Extracting common features hidden in data is one of the most important tasks in data sciences. By acquiring a lot of data and extracting the underlying structure, we can obtain global information that cannot be found from each datum. To realize this extraction of information, we must formulate the problem mathematically and solve it. Simultaneous diagonalization (SD) is one of the important problems that can be applied in various research fields. SD is the problem of finding a common transformation matrix that simultaneously diagonalizes all matrices obtained from data. This paper handles a variant of SD called approximate SD (ASD) that approximately diagonalizes the matrices obtained from noisy data.

ASD is a difficult problem owing to its non-convexity. A complete solution has not been known and will probably not be found. Therefore, many approximate algorithms have been proposed by focusing on approximate solutions that are valuable enough for engineering. However, the conventional non-convex optimization methods have no guarantee to obtain an appropriate solution even for SD. In contrast, the authors proposed a two-step framework called Approximate-Then-Diagonalize-Simultaneously (ATDS) that has a guarantee to obtain an appropriate solution. It decomposes ASD into (1) a problem of approximating the data by simultaneously diagonalizable matrices and (2) algebraic SD. In addition, they characterized simultaneous diagonalizability by a certain linear mapping to further decompose ATDS. As a result, a novel three-step algorithm was derived. Since this paper proposed not only an algorithm but also the ATDS framework that provided a new approach to this important problem in engineering, this is one of the best papers in the journal and is perfectly appropriate to receive the prize.