Several tensor eigenpair definitions have been put forth in the past decade, but these can all be unified under generalized tensor eigenpair framework, introduced by Chang, Pearson, and Zhang [J. Math. Anal. Appl., 350 (2009), pp. 416--422]. Given mth-order, n-dimensional real-valued symmetric tensors @math and @math , the goal is to find @math and @math such that @math . Different choices for @math yield different versions of the tensor eigenvalue problem. We present our generalized eigenproblem adaptive power (GEAP) method for solving the problem, which is an extension of the shifted symmetric higher-order power method (SS-HOPM) for finding Z-eigenpairs. A major drawback of SS-HOPM is that its performance depended on choosing an appropriate shift, but our GEAP method also includes an adaptive method for choosing the shift automatically.
In general, Han's method represents an alternative approach to solving the generalized tensor eigenpair problem. In @cite_9 , Han's method was compared to SS-HOPM with a fixed shift (for Z-eigenpairs only) and was superior. However, GEAP is usually as fast as Han's method and perhaps a little more robust in terms of its convergence behavior. The speed being similar is thanks to the adaptive shift in GEAP. It may be that Han's method could avoid problems of converging to incorrect solutions with tighter tolerances, but then the speed would be slower.
@cite_9
2070028074 2051434435 2073599217 2963626304
Recent work on eigenvalues and eigenvectors for tensors of order @math has been motivated by applications in blind source separation, magnetic resonance imaging, molecular conformation, and more. In this paper, we consider methods for computing real symmetric-tensor eigenpairs of the form @math subject to @math , which is closely related to optimal rank-1 approximation of a symmetric tensor. Our contribution is a shifted symmetric higher-order power method (SS-HOPM), which we show is guaranteed to converge to a tensor eigenpair. SS-HOPM can be viewed as a generalization of the power iteration method for matrices or of the symmetric higher-order power method. Additionally, using fixed point analysis, we can characterize exactly which eigenpairs can and cannot be found by the method. Numerical examples are presented, including examples from an extension of the method to finding complex eigenpairs. We study the numerical performance of a limited memory quasi-Newton method for large scale optimization, which we call the L-BFGS method. We compare its performance with that of the method developed by Buckley and LeNir (1985), which combines cycles of BFGS steps and conjugate direction steps. Our numerical tests indicate that the L-BFGS method is faster than the method of Buckley and LeNir, and is better able to use additional storage to accelerate convergence. We show that the L-BFGS method can be greatly accelerated by means of a simple scaling. We then compare the L-BFGS method with the partitioned quasi-Newton method of Griewank and Toint (1982a). The results show that, for some problems, the partitioned quasi-Newton method is clearly superior to the L-BFGS method. However we find that for other problems the L-BFGS method is very competitive due to its low iteration cost. We also study the convergence properties of the L-BFGS method, and prove global convergence on uniformly convex problems. A method for solving a partial algebraic eigenvalues problem is constructed. It exploits tensor structure of eigenvectors in two-dimensional case. For a symmetric matrix represented in tensor format, the method finds low-rank approximations to the eigenvectors corresponding to the smallest eigenvalues. For sparse matrices, execution time and required memory for the proposed method are proportional to the square root of miscellaneous overall number of unknowns, whereas this dependence is usually linear. To maintain tensor structure of vectors at each iteration step, low-rank approximations are performed, which introduces errors into the original method. Nevertheless, the new method was proved to converge. Convergence rate estimates are obtained for various tensor modifications of the abstract one-step method. It is shown how the convergence of a multistep method can be derived from the convergence of the corresponding one-step method. Several modifications of the method with an low-rank approximation techniques were implemented on the basis of the block conjugate gradient method. Their performance is compared on numerical examples. Summary Alternating least squares (ALS) is often considered the workhorse algorithm for computing the rank-R canonical tensor approximation, but for certain problems, its convergence can be very slow. The nonlinear conjugate gradient (NCG) method was recently proposed as an alternative to ALS, but the results indicated that NCG is usually not faster than ALS. To improve the convergence speed of NCG, we consider a nonlinearly preconditioned NCG (PNCG) algorithm for computing the rank-R canonical tensor decomposition. Our approach uses ALS as a nonlinear preconditioner in the NCG algorithm. Alternatively, NCG can be viewed as an acceleration process for ALS. We demonstrate numerically that the convergence acceleration mechanism in PNCG often leads to important pay-offs for difficult tensor decomposition problems, with convergence that is significantly faster and more robust than for the stand-alone NCG or ALS algorithms. We consider several approaches for incorporating the nonlinear preconditioner into the NCG algorithm that have been described in the literature previously and have met with success in certain application areas. However, it appears that the nonlinearly PNCG approach has received relatively little attention in the broader community and remains underexplored both theoretically and experimentally. Thus, this paper serves several additional functions, by providing in one place a concise overview of several PNCG variants and their properties that have only been described in a few places scattered throughout the literature, by systematically comparing the performance of these PNCG variants for the tensor decomposition problem, and by drawing further attention to the usefulness of nonlinearly PNCG as a general tool. In addition, we briefly discuss the convergence of the PNCG algorithm. In particular, we obtain a new convergence result for one of the PNCG variants under suitable conditions, building on known convergence results for non-preconditioned NCG. Copyright © 2014 John Wiley & Sons, Ltd.