Optimal transport (OT), which provides a distance between two probability distributions by considering their spatial locations, has been applied to widely diverse applications. Computing an OT problem requires solution of linear programming with tight mass-conservation constraints. This requirement hinders its application to large-scale problems. To alleviate this issue, the recently proposed relaxed-OT approach uses a faster algorithm by relaxing such constraints. Its effectiveness for practical applications has been demonstrated. Nevertheless, it still exhibits slow convergence. To this end, addressing a convex semi-relaxed OT, we propose a fast blockcoordinate Frank-Wolfe (BCFW) algorithm, which gives sparse solutions. Specifically, we provide their upper bounds of the worst convergence iterations, and equivalence between the linearization duality gap and the Lagrangian duality gap. Three fast variants of the proposed BCFW are also proposed. Numerical evaluations in color transfer problem demonstrate that the proposed algorithms outperform state-of-the-art algorithms across different settings.
The optimal transport (OT) problem has been used widely for machine learning. It is necessary for computation of an OT problem to solve linear programming with tight mass-conservation constraints. These constraints prevent its application to large-scale problems. To address this issue, loosening such constraints enables us to propose the relaxed-OT method using a faster algorithm. This approach has demonstrated its effectiveness for applications. However, it remains slow. As a superior alternative, we propose a fast block-coordinate Frank-Wolfe (BCFW) algorithm for a convex semi-relaxed OT. Specifically, we prove their upper bounds of the worst convergence iterations, and equivalence between the linearization duality gap and the Lagrangian duality gap. Additionally, we develop two fast variants of the proposed BCFW. Numerical experiments have demonstrated that our proposed algorithms are effective for color transfer and surpass state-of-the-art algorithms. This report presents a short version of [1]. The source code is available at https://github.com/hiroyuki-kasai/srot.
This paper presents a proposal of a faster Wasserstein k-means algorithm for histogram data by reducing Wasserstein distance computations and exploiting sparse simplex projection. We shrink data samples, centroids, and the ground cost matrix, which leads to considerable reduction of the computations used to solve optimal transport problems without loss of clustering quality. Furthermore, we dynamically reduced the computational complexity by removing lowervalued data samples and harnessing sparse simplex projection while keeping the degradation of clustering quality lower. We designate this proposed algorithm as sparse simplex projection based Wasserstein k-means, or SSPW k-means. Numerical evaluations conducted with comparison to results obtained using Wasserstein k-means algorithm demonstrate the effectiveness of the proposed SSPW k-means for real-world datasets 1 .
We construct detection systems for lung cancer in chest X-ray images with neural networks. The exact position of abnormal area in the X-ray images, namely lung cancer is determined with the result from CT images, and then, the one-dimensional numeric sequences including the abnormal area are used as a teacher signal. In order to reduce noise included in the images, the two-dimensional median filters with various sizes are used in the pre-processing unit of this system, and their sizes are optimized with a view of the performance of the system. At the same time, the ensemble learning method is applied to the detection system. As a result, we were able to construct the detection system with high performance for lung cancer in chest X-ray images.
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