Nonnegative Tensor Completion via Integer Optimization

Unlike matrix completion, tensor completion does not have an algorithm that is known to achieve the information-theoretic sample complexity rate. This paper develops a new algorithm for the special case of completion for nonnegative tensors. We prove that our algorithm converges in a linear (in numerical tolerance) number of oracle steps, while achieving the information-theoretic rate. Our approach is to define a new norm for nonnegative tensors using the gauge of a particular 0-1 polytope; integer linear programming can, in turn, be used to solve linear separation problems over this polytope. We combine this insight with a variant of the Frank-Wolfe algorithm to construct our numerical algorithm, and we demonstrate its effectiveness and scalability through computational experiments using a laptop on tensors with up to one-hundred million entries

Hybrid Parallel Imaging and Compressed Sensing MRI Reconstruction with GRAPPA Integrated Multi-loss Supervised GAN

Objective: Parallel imaging accelerates the acquisition of magnetic resonance imaging (MRI) data by acquiring additional sensitivity information with an array of receiver coils resulting in reduced phase encoding steps. Compressed sensing magnetic resonance imaging (CS-MRI) has achieved popularity in the field of medical imaging because of its less data requirement than parallel imaging. Parallel imaging and compressed sensing (CS) both speed up traditional MRI acquisition by minimizing the amount of data captured in the k-space. As acquisition time is inversely proportional to the number of samples, the inverse formation of an image from reduced k-space samples leads to faster acquisition but with aliasing artifacts. This paper proposes a novel Generative Adversarial Network (GAN) namely RECGAN-GR supervised with multi-modal losses for de-aliasing the reconstructed image. Methods: In contrast to existing GAN networks, our proposed method introduces a novel generator network namely RemU-Net integrated with dual-domain loss functions including weighted magnitude and phase loss functions along with parallel imaging-based loss i.e., GRAPPA consistency loss. A k-space correction block is proposed as refinement learning to make the GAN network self-resistant to generating unnecessary data which drives the convergence of the reconstruction process faster. Results: Comprehensive results show that the proposed RECGAN-GR achieves a 4 dB improvement in the PSNR among the GAN-based methods and a 2 dB improvement among conventional state-of-the-art CNN methods available in the literature. Conclusion and significance: The proposed work contributes to significant improvement in the image quality for low retained data leading to 5x or 10x faster acquisition.Comment: 12 pages, 11 figure

Avoiding bad steps in Frank Wolfe variants

The analysis of Frank Wolfe (FW) variants is often complicated by the presence of different kinds of "good" and "bad" steps. In this article we aim to simplify the convergence analysis of some of these variants by getting rid of such a distinction between steps, and to improve existing rates by ensuring a sizable decrease of the objective at each iteration. In order to do this, we define the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper stopping conditions are satisfied. This technique allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as a linear convergence rate under a Kurdyka-Lojasiewicz (KL) property. While this setting has been widely studied for proximal gradient type methods, to our knowledge, it has not been analyzed before for the Frank Wolfe variants under study. An angle condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant, is used to carry out our analysis. We prove that this condition is satisfied on polytopes by the away step Frank-Wolfe (AFW), the pairwise Frank-Wolfe (PFW), and the Frank-Wolfe method with in face directions (FDFW).Comment: See arXiv:2008.09781 for an extended version of the pape

A unifying framework for the analysis of projection-free first-order methods under a sufficient slope condition

The analysis of projection-free first order methods is often complicated by the presence of different kinds of "good" and "bad" steps. In this article, we propose a unifying framework for projection-free methods, aiming to simplify the converge analysis by getting rid of such a distinction between steps. The main tool employed in our framework is the Short Step Chain (SSC) procedure, which skips gradient computations in consecutive short steps until proper stopping conditions are satisfied. This technique allows us to give a unified analysis and converge rates in the general smooth non convex setting, as well as convergence rates under a Kurdyka-Lojasiewicz (KL) property, a setting that, to our knowledge, has not been analyzed before for the projection-free methods under study. In this context, we prove local convergence rates comparable to those of projected gradient methods under the same conditions. Our analysis relies on a sufficient slope condition, ensuring that the directions selected by the methods have the steepest slope possible up to a constant among feasible directions. This condition is satisfied, among others, by several Frank-Wolfe (FW) variants on polytopes, and by some projection-free methods on convex sets with smooth boundary.Comment: 36 pages, 4 figure

Second-order Conditional Gradient Sliding

Constrained second-order convex optimization algorithms are the method of choice when a high accuracy solution to a problem is needed, due to their local quadratic convergence. These algorithms require the solution of a constrained quadratic subproblem at every iteration. We present the \emph{Second-Order Conditional Gradient Sliding} (SOCGS) algorithm, which uses a projection-free algorithm to solve the constrained quadratic subproblems inexactly. When the feasible region is a polytope the algorithm converges quadratically in primal gap after a finite number of linearly convergent iterations. Once in the quadratic regime the SOCGS algorithm requires $\mathcal{O}(\log(\log 1/\varepsilon))$ first-order and Hessian oracle calls and $\mathcal{O}(\log (1/\varepsilon) \log(\log1/\varepsilon))$ linear minimization oracle calls to achieve an $\varepsilon$-optimal solution. This algorithm is useful when the feasible region can only be accessed efficiently through a linear optimization oracle, and computing first-order information of the function, although possible, is costly

Column Generation-Based Techniques for Intensity-Modulated Radiation Therapy (IMRT) and Volumetric Modulated Arc Therapy (VMAT) Treatment Planning

RÉSUMÉ: Les statistiques ont estimé à environ 14,1 millions le nombre de cas de cancer en 2018 dans le monde, et qui devrait passer à 24 millions d’ici 2035. La radiothérapie est l’une des premières méthodes de traitement du cancer, qu’environ 50% des patients reçoivent au cours de leur maladie. Cette méthode endommage le matériel génétique des cellules cancéreuses, détruisant ainsi leur capacité de reproduction. Cependant, les cellules normales sont également affectées par le rayonnement ; par conséquent, le traitement doit être effectué de manière à maximiser la dose de rayonnement aux tumeurs, tout en minimisant les effets néfastes des radiations sur les tissus sains. Les techniques d’optimisation sont utilisées afin de déterminer la dose et la position du rayonnement à administrer au corps du patient. Ce projet aborde la radiothérapie externe à travers la radiothérapie par modulation d’intensité (IMRT), ainsi qu’une nouvelle forme appelée modulation d’intensité volumétrique par thérapie par arcs (VMAT). En IMRT, un nombre fini de directions sont déterminées pour le rayonnement du faisceau, tandis qu’en VMAT l’accélérateur linéaire tourne autour du corps du patient alors que le faisceau est allumé. Cette technologie permet de modifier dynamiquement la forme du faisceau et le débit de dose pendant le traitement. Le problème de planification du traitement consiste à choisir une séquence de distribution des formes de faisceaux, à optimiser le dé bit de dose du faisceau et à déterminer la vitesse de rotation du portique, si nécessaire. Cette recherche tire profit de la méthode de génération de colonnes, en tant que méthode d’optimisation efficace en particulier pour les problèmes à grande échelle. Cette technique permet d’améliorer le temps de traitement et les objectifs cliniques non linéaires et non convexes, dans la planification de traitement en VMAT. Un nouveau modèle multi-objectif de génération de colonnes pour l’IMRT est également développé. Dans le premier essai, nous développons un nouvel algorithme de génération de colonnes qui optimise le compromis entre le temps et la qualité du traitement délivré pour la planification de traitement en VMAT. Pour ce faire, une génération simultanée de colonnes et de rangées est développée, afin de relier les colonnes, contenant la configuration des ouvertures de faisceaux, aux rangées du modèle, représentant la restriction de temps de traitement. De plus, nous proposons une technique de regroupement par grappe modifiée, afin d’agréger des éléments de volume similaires du corps du patient, et de réduire efficacement le nombre de contraintes dans le modèle. Les résultats de calcul montrent qu’il est possible d’obtenir un traitement de haute qualité sur quatre processeurs en parallèle. Dans le deuxième essai, nous développons une approche de planification automatique intégrant les critères de l’histogramme dose-volume (DVH). Les DVH sont la représentation de dose la plus courante pour l’évaluation de la qualité de traitement en technologie VMAT. Nous profitons de la procédure itérative de génération de colonnes pour ajuster les paramètres du modèle lors de la génération d’ouverture, et répondre aux critères DVH non linéaires, sans tenir compte des contraintes dures dans le modèle. Les résultats sur les cas cliniques montrent que notre méthodologie a été significativement améliorée, pour obtenir des plans cliniquement acceptables sans intervention humaine par rapport à une simple optimisation VMAT. De plus, la comparaison avec un système de planification de traitement commercial existant montre que la qualité des plans obtenus à partir de la méthode proposée, en particulier pour les tissus sains, est largement meilleure alors que le temps de calcul est moindre. Dans le troisième essai, nous abordons la planification de traitement en IMRT, qui est formulée comme un problème d’optimisation convexe à grande échelle, avec un espace de faisabilité simplex. Nous intégrons d’abord une nouvelle approche de solution basée sur la méthode Frank-Wolfe, appelée Blended Conditional Gradients, dans la génération de colonnes, pour améliorer les performances de calcul de la méthode. Nous proposons ensuite une technique de génération de colonnes multi-objectif, pour obtenir directement des ouvertures qui se rapprochent d’un ensemble efficace de plans de traitement non dominés. A cette fin, nous trouvons les limites inférieure et supérieure du front de Pareto, et générons une colonne avec un vecteur de poids des objectifs pré-assigné ou nouveau, réduisant la distance maximale de deux bornes. Nous prouvons que cet algorithme converge vers le front de Pareto. Les résultats de recherche d’un bon compromis de traitement entre la destruction des volumes cibles et la protection des structures saines dans un espace objectif bidimensionnel, montrent l’efficacité de l’algorithme dans l’approche du front de Pareto, avec des plans de traitement livrables en 3 minutes environ, et évitant un grand nombre de colonnes. Cette méthode s’applique également à d’autres classes de problèmes d’optimisation convexe, faisant appel à la fois à une génération de colonnes et à une optimisation multi-objectifs.----------ABSTRACT: The statistics have estimated about 18.1 million cancer cases in 2018 around the world, which is expected to increase to 24 million by 2035. Radiation therapy is one of the most important cancer treatment methods, which about 50% of patients receive during their illness. This method works by damaging the genetic material within cancerous cells and destroying their ability to reproduce. However, normal cells are also affected by radiation; therefore, the treatment should be performed in such a way that it maximizes the dose of radiation to tumors, while simultaneously minimizing the adverse effects of radiations to healthy tissues. The optimization techniques are useful to determine where and how much radiation should be delivered to patient’s body. In this project, we address the intensity-modulated radiation therapy (IMRT) as a widelyused external radiotherapy method and also a novel form called volumetric modulated arc therapy (VMAT). In IMRT, a finite number of directions are determined for the beam radiation, while in VMAT, the linear accelerator rotates around the patient’s body while the beam is on. These technologies give us the ability of changing the beam shape and the dose rate dynamically during the treatment. The treatment planning problem consists of selecting a delivery sequence of beam shapes, optimizing the dose rate of the beam, and determining the rotation speed of the gantry, if required. In this research, we take advantages of the column generation technique, as a leading optimization method specifically for large-scale problems, to improve the treatment time and non-linear non-convex clinical objectives in VMAT treatment planning, and also develop a new multi-objective column generation framework for IMRT. In the first essay, we develop a novel column generation algorithm optimizing the trade-off between delivery time and treatment quality for VMAT treatment planning. To this end, simultaneous column-and-row generation is developed to relate the configuration of beam apertures in columns to the treatment time restriction in the rows of the model. Moreover, we propose a modified clustering technique to aggregate similar volume elements of the patient’s body and efficiently reduce the number of constraints in the model. The computational results show that a high-quality treatment is achievable using a four-thread CPU. In the second essay, we develop an automatic planning approach integrating dose-volume histogram (DVH) criteria, the most common method of treatment evaluation in practice, for VMAT treatment planning. We take advantage of the iterative procedure of column generation to adjust the model parameters during aperture generation and meet nonlinear DVH criteria without considering hard constraints in the model. The results on clinical cases show that our methodology had significant improvement to obtain clinically acceptable plans without human intervention in comparison to simple VMAT optimization. In addition, the comparison to an existing commercial treatment planning system shows the quality of the obtained plans from the proposed method, especially for the healthy tissues, is significantly better while the computational time is less. In the third essay, we address the IMRT treatment planning, which is formulated as a large scale convex optimization problem with simplex feasibility space. We first integrate a novel Frank-Wolfe-based solution approach, so-called Blended Conditional Gradients, into the column generation to improve the computational performance for the method. We then propose a multi-objective column generation technique to directly obtain apertures that approximate an efficient non-dominated set of treatment plans. To this end, we find lower and upper bounds for the Pareto front and generate a column with a pre-assigned or new weight-vector of the objectives, reducing the maximum distance of two bounds. We prove this algorithm converges to the Pareto front. The results in a two-dimensional objective space to find the trade-off plans between the treat of target volumes and sparing the healthy structures show the efficiency of the algorithm to approximate the Pareto front with deliverable treatment plans in about 3 minutes, avoiding a large number of columns. This method is also applicable for other classes of convex optimization problems requiring both column generation and multi-objective optimization