A derivation of the Levinson algorithm for solving linear systems with symmetric positive definite Toeplitz matrix

A Schur-based algorithm for computing bounds to the smallest eigenvalue of a symmetric positive definite Toeplitz matrix

Bounds for the condition number are obtained which indicate the dependence of the conditioning of a symmetric positive definite Toeplitz matrix on the sign distribution of its reflection coefficients. This dependence is illustrated with numerical examples. Similar bounds for symmetric positive definite block Toeplitz matrices are also presented.

We study an iterative, locally quadratically convergent algorithm for solving Toeplitz systems of equations from [R. P. Brent, F. G. Gustavson and D. Y. Y. Yun. “Fast solution of Toeplitz systems of equations and computation of Padé approximations”,J. Algorithms, 1:259–295, 1980]. We introduce a new iterative algorithm that is locally quadratically convergent when used to solve symmetric positive definite Toeplitz systems. We present a set of numerical experiments on randomly generated symmetric positive definite Toeplitz matrices. In these experiments, our algorithm performed significantly better than the previously proposed algorithm.

A novel method for computing the minimal eigenvalue of a symmetric positive-definite Toeplitz matrix is presented. Similar to the algorithm of Cybenko and Van Loan, it is a combination of bisection and a root finding method. Both phases of the method are accelerated considerably by rational Hermitian interpolation of the secular equation. For randomly generated test problems of dimension 800 the average number of linear systems which must be solved to determine the smallest eigenvalue is 6.6, which reduces the computational cost of the method of Cybenko and Van Loan to approximately 35%. The method includes a rigorous error bound.

Symmetric positive definite (SPD) matrices used as feature descriptors in image recognition are usually high dimensional. Traditional manifold learning is only applicable for reducing the dimension of high-dimensional vector-form data.For high-dimensional SPD matrices, directly using manifold learning algorithms to reduce the dimension of matrix-form data is impossible. The SPD matrix must first be transformed into a long vector, and then the dimension of this vector must be reduced. However, this approach breaks the spatial structure of the SPD matrix space. To overcome this limitation, we propose a new dimension reduction algorithm on SPD matrix space to transform high-dimensional SPD matrices into low-dimensional SPD matrices. Our work is based on the fact that the set of all SPD matrices with the same size has a Lie group structure, and we aim to transform the manifold learning to the SPD matrix Lie group. We use the basic idea of the manifold learning algorithm called locality preserving projection (LPP) to construct the corresponding Laplacian matrix on the SPD matrix Lie group. Thus, we call our approach Lie-LPP to emphasize its Lie group character. We present a detailed algorithm analysis and show through experiments that Lie-LPP achieves effective results on human action recognition and human face recognition.

In this paper, we analyze the spectra of the preconditioned matrices arising from discretized multi-dimensional Riesz spatial fractional diffusion equations. The finite difference method is employed to approximate the multi-dimensional Riesz fractional derivatives, which will generate symmetric positive definite ill-conditioned multi-level Toeplitz matrices. The preconditioned conjugate gradient method with a preconditioner based on the sine transform is employed to solve the resulting linear system. Theoretically, we prove that the spectra of the preconditioned matrices are uniformly bounded in the open interval (1/2,3/2) and thus the preconditioned conjugate gradient method converges linearly. The proposed method can be extended to multi-level Toeplitz matrices generated by functions with zeros of fractional order. Our theoretical results fill in a vacancy in the literature. Numerical examples are presented to demonstrate our new theoretical results in the literature and show the convergence performance of the proposed preconditioner that is better than other existing preconditioners.

Symmetric Positive Definite (SPD) matrices have been used in many fields of medical data analysis. Many Riemannian metrics have been defined on this manifold but the choice of the Riemannian structure lacks a set of principles that could lead one to choose properly the metric. This drives us to introduce the principle of balanced metrics that relate the affine-invariant metric with the Euclidean and inverse-Euclidean metric, or the Bogoliubov-Kubo-Mori metric with the Euclidean and log-Euclidean metrics. We introduce two new families of balanced metrics, the mixed-power-Euclidean and the mixed-power-affine metrics and we discuss the relation between this new principle of balanced metrics and the concept of dual connections in information geometry.

The geometry of mixed-Euclidean metrics on symmetric positive definite matrices

<p><b>Metric Learning has proved valuable in information retrieval and classification problems, with many algorithms using statistical formulations to their advantage. Optimal Transport, a powerful framework for computing distances between probability distributions, has seen significant uptake in Machine Learning and some applications in Metric Learning. The Bures-Wasserstein distance, a particular formulation of Optimal Transport, has seen limited application in Metric Learning despite being well suited to the field. In this thesis Linear Metric Learning algorithms incorporating the Bures-Wasserstein distance are developed, utilising its novel properties. The first set of algorithms we contribute use the Bures-Wasserstein distance for learning linear metrics using Cyclic Projections, Riemannian barycenters and gradient-based algorithms. The convergence properties and classification performance of these algorithms are then compared to benchmark Linear Metric Learning algorithms. Our algorithms achieve competitive classification performance on feature and image datasets and converge reliability.</b></p> <p>The second set of contributions extend Linear Metric Learning with the Bures-Wasserstein distance to a low-rank context to address issues related to scalability. An extension of the previous Cyclic Projections algorithm and a Riemannian optimisation method are developed. Convergence and classification experiments analogous to the full-rank case are conducted to examine how the reduced rank affects both of these performance metrics. Both algorithms converge reliably under a variety of synthetic conditions. Reduction in rank generally decreased the training time for algorithms until effects caused by the size of the training data dominated. Reducing rank either left classification error unchanged or increased it. With respect to other low-rank Linear Metric Learning algorithms our algorithms again performed competitively with respect to classification error. </p> <p>The last set of contributions consider Linear Metric Learning on Symmetric Positive Definite (SPD) matrices. Some datasets are expressed as or are suited to a SPD representation. As such, Metric Learning algorithms that respect the structure of these matrices have found success when adapted to this kind of data. Another extension of the Cyclic Projections algorithm is developed to use the Generalised Bures-Wasserstein distance on SPD matrices. Two other methods based on a ground metric formulation of the Bures-Wasserstein distance for learning linear metrics on SPD data are also introduced. One ground metric formulation uses an approximation (inspired by the Sinkhorn distance) to increase its scalability. Convergence experiments on synthetic problems adapted to SPD matrices are conducted on the ground metric algorithms to establish their convergence properties. Classification experiments are also conducted with respect to several types of SPD data. While the Cyclic Projections method is competitive, the ground metric algorithms have poor performance. Despite their poor classification performance, the approximation method does evaluate faster than the Generalised Bures-Wasserstein distance, potentially offering some scalability improvements.</p>

We present a distributed memory algorithm for the approximate hierarchical of symmetric positive definite (SPD) matrices. Our method is based on the distributed memory GOFMM, an algorithm that appeared in SC18 (doi:10.1109/SC.2018.00018). GOFMM constructs a hierarchical matrix approximation of an arbitrary SPD matrix that compresses the matrix by creating low-rank approximations of the off-diagonal blocks. GOFMM method has no guarantees of success for arbitrary SPD matrices. (This is similar to the SVD; not every matrix admits a good low-rank approximation.) But for many SPD matrices, GOFMM does enable compression that results in fast matrix-vector multiplication that can reach N logN time—as opposed to N2 required for a dense matrix. GOFMM supports shared and distributed memory parallelism. In this paper, we build an approximate based on the Hierarchically Semi-Separable (HSS) compression of the GOFMM. This requires O(N) work (given the compressed matrix) and O(N=p) + O(log p) time on p MPI processes (assuming a hypercube topology). The previous state-of-the-art required O(N logN) work. We present the algorithm, discuss its complexity, and present weak and strong scaling results for the factorization and phases of our algorithm. We also discuss the performance of the inexact ULV as a preconditioner for a few exemplary large dense linear systems. In our largest run, we were able to factorize a 67M-by-67M matrix in less than one second; and solve a system with 64 right-hand sides in less than one-tenth of a second. This run was on 6,144 Intel Skylake cores on the SKX partition of the Stampede2 system at the Texas Advanced Computing Center.

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix $$T$$ satisfies property A, one can easily construct a diagonal matrix $$\hat{D}$$ such that $$\hat{D}T\hat{D}$$ has the lowest condition number over all matrices $$DTD$$ , for any choice of diagonal matrix $$D$$ . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form.

Representing images and videos with Symmetric Positive Definite (SPD) matrices, and considering the Riemannian geometry of the resulting space, has been shown to yield high discriminative power in many visual recognition tasks. Unfortunately, computation on the Riemannian manifold of SPD matrices -especially of high-dimensional ones- comes at a high cost that limits the applicability of existing techniques. In this paper, we introduce algorithms able to handle high-dimensional SPD matrices by constructing a lower-dimensional SPD manifold. To this end, we propose to model the mapping from the high-dimensional SPD manifold to the low-dimensional one with an orthonormal projection. This lets us formulate dimensionality reduction as the problem of finding a projection that yields a low-dimensional manifold either with maximum discriminative power in the supervised scenario, or with maximum variance of the data in the unsupervised one. We show that learning can be expressed as an optimization problem on a Grassmann manifold and discuss fast solutions for special cases. Our evaluation on several classification tasks evidences that our approach leads to a significant accuracy gain over state-of-the-art methods.

In this paper, we will derive a new formula for the Morse index for the linear Hamiltonian systems:$$ J \dot x+Ax=0, $$where matrix $J$ is the standard symplectic structure matrix of order $2N$ and $A$ is a constant, symmetric positive definite matrix of order $2N$, $N$ is any natural number.

An extension of Levinson’s algorithm for solving linear systems with symmetric indefinite Toeplitz matrices is presented. This new algorithm is able to “look ahead” and, if necessary, use block Gaussian elimination to skip all ill-conditioned leading principal submatrices encountered during the recursive processing. This makes the new algorithm numerically stable for a broader class of symmetric Toeplitz matrices than the standard Levinson algorithm. In addition, a reliable condition number estimate is produced. The overhead is typically small.

The electroencephalography (EEG) signals are easily contaminated by various artifacts and noise, which induces a domain shift in each subject and significant pattern variability among different subjects. Therefore, it hinders the improvement of EEG classification accuracy in the cross-subject learning scenario. Convolutional neural networks (CNNs) have been extensively applied to EEG-based Brain-Computer Interfaces (BCIs) by virtue of the capability of performing automatic feature extraction and classification. However, they have been mainly applied to the within-subject classification which would consume lots of time for training and calibration. Thus, it limits the further applications of CNNs in BCIs. In order to build a robust classification algorithm for a calibration-less BCI system, we propose an end-to-end model that transforms the EEG signals into symmetric positive definite (SPD) matrices and captures the features of SPD matrices by using a CNN. To avoid the time-consuming calibration and ensure the application of the proposed model, we use the meta-transfer-learning (MTL) method to learn the essential features from different subjects. We validate our model by making extensive experiments on three public motor-imagery datasets. The experimental results demonstrate the effectiveness of our proposed method in the cross-subject learning scenario.