Optimal mass transport (OMT) theory is used to preprocess brain tumor datasets for the first time. The goal is to move any irregular 3D object (i.e., the brain) without causing significant distortion. The first stage of a two-stage OMT (TSOMT) procedure transforms the brain onto the unit solid ball. The sphere shape boundary constraint is necessary to ensure OMT convergence. The second stage is to transform the unit ball to a unit cube, as it is easier to apply a 3D convolutional neural network (CNN) to rectangular coordinates. Small variations in the local mass-measured stretch ratio among all the brain tumor datasets show the robustness of the transform. Additionally, the distortion is kept at a minimum (<0.1) with a reasonable transport cost. The original 240 × 240 × 155 × 4 dataset is thus reduced to a cube of 128 × 128 × 128 × 4, which is a 76.6% reduction in the total number of voxels, without losing much detail. Two mass-preserving OMT functions (one is fluid-attenuated inversion recovery (FLAIR) only and the other is 4 modalities on average) are used to double the amount of training data, which helps to reduce overfitting when training a vast machine learning (ML) model. Three typical U-Nets are trained separately to predict the whole tumor (WT), tumor core (TC), and contrast-enhancing tumor (CET) from the cube. An impressive 0.9901 training accuracy in the WT cube is achieved at 1000 epochs. Inverse TSOMT is performed on the predicted cube so that brain results are obtained. The conversion loss from OMT to inverse OMT is found to be less than one percent. For training, the Dice scores (0.9852 for the WT, 0.9743 for the TC, and 0.9433 for the CET) can be obtained. Significant improvements in brain tumor detection and the segmentation accuracy have been achieved. For validation, postprocessing using rotating techniques is added to the TSOMT, U-Net prediction, and inverse TSOMT method for an accuracy improvement of a few percent. It takes 200 seconds to complete the whole segmentation process on each new brain tumor dataset. Finally, the false prediction for the worst WT, TC and CET validation cases is discussed. Again, the TSOMT and inverse TSOMT do not induce any new failure mode because of the small conversion loss.

We study the problem of globally recovering a dictionary from a set of signals via ℓ1-minimization. We assume that the signals are generated as i.i.d. random linear combinations of the K atoms from a complete reference dictionary D∗∈RK×K, where the linear combination coefficients are from either a Bernoulli type model or exact sparse model. First, we obtain a necessary and sufficient norm condition for the reference dictionary D∗ to be a sharp local minimum of the expected ℓ1 objective function. Our result substantially extends that of Wu and Yu (2015) and allows the combination coefficient to be non-negative. Secondly, we obtain an explicit bound on the region within which the objective value of the reference dictionary is minimal. Thirdly, we show that the reference dictionary is the unique sharp local minimum, thus establishing the first known global property of ℓ1-minimization dictionary learning. Motivated by the theoretical results, we introduce a perturbation based test to determine whether a dictionary is a sharp local minimum of the objective function. In addition, we also propose a new dictionary learning algorithm based on Block Coordinate Descent, called DL-BCD, which is guaranteed to decrease the obective function monotonically. Simulation studies show that DL-BCD has competitive performance in terms of recovery rate compared to other state-of-the-art dictionary learning algorithms when the reference dictionary is generated from random Gaussian matrices.

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Nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem have attracted arising attention due to their empirical efficiency and scalability. Compared with the original convex formulations, the nonconvex ones typically involve much fewer variables, allowing them to scale to scenarios with millions of variables. However, it opens a new challenge that under what conditions the nonconvex stochastic algorithms may find the population minimizer within the optimal statistical precision despite their empirical success in applications. In this paper, we provide an answer that the stochastic gradient descent (SGD) method can be adapted to solve the nonconvex reformulation of the original convex problem, with a global linear convergence when using a fixed step size, i.e., converging exponentially fast to the population minimizer within an optimal statistical precision in the restricted strongly convex case. If a diminishing step size is adopted, the bad effect caused by the variance of gradients on the optimization error can be eliminated but the rate is dropped to be sublinear. The core of our treatment relies on a novel second-order descent lemma, which is more general than the existing best result in the literature and improves the analysis on both online and batch algorithms. The developed theoretical results and effectiveness of the suggested SGD are also verified by a series of experiments.

We formulate the Riemannian calculus of the probability set embedded with L^2-Wasserstein metric. This is an initial work of transport information geometry. Our investigation starts with the probability simplex (probability manifold) supported on vertices of a finite graph. The main idea is to embed the probability manifold as a submanifold of the positive measure space with a nonlinear metric tensor. Here the nonlinearity comes from the linear weighted Laplacian operator. By this viewpoint, we establish torsion-free Christoffel symbols, Levi-Civita connections, curvature tensors, and volume forms in the probability manifold by Euclidean coordinates. As a consequence, the Jacobi equation, Laplace-Beltrami and Hessian operators on the probability manifold are derived. These geometric computations are also provided in the infinite-dimensional density space (density manifold) supported on a finite-dimensional manifold. In particular, an identity is given connecting the Baker-{É}mery Γ2 operator (carr{é} du champ it{é}r{é}) by connecting Fisher-Rao information metric and optimal transport metric. Several examples are demonstrated.