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The CP tensor decomposition is used in applications such as machine learning and signal processing to discover latent low-rank structure in multidimensional data. Computing a CP decomposition via an alternating least squares (ALS) method reduces the problem to several linear least squares problems. The standard way to solve these linear least squares subproblems is to use the normal equations, which inherit special tensor structure that can be exploited for computational efficiency. However, the normal equations are sensitive to numerical ill-conditioning, which can compromise the results of the decomposition. In this paper, we develop versions of the CP-ALS algorithm using the QR decomposition and the singular value decomposition, which are more numerically stable than the normal equations, to solve the linear least squares problems. Our algorithms utilize the tensor structure of the CP-ALS subproblems efficiently, have the same complexity as the standard CP-ALS algorithm when the input is dense and the rank is small, and are shown via examples to produce more stable results when ill-conditioning is present. Our MATLAB implementation achieves the same running time as the standard algorithm for small ranks, and we show that the new methods can obtain lower approximation error.
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Rapid simulation of two-dimensional spectra with correlated anisotropic dimensions
A new algorithm has been developed to simulate two-dimensional (2D) spectra with correlated anisotropic frequencies faster and more accurately than previous methods. The technique uses finite-element numerical integration on the sphere and an interpolation scheme based on the Alderman–Solum–Grant algorithm. This method is particularly useful for numerical calculations of joint probability distribution functions involving quantities with a parametric orientation dependence. The technique’s efficiency also allows for practical least-squares fitting of experimental 2D solid-state nuclear magnetic resonance (NMR) datasets. The simulation method is illustrated for select 2D NMR methods, and a least-squares analysis is demonstrated in the extraction of paramagnetic shift and quadrupolar coupling tensors and their relative orientation from the experimental shifting-d echo 2H NMR spectrum of a NiCl2 · 2D2O salt.
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- Award ID(s):
- 2107636
- PAR ID:
- 10518957
- Publisher / Repository:
- American Institute of Physics
- Date Published:
- Journal Name:
- The Journal of Chemical Physics
- Volume:
- 160
- Issue:
- 13
- ISSN:
- 0021-9606
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1063/5.0200042
