59,415 research outputs found
Householder triangularization of a quasimatrix
A standard algorithm for computing the QR factorization of a matrix A is Householder triangularization. Here this idea is generalized to the situation in which A is a quasimatrix, that is, a “matrix” whose “columns” are functions defined on an interval [a,b]. Applications are mentioned to quasimatrix leastsquares fitting, singular value decomposition, and determination of ranks, norms, and condition numbers, and numerical illustrations are presented using the chebfun system
On the formulation and uses of SVD-based generalized curvatures
2016 Summer.Includes bibliographical references.In this dissertation we consider the problem of computing generalized curvature values from noisy, discrete data and applications of the provided algorithms. We first establish a connection between the Frenet-Serret Frame, typically defined on an analytical curve, and the vectors from the local Singular Value Decomposition (SVD) of a discretized time-series. Next, we expand upon this connection to relate generalized curvature values, or curvatures, to a scaled ratio of singular values. Initially, the local singular value decomposition is centered on a point of the discretized time-series. This provides for an efficient computation of curvatures when the underlying curve is known. However, when the structure of the curve is not known, for example, when noise is present in the tabulated data, we propose two modifications. The first modification computes the local singular value decomposition on the mean-centered data of a windowed selection of the time-series. We observe that the mean-center version increases the stability of the curvature estimations in the presence of signal noise. The second modification is an adaptive method for selecting the size of the window, or local ball, to use for the singular value decomposition. This allows us to use a large window size when curvatures are small, which reduces the effects of noise thanks to the use of a large number of points in the SVD, and to use a small window size when curvatures are large, thereby best capturing the local curvature. Overall we observe that adapting the window size to the data, enhances the estimates of generalized curvatures. The combination of these two modifications produces a tool for computing generalized curvatures with reasonable precision and accuracy. Finally, we compare our algorithm, with and without modifications, to existing numerical curvature techniques on different types of data such as that from the Microsoft Kinect 2 sensor. To address the topic of action segmentation and recognition, a popular topic within the field of computer vision, we created a new dataset from this sensor showcasing a pose space skeletonized representation of individuals performing continuous human actions as defined by the MSRC-12 challenge. When this data is optimally projected onto a low-dimensional space, we observed each human motion lies on a distinguished line, plane, hyperplane, etc. During transitions between motions, either the dimension of the optimal subspace significantly, or the trajectory of the curve through pose space nearly reverses. We use our methods of computing generalized curvature values to identify these locations, categorized as either high curvatures or changing curvatures. The geometric characterization of the time-series allows us to segment individual,or geometrically distinct, motions. Finally, using these segments, we construct a methodology for selecting motions to conjoin for the task of action classification
Two harmonic Jacobi--Davidson methods for computing a partial generalized singular value decomposition of a large matrix pair
Two harmonic extraction based Jacobi--Davidson (JD) type algorithms are
proposed to compute a partial generalized singular value decomposition (GSVD)
of a large regular matrix pair. They are called cross product-free (CPF) and
inverse-free (IF) harmonic JDGSVD algorithms, abbreviated as CPF-HJDGSVD and
IF-HJDGSVD, respectively. Compared with the standard extraction based JDGSVD
algorithm, the harmonic extraction based algorithms converge more regularly and
suit better for computing GSVD components corresponding to interior generalized
singular values. Thick-restart CPF-HJDGSVD and IF-HJDGSVD algorithms with some
deflation and purgation techniques are developed to compute more than one GSVD
components. Numerical experiments confirm the superiority of CPF-HJDGSVD and
IF-HJDGSVD to the standard extraction based JDGSVD algorithm.Comment: 24 pages, 5 figure
Preconditioned Algorithm for Difference of Convex Functions with applications to Graph Ginzburg-Landau Model
In this work, we propose and study a preconditioned framework with a graphic
Ginzburg-Landau functional for image segmentation and data clustering by
parallel computing. Solving nonlocal models is usually challenging due to the
huge computation burden. For the nonconvex and nonlocal variational functional,
we propose several damped Jacobi and generalized Richardson preconditioners for
the large-scale linear systems within a difference of convex functions
algorithms framework. They are efficient for parallel computing with GPU and
can leverage the computational cost. Our framework also provides flexible step
sizes with a global convergence guarantee. Numerical experiments show the
proposed algorithms are very competitive compared to the singular value
decomposition based spectral method
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