text: We present Nesterov-type acceleration techniques for Alternating Least Squares (ALS) methods applied to canonical tensor decomposition. While Nesterov acceleration turns gradient descent into an optimal first-order method for convex problems by adding a momentum term with a specific weight sequence, a direct application of this method and weight sequence to ALS results in erratic convergence behaviour or divergence. This is so because the tensor decomposition problem is non-convex and ALS is accelerated instead of gradient descent. We investigate how line search or restart mechanisms can be used to obtain effective acceleration. We first consider a cubic line search (LS) strategy for determining the momentum weight, showing numerically that the combined Nesterov-ALS-LS approach is competitive with or superior to other recently developed nonlinear acceleration techniques for ALS, including acceleration by nonlinear conjugate gradients (NCG) and LBFGS. As an alternative, we consider various restarting techniques, some of which are inspired by previously proposed restarting mechanisms for Nesterov's accelerated gradient method. We study how two key parameters, the momentum weight and the restart condition, should be set. Our extensive empirical results show that the Nesterov-accelerated ALS methods with restart can be dramatically more efficient than the stand-alone ALS or Nesterov accelerated gradient method, when problems are ill-conditioned or accurate solutions are required. The resulting methods perform competitively with or superior to existing acceleration methods for ALS, and additionally enjoy the benefit of being much simpler and easier to implement. On a large and ill-conditioned 71 x 1000 x 900 tensor consisting of readings from chemical sensors used for tracking hazardous gases, the restarted Nesterov-ALS method outperforms any of the existing methods by a large factor.
  @cite_8: 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.
  @cite_7: 
  @cite_11: A new algorithm is presented for computing a canonical rank- @math tensor approximation that has minimal distance to a given tensor in the Frobenius norm, where the canonical rank- @math tensor consists of the sum of @math rank-one tensors. Each iteration of the method consists of three steps. In the first step, a tentative new iterate is generated by a stand-alone one-step process, for which we use alternating least squares (ALS). In the second step, an accelerated iterate is generated by a nonlinear generalized minimal residual (GMRES) approach, recombining previous iterates in an optimal way, and essentially using the stand-alone one-step process as a preconditioner. In particular, the nonlinear extension of GMRES we use that was proposed by Washio and Oosterlee in [Electron. Trans. Numer. Anal., 15 (2003), pp. 165--185] for nonlinear partial differential equation problems (which is itself related to other existing acceleration methods for nonlinear equation systems). In the third step, a line search is perfor...
summary: Several ALS-specific nonlinear acceleration techniques have been developed recently as discussed in the introduction @cite_11 @cite_8 @cite_7 . These algorithms often have complex forms and incur significant computational overhead. Our Nesterov-ALS scheme is simple and straightforward to implement, and only incurs a small amount of computational overhead.