Predicting the structure of sparse orthogonal factors

AbstractThe problem of correctly predicting the structures of the orthogonal factors Q and R from the structure of a matrix A with full column rank is considered. Recently Hare, Johnson, Olesky, and van den Driessche have described a method to predict these structures, and they have shown that corresponding to any specified nonzero element in the predicted structures of Q or R, there exists a matrix with the given structure whose factor has a nonzero in that position. In this paper this method is shown to satisfy a stronger property: there exist matrices with the structure of A whose factors have exactly the predicted structures. These results use matching theory, the Dulmage-Mendelsohn decomposition of bipartite graphs, and techniques from algebra. The proof technique shows that if values are assigned randomly to the nonzeros in A, then with high probability the elements predicted to be nonzero in the factors have nonzero values. It is shown that this stronger requirement cannot be satisfied for orthogonal factorization with column pivoting. In addition, efficient algorithms for computing the structures of the factors are designed, and the relationship between the structure of Q and the Householder array is described

Matching Phosphorylation Response Patterns of Antigen-Receptor-Stimulated T Cells Via Flow Cytometry

Background When flow cytometric data on mixtures of cell populations are collected from samples under different experimental conditions, computational methods are needed (a) to classify the samples into similar groups, and (b) to characterize the changes within the corresponding populations due to the different conditions. Manual inspection has been used in the past to study such changes, but high-dimensional experiments necessitate developing new computational approaches to this problem. A robust solution to this problem is to construct distinct templates to summarize all samples from a class, and then to compare these templates to study the changes across classes or conditions. Results We designed a hierarchical algorithm, flowMatch, to first match the corresponding clusters across samples for producing robust meta-clusters, and to then construct a high-dimensional template as a collection of meta-clusters for each class of samples. We applied the algorithm on flow cytometry data obtained from human blood cells before and after stimulation with anti-CD3 monoclonal antibody, which is reported to change phosphorylation responses of memory and naive T cells. TheflowMatch algorithm is able to construct representative templates from the samples before and after stimulation, and to match corresponding meta-clusters across templates. The templates of the pre-stimulation and post-stimulation data corresponding to memory and naive T cell populations clearly show, at the level of the meta-clusters, the overall phosphorylation shift due to the stimulation. Conclusions We concisely represent each class of samples by a template consisting of a collection of meta-clusters (representative abstract populations). Using flowMatch, the meta-clusters across samples can be matched to assess overall differences among the samples of various phenotypes or time-points

Protocols for Disease Classification from Mass Spectrometry Data

We report our results in classifying protein matrix-assisted laser desorption/ionizationtime of flight mass spectra obtained from serum samples into diseased and healthy groups. We discuss in detail five of the steps in preprocessing the mass spectral data for biomarker discovery, as well as our criterion for choosing a small set of peaks for classifying the samples. Cross-validation studies with four selected proteins yielded misclassification rates in the 10-15% range for all the classification methods. Three of these proteins or protein fragments are down-regulated and one up-regulated in lung cancer, the disease under consideration in this data set. When cross-validation studies are performed, care must be taken to ensure that the test set does not influence the choice of the peaks used in the classification. Misclassification rates are lower when both the training and test sets are used to select the peaks used in classification versus when only the training set is used. This expectation was validated for various statistical discrimination methods when thirteen peaks were used in cross-validation studies. One particular classification method, a linear support vector machine, exhibited especially robust performance when the number of peaks was varied from four to thirteen, and when the peaks were selected from the training set alone. Experiments with the samples randomly assigned to the two classes confirmed that misclassification rates were significantly higher in such cases than those observed with the true data. This indicates that our findings are indeed significant. We found closely matching masses in a database for protein expression in lung cancer for three of the four proteins we used to classify lung cancer. Data from additional samples, increased experience with the performance of various preprocessing techniques, and affirmation of the biological roles of the proteins that help in classification, will strengthen our conclusions in the future

Combinatorial Algorithms for Computing Column Space Bases That Have Sparse Inverses

Abstract. This paper presents a new combinatorial approach towards constructing a sparse, implicit basis for the null space of a sparse, under-determined matrix. Our approach is to compute a column space basis of that has a sparse inverse, which could be used to represent a null space basis in implicit form. We investigate three different algorithms for computing column space bases: two greedy algorithms implemented using graph matchings, and a third, which employs a divide and conquer strategy implemented with hypergraph partitioning followed by a matching. Our results show that for many matrices from linear programming, structural analysis, and circuit simulation, it is possible to compute column space bases having sparse inverses, contrary to conventional wisdom. The hypergraph partitioning method yields sparser basis inverses and has low computational time requirements, relative to the greedy approaches. We also discuss the complexity of selecting a column space basis when it is known that such a basis exists in block diagonal form with a given small block size. Key words. sparse column space basis, sparse null space basis, block angular matrix, block diagonal matrix, matching, hypergraph partitioning, inverse of a basis AMS subject classifications. 65F50, 68R10, 90C20 1. Introduction. Man

Interactively Cutting and Constraining Vertices in Meshes Using Augmented Matrices

We present a finite-element solution method that is well suited for interactive simulations of cutting meshes in the regime of linear elastic models. Our approach features fast updates to the solution of the stiffness system of equations to account for real-time changes in mesh connectivity and boundary conditions. Updates are accomplished by augmenting the stiffness matrix to keep it consistent with changes to the underlying model, without refactoring the matrix at each step of cutting. The initial stiffness matrix and its Cholesky factors are used to implicitly form and solve a Schur complement system using an iterative solver. As changes accumulate over many simulation timesteps, the augmented solution method slows down due to the size of the augmented matrix. However, by periodically refactoring the stiffness matrix in a concurrent background process, fresh Cholesky factors that incorporate recent model changes can replace the initial factors. This controls the size of the augmented matrices and provides a way to maintain a fast solution rate as the number of changes to a model grows. We exploit sparsity in the stiffness matrix, the right-hand-side vectors and the solution vectors to compute the solutions fast, and show that the time complexity of the update steps is bounded linearly by the size of the Cholesky factor of the initial matrix. Our complexity analysis and experimental results demonstrate that this approach scales well with problem size. Results for cutting and deformation of 3D linear elastic models are reported for meshes representing the brain, eye, and model problems with element counts up to 167,000; these show the potential of this method for real-time interactivity. An application to limbal incisions for surgical correction of astigmatism, for which linear elastic models and small deformations are sufficient, is included