Jennifer B. Erway

Bio: Jennifer B. Erway is an academic researcher from Wake Forest University. The author has contributed to research in topics: Trust region & Matrix (mathematics). The author has an hindex of 9, co-authored 34 publications receiving 312 citations.

Iterative Methods for Finding a Trust-region Step

Abstract: We consider methods for large-scale unconstrained minimization based on finding an approximate minimizer of a quadratic function subject to a two-norm trust-region inequality constraint. The Steihaug-Toint method uses the conjugate-gradient algorithm to minimize the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary. The benefit of this approach is that an approximate solution may be obtained with minimal work and storage. However, the method does not allow the accuracy of a constrained solution to be specified. We propose an extension of the Steihaug-Toint method that allows a solution to be calculated to any prescribed accuracy. If the Steihaug-Toint point lies on the boundary, the constrained problem is solved on a sequence of evolving low-dimensional subspaces. Each subspace includes an accelerator direction obtained from a regularized Newton method applied to the constrained problem. A crucial property of this direction is that it can be computed by applying the conjugate-gradient method to a positive-definite system in both the primal and dual variables of the constrained problem. The method includes a parameter that allows the user to take advantage of the tradeoff between the overall number of function evaluations and matrix-vector products associated with the underlying trust-region method. At one extreme, a low-accuracy solution is obtained that is comparable to the Steihaug-Toint point. At the other extreme, a high-accuracy solution can be specified that minimizes the overall number of function evaluations at the expense of more matrix-vector products.

A Subspace Minimization Method for the Trust-Region Step

Abstract: We consider methods for large-scale unconstrained minimization based on finding an approximate minimizer of a quadratic function subject to a two-norm trust-region constraint. The Steihaug-Toint method uses the conjugate-gradient method to minimize the quadratic over a sequence of expanding subspaces until the iterates either converge to an interior point or cross the constraint boundary. However, if the conjugate-gradient method is used with a preconditioner, the Steihaug-Toint method requires that the trust-region norm be defined in terms of the preconditioning matrix. If a different preconditioner is used for each subproblem, the shape of the trust-region can change substantially from one subproblem to the next, which invalidates many of the assumptions on which standard methods for adjusting the trust-region radius are based. In this paper we propose a method that allows the trust-region norm to be defined independently of the preconditioner. The method solves the inequality constrained trust-region subproblem over a sequence of evolving low-dimensional subspaces. Each subspace includes an accelerator direction defined by a regularized Newton method for satisfying the optimality conditions of a primal-dual interior method. A crucial property of this direction is that it can be computed by applying the preconditioned conjugate-gradient method to a positive-definite system in both the primal and dual variables of the trust-region subproblem. Numerical experiments on problems from the CUTEr test collection indicate that the method can require significantly fewer function evaluations than other methods. In addition, experiments with general-purpose preconditioners show that it is possible to significantly reduce the number of matrix-vector products relative to those required without preconditioning.

On solving L-SR1 trust-region subproblems

Abstract: In this article, we consider solvers for large-scale trust-region subproblems when the quadratic model is defined by a limited-memory symmetric rank-one (L-SR1) quasi-Newton matrix. We propose a solver that exploits the compact representation of L-SR1 matrices. Our approach makes use of both an orthonormal basis for the eigenspace of the L-SR1 matrix and the Sherman---Morrison---Woodbury formula to compute global solutions to trust-region subproblems. To compute the optimal Lagrange multiplier for the trust-region constraint, we use Newton's method with a judicious initial guess that does not require safeguarding. A crucial property of this solver is that it is able to compute high-accuracy solutions even in the so-called hard case. Additionally, the optimal solution is determined directly by formula, not iteratively. Numerical experiments demonstrate the effectiveness of this solver.

Randomized SVD methods in hyperspectral imaging

Abstract: We present a randomized singular value decomposition (rSVD) method for the purposes of lossless compression, reconstruction, classification, and target detection with hyperspectral (HSI) data. Recent work in low-rank matrix approximations obtained from random projections suggests that these approximations are well suited for randomized dimensionality reduction. Approximation errors for the rSVD are evaluated on HSI, and comparisons aremade to deterministic techniques and as well as to other randomized low-rank matrix approximation methods involving compressive principal component analysis. Numerical tests on real HSI data suggest that the method is promising and is particularly effective for HSI data interrogation.

On Efficiently Computing the Eigenvalues of Limited-Memory Quasi-Newton Matrices

Abstract: In this paper, we consider the problem of efficiently computing the eigenvalues of limited-memory quasi-Newton matrices that exhibit a compact formulation. In addition, we produce a compact formula for quasi-Newton matrices generated by any member of the Broyden convex class of updates. Our proposed method makes use of efficient updates to the QR factorization that substantially reduce the cost of computing the eigenvalues after the quasi-Newton matrix is updated. Numerical experiments suggest that the proposed method is able to compute eigenvalues to high accuracy. Applications for this work include modified quasi-Newton methods and trust-region methods for large-scale optimization, the efficient computation of condition numbers and singular values, and sensitivity analysis.

Linear And Nonlinear Programming

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SVD Compression for Magnetic Resonance Fingerprinting in the Time Domain

Abstract: Magnetic resonance (MR) fingerprinting is a technique for acquiring and processing MR data that simultaneously provides quantitative maps of different tissue parameters through a pattern recognition algorithm. A predefined dictionary models the possible signal evolutions simulated using the Bloch equations with different combinations of various MR parameters and pattern recognition is completed by computing the inner product between the observed signal and each of the predicted signals within the dictionary. Though this matching algorithm has been shown to accurately predict the MR parameters of interest, one desires a more efficient method to obtain the quantitative images. We propose to compress the dictionary using the singular value decomposition, which will provide a low-rank approximation. By compressing the size of the dictionary in the time domain, we are able to speed up the pattern recognition algorithm, by a factor of between 3.4-4.8, without sacrificing the high signal-to-noise ratio of the original scheme presented previously.

On the Complexity of Steepest Descent, Newton's and Regularized Newton's Methods for Nonconvex Unconstrained Optimization Problems

Abstract: It is shown that the steepest-descent and Newton's methods for unconstrained nonconvex optimization under standard assumptions may both require a number of iterations and function evaluations arbitrarily close to $O(\epsilon^{-2})$ to drive the norm of the gradient below $\epsilon$. This shows that the upper bound of $O(\epsilon^{-2})$ evaluations known for the steepest descent is tight and that Newton's method may be as slow as the steepest-descent method in the worst case. The improved evaluation complexity bound of $O(\epsilon^{-3/2})$ evaluations known for cubically regularized Newton's methods is also shown to be tight.

Newton-type methods for non-convex optimization under inexact Hessian information

Abstract: We consider variants of trust-region and adaptive cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under certain condition on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve $$\varepsilon $$ -approximate second-order optimality which have been shown to be tight. Our Hessian approximation condition offers a range of advantages as compared with the prior works and allows for direct construction of the approximate Hessian with a priori guarantees through various techniques, including randomized sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding sub-sampled trust-region and adaptive cubic regularization methods.

Newton-Type Methods for Non-Convex Optimization Under Inexact Hessian Information

Abstract: We consider variants of trust-region and cubic regularization methods for non-convex optimization, in which the Hessian matrix is approximated. Under mild conditions on the inexact Hessian, and using approximate solution of the corresponding sub-problems, we provide iteration complexity to achieve $ \epsilon $-approximate second-order optimality which have shown to be tight. Our Hessian approximation conditions constitute a major relaxation over the existing ones in the literature. Consequently, we are able to show that such mild conditions allow for the construction of the approximate Hessian through various random sampling methods. In this light, we consider the canonical problem of finite-sum minimization, provide appropriate uniform and non-uniform sub-sampling strategies to construct such Hessian approximations, and obtain optimal iteration complexity for the corresponding sub-sampled trust-region and cubic regularization methods.