Conjugate gradient algorithm for optimization under unitary matrix constraint

Abrudan, Traian E.; Eriksson, Jan; Koivunen, Visa

doi:10.1016/j.sigpro.2009.03.015

Cited by 92 publications

(98 citation statements)

References 20 publications

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“…Because L is a quasiperiodic function along [90,91], the period of its fastest oscillation λ0 can be estimated from the largest absolute eigenvalue |ω|max of θ as λ0 " π{2|ω|max. This expression can be derived when the Lagrangian has fourth-order dependence on the orbitals [92], as is the case here due to the two-electron integrals.…”

Section: Computational Detailsmentioning

confidence: 99%

Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

2017

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We describe the implementation of orbital optimization for the models in the perfect pairing hierarchy [Lehtola et al, J. Chem. Phys. 145, 134110 (2016)]. Orbital optimization, which is generally necessary to obtain reliable results, is pursued at perfect pairing (PP) and perfect quadruples (PQ) levels of theory for applications on linear polyacenes, which are believed to exhibit strong correlation in the π space. While local minima and σ-π symmetry breaking solutions were found for PP orbitals, no such problems were encountered for PQ orbitals. The PQ orbitals are used for single-point calculations at PP, PQ and perfect hextuples (PH) levels of theory, both only in the π subspace, as well as in the full σπ valence space. It is numerically demonstrated that the inclusion of single excitations is necessary also when optimized orbitals are used. PH is found to yield good agreement with previously published density matrix renormalization group (DMRG) data in the π space, capturing over 95% of the correlation energy. Full-valence calculations made possible by our novel, efficient code reveal that strong correlations are weaker when larger bases or active spaces are employed than in previous calculations. The largest full-valence PH calculations presented correspond to a (192e,192o) problem.

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Section: Computational Detailsmentioning

confidence: 99%

Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

2017

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“…for the isometry g s using a conjugate gradient algorithm adapted to unitary manifolds 37 , we then iteratively update the left and right fixed points of Eq. (7) until convergence of Eq.…”

Section: Coarse-graining Transfer Matricesmentioning

confidence: 99%

Matrix product state renormalization

et al. 2016

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The truncation or compression of the spectrum of Schmidt values is inherent to the matrix product state (MPS) approximation of one-dimensional quantum ground states. We provide a renormalization group picture by interpreting this compression as an application of Wilson's numerical renormalization group along the imaginary time direction appearing in the path integral representation of the state. The location of the physical index is considered as an impurity in the transfer matrix and static MPS correlation functions are reinterpreted as dynamical impurity correlations. Coarse-graining the transfer matrix is performed using a hybrid variational ansatz based on matrix product operators, combining ideas of MPS and the multi-scale entanglement renormalization ansatz. Through numerical comparison with conventional MPS algorithms, we explicitly verify the impurity interpretation of MPS compression, as put forward by V. Zauner et al. [New J. Phys. 17, 053002 (2015)] for the transverse-field Ising model. Additionally, we motivate the conceptual usefulness of endowing MPS with an internal layered structure by studying restricted variational subspaces to describe elementary excitations on top of the ground state, which serves to elucidate a transparent renormalization group structure ingrained in MPS descriptions of ground states.

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“…As described in [3], this is the gradient update on a so-called Grassmann Manifold and forms the foundation of conjugate gradient algorithms. See [1] for signal processing optimization applications with unitary matrix constraint.…”

Section: Input Estimationmentioning

confidence: 99%

On estimation of the gain of a dynamical system

Wahlberg

Hjalmarsson

Stoica

2011

2011 Digital Signal Processing and Signal Processing Education Meeting (DSP/SPE)

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We analyze and compare methods to estimate the L 2 -gain (H ∞ -norm) of a stable linear dynamical system, that is the maximum of the absolute value of the corresponding frequency response. The standard approach is to estimate a parametric model of the system, which then is used for gain calculation. An asymptotic error variance expression for H ∞ -norm estimates based on finite impulse response (FIR) models is presented. We then study the problem of finding the minimum variance excitation signal that satisfies a given error variance bound for the FIR model gain estimate. It is shown that a sinusoidal signal with frequency equal to the peak frequency of the system is minimum variance optimal. The second approach to gain estimation is based on iterative experiments, and is inspired by numerical methods eigenvalue calculation. The asymptotic statistical properties of such gain estimation methods are compared to the FIR model approach. A transparent expression for the additional variance cost of using iterative experiments is derived. Finally, we present some ideas for input sequence estimation based on iterative experiments.

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Conjugate gradient algorithm for optimization under unitary matrix constraint

Cited by 92 publications

References 20 publications

Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

Orbital optimisation in the perfect pairing hierarchy: applications to full-valence calculations on linear polyacenes

Matrix product state renormalization

On estimation of the gain of a dynamical system

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