An Implicit Riemannian Trust-Region Method for the Symmetric Generalized Eigenproblem

Baker, Christopher G.; Absil, Pierre-Antoine; Gallivan, Kyle A.

doi:10.1007/11758501_32

Cited by 12 publications

(5 citation statements)

References 15 publications

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“…The solution to (25) turns out to be closely related to a more general RayleighRitz form, as we now explain [6]. To see this, observe that (25) can be written as…”

Section: B Joint Mmse Solution For {Kf }mentioning

confidence: 86%

On Signal Processing Methods for MIMO Relay Architectures

Behbahani

Merched

Eltawil

2007

IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference

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Relay networks have received considerable attention recently, especially when limited size and power resources impose constraints on the number of antennas within a wireless sensor network. In this context, signal processing techniques play a fundamental role, and optimality within a given relay architecture can be achieved under several design criteria. In this paper, we extend recent optimal minimum-mean-square-error (MMSE) and SNR designs of relay networks to the corresponding multipleinput-multiple-output (MIMO) scenarios, whereby the source, relays and destination comprise multiple antennas. We shall investigate maximum SNR solutions subject to power constraints and zero-forcing (ZF) criteria, as well as approximate MMSE equalizers with specified target SNR and global power constraint.

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“…The solution to (25) turns out to be closely related to a more general RayleighRitz form, as we now explain [6]. To see this, observe that (25) can be written as…”

Section: B Joint Mmse Solution For {Kf }mentioning

confidence: 86%

On Signal Processing Methods for MIMO Relay Architectures

Behbahani

Merched

Eltawil

2007

IEEE GLOBECOM 2007-2007 IEEE Global Telecommunications Conference

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“…The problem of optimizing a function on a matrix manifold has received much attention in the scientific and engineering fields due to its peculiarity and capacity. Its applications include, but are not limited to, the study of eigenvalue problems [12,13,7,1,2,14,10,50,52,48,46,51], matrix low rank approximation [4,27], and nonlinear matrix equations [44,11]. Numerical methods for solving problems involving matrix manifolds rely on interdisciplinary inputs from differential geometry, optimization theory, and gradient flows.…”

Section: Riemannian Inexact Newton Methodsmentioning

confidence: 99%

Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems

2019

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Inverse eigenvalue and singular value problems have been widely discussed for decades. The well-known result is the Weyl-Horn condition, which presents the relations between the eigenvalues and singular values of an arbitrary matrix. This result by Weyl-Horn then leads to an interesting inverse problem, i.e., how to construct a matrix with desired eigenvalues and singular values. In this work, we do that and more. We propose an eclectic mix of techniques from differential geometry and the inexact Newton method for solving inverse eigenvalue and singular value problems as well as additional desired characteristics such as nonnegative entries, prescribed diagonal entries, and even predetermined entries. We show theoretically that our method converges globally and quadratically, and we provide numerical examples to demonstrate the robustness and accuracy of our proposed method. Having theoretical interest, we provide in the appendix a necessary and sufficient condition for the existence of a 2 × 2 real matrix, or even a nonnegative matrix, with prescribed eigenvalues, singular values, and main diagonal entries.Mathematics Subject Classification (2010) 15A29 · 65H17.Keywords Inverse eigenvalue and singular value problems, nonnegative matrices, Riemannian inexact Newton method.

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“…As the dimensionality of the Hilbert space increases, the optimization task of exact operator diagonalization formulated in Eq. (64) becomes numerically challenging despite active efforts to exploit existing Riemannian tools [56][57][58] for tractable solutions. The Krylov method overcomes this numerical difficulty by restricting the optimization to the lower-dimensional Krylov subspace,…”

Section: Appendixmentioning

confidence: 99%

Real-Time Krylov Theory for Quantum Computing Algorithms

Yizhi¹,

Klymko²,

Sud³

et al. 2022

Preprint

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Quantum computers provide alternative avenues to access ground and excited state properties of systems difficult to simulate on classical hardware. New approaches using subspaces generated by real-time evolution have shown efficiency in extracting eigenstate information, but the full capabilities of such approaches are still not understood. In recent work, we developed the variational quantum phase estimation (VQPE) method, a compact and efficient real-time algorithm to extract eigenvalues using quantum hardware. Here we build on that work by theoretically and numerically exploring a generalized Krylov scheme where the Krylov subspace is constructed through a parametrized real-time evolution, applicable to the VQPE algorithm as well as others. We establish an error bound that justifies the fast convergence of our spectral approximation. We also derive how the overlap with high energy eigenstates becomes suppressed from real-time subspace diagonalization and we visualize the process that shows the signature phase cancellations at specific eigenenergies. We investigate various algorithm implementations and consider performance when stochasticity is added to the target Hamiltonian in the form of spectral statistics. To demonstrate the practicality of such real-time evolution methods, we discuss its application to fundamental problems in quantum computation such as electronic structure predictions for strongly correlated systems.

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An Implicit Riemannian Trust-Region Method for the Symmetric Generalized Eigenproblem

Cited by 12 publications

References 15 publications

On Signal Processing Methods for MIMO Relay Architectures

On Signal Processing Methods for MIMO Relay Architectures

Riemannian inexact Newton method for structured inverse eigenvalue and singular value problems

Real-Time Krylov Theory for Quantum Computing Algorithms

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