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In recent years, a vast amount of data has been accumulated across various fields in industry and academia, and with the rise of artificial intelligence and machine learning technologies, knowledge discovery and high-precision predictions through such data have been demanded. However, real-world data is diverse, including network data that represent relationships, data with multiple modalities or views, data that is distributed across multiple institutions and requires a certain level of information confidentiality. There is also data that requires extracting latent features in complex subspaces for analysis. Therefore, analysis methods that can handle such diversity are needed. In this chapter, we introduce effective methods for such data using novel numerical analysis techniques.

This chapter is organized as follows. Section 4.1 gives an overview of several spectral methods for unsupervised dimensionality reduction and clustering. Section 4.2 describes a recent advanced dimensionality reduction method based on complex moment-based subspace and matrix trace optimization. Section 4.3 shows methods that can utilize data relationships with multiple views simultaneously. In Sect. 4.4, we describe so-called data collaboration analysis that can securely utilize data distributed across multiple institutions.