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 standalone 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 71×1000×900 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.

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We evaluated a physics-based model for planning for magnetic resonance-guided laser interstitial thermal therapy for focal brain lesions. Linear superposition of analytical point source solutions to the steady-state Pennes bioheat transfer equation simulates laser-induced heating in brain tissue. The line integral of the photon attenuation from the laser source enables computation of the laser interaction with heterogeneous tissue. Magnetic resonance thermometry data sets (n = 31) were used to calibrate and retrospectively validate the model's thermal ablation prediction accuracy, which was quantified by the Dice similarity coefficient (DSC) between model-predicted and measured ablation regions (T > 57 °C). A Gaussian mixture model was used to identify independent tissue labels on pre-treatment anatomical magnetic resonance images. The tissue-dependent optical attenuation coefficients within these labels were calibrated using an interior point method that maximises DSC agreement with thermometry. The distribution of calibrated tissue properties formed a population model for our patient cohort. Model prediction accuracy was cross-validated using the population mean of the calibrated tissue properties. A homogeneous tissue model was used as a reference control. The median DSC values in cross-validation were 0.829 for the homogeneous model and 0.840 for the heterogeneous model. In cross-validation, the heterogeneous model produced a DSC higher than that produced by the homogeneous model in 23 of the 31 brain lesion ablations. Results of a paired, two-tailed Wilcoxon signed-rank test indicated that the performance improvement of the heterogeneous model over that of the homogeneous model was statistically significant (p < 0.01).

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I thank my wife, Julika Kaplan, for her constant love and encouragement. She is an inspiration and an unrelenting force of good. I thank my mother, Melanie Mitchell, who has always been first in supporting me through every difficulty and every success in my life. I thank my father, Randall Mitchell. I aspire to be as genuine and kind as he always was. I thank my both of my parents for enabling me to freely pursue my academic interests throughout my life and for encouraging my curiousity always. I thank my brothers, Matthew and Kyle Mitchell. They are true friends. I could not have done this without my family. They have my eternal love and gratitude. v