Sparse modeling is an emerging field in applied mathematics. It is a collection of fundamental theories, optimization methods, and practical applications. We believe this field provides new efficient approaches for many applications, and will be a standard methodology in a near future. In the following three articles, we show a part of the projects "Initiative for high-dimensional data-driven science based on sparse modeling" (MEXT grant-in-aid for scientific research on innovative areas, 2013-2018). In the present article, we explain the idea of sparse modeling.

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Feature selection problem has been widely used for various fields. In particular, the sparse estimation has the advantage that its computational cost is the polynomial order of the number of features. However, it has the problem that the obtained solution varies as the dataset has changed a little. The goal of this article is to exhaustively search the solutions which minimize the generalization error for feature selection problem. This article focuses on the feature selection problem for the binary classification with linear discriminant. We calculate the generalization errors for all combinations of features in order to get the histogram of generalization error by using the cross validation method. By using this histogram, we propose a method to verify whether the given data include information for binary classification by comparing the histogram of predictive error for random guessing.

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Magnetic resonance imaging (MRI) is essential in modern medicine and has been used to visualize a variety of biological phenomena noninvasively. Recent advances in data processing have enabled accelerated undersampling of the MRI data by taking advantages of compressed sensing, which utilizes sparsity of the object. Within a few years, a variety of techniques have been introduced to deal with fast and accurate recovery of the MR images. Sparsity and low-rankedness are sought and exploited not only in the image domain but also in the frequency domain.

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In this paper, we show some examples of sparse modeling in astronomy. In many cases, astronomy data has sparsity. If we can utilize it, we will have better results. What is measured in astronomy is the electromagnetic wave of various wavelength. The technology used for each wavelength is different. We show three examples. For each of them, the sparse modeling plays an important role.

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