistic SLAM methodology from a rigid to a deformabl ..."
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Cited by 8 (3 self)
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We present an algorithm to simultaneously recover nonrigid shape and camera poses from point correspondences between a reference shape and a sequence of input images. The key novel contribution of our approach is in bringing the tools of the probabilistic SLAM methodology from a rigid to a deformable domain. Under the assumption that the shape may be represented as a weighted sum of deformation modes, we show that the problem of estimating the modal weights along with the camera poses, may be probabilistically formulated as a maximum a posterior estimate and solved using an iterative least squares optimization. An extensive evaluation on synthetic and real data, shows that our approach has several significant advantages over current approaches, such as performing robustly under large amounts of noise and outliers, and neither requiring to track points over the whole sequence nor initializations close from the ground truth solution. 1.
1Sparse reconstruction of sharp point set surfaces
 ACM T. Graphic
"... We introduce an 1sparse method for the reconstruction of a piecewise smooth point set surface. The technique is motivated by recent advancements in sparse signal reconstruction. The assumption underlying our work is that common objects, even geometrically complex ones, can typically be characterize ..."
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Cited by 6 (1 self)
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We introduce an 1sparse method for the reconstruction of a piecewise smooth point set surface. The technique is motivated by recent advancements in sparse signal reconstruction. The assumption underlying our work is that common objects, even geometrically complex ones, can typically be characterized by a rather small number of features. This, in turn, naturally lends itself to incorporating the powerful notion of sparsity into the model. The sparse reconstruction principle gives rise to a reconstructed point set surface that consists mainly of smooth modes, with the residual of the objective function strongly concentrated near sharp features. Our technique is capable of recovering orientation and positions of highly noisy point sets. The global nature of the optimization yields a sparse solution and avoids local minima. Using an interiorpoint logbarrier solver with a customized preconditioning scheme, the solver for the corresponding convex optimization problem is competitive and the results are of high quality.
User’s Guide for SuiteSparseQR, a multifrontal multithreaded sparse QR factorization package
, 2009
"... SuiteSparseQR is an implementation of the multifrontal sparse QR factorization method. Parallelism is exploited both in the BLAS and across different frontal matrices using Intel’s Threading Building Blocks, a sharedmemory programming model for modern multicore architectures. It can obtain a substa ..."
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SuiteSparseQR is an implementation of the multifrontal sparse QR factorization method. Parallelism is exploited both in the BLAS and across different frontal matrices using Intel’s Threading Building Blocks, a sharedmemory programming model for modern multicore architectures. It can obtain a substantial fraction of the theoretical peak performance of a multicore computer. The package is written in C++ with user interfaces for MATLAB, C, and C++. Both real and complex sparse matrices are supported. 1
ACKNOWLEDGMENTS
, 2009
"... I am fortunate to have the support of some truly wonderful people. First and foremost, I would like to thank my advisor Dr.Timothy Davis who has been an excellent mentor. I will be indebted to him forever for the trust he placed in me even before I started graduate studies and while doing this study ..."
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I am fortunate to have the support of some truly wonderful people. First and foremost, I would like to thank my advisor Dr.Timothy Davis who has been an excellent mentor. I will be indebted to him forever for the trust he placed in me even before I started graduate studies and while doing this study. He has not only taught me technical nuances, but also helped me appreciate the beauty in everything from perfect software to poetry. I am grateful for his valuable advice, patience and for making me a better person. I would like to acknowledge Dr.Gene Golub who was in my PhD committee for his kind and encouraging words. He gave the motivation to solve the sparse singular value decomposition problem. I would also like to thank Dr.Alin Dobra, Dr.William Hager, Dr.Jorg Peters, and Dr.Meera Sitharam for serving in my PhD committee. On the personal side, I would like to thank my wife Sridevi whose sacrifices and constant support made this whole effort possible. She has always been with me whenever I needed her. I could have never completed this study without her beside me. My son Akilan and daughter Sowmya deserve a special mention for the time spent with them rejuvenated me while working on this dissertation.