Nickolay Trendafilov - Multivariate Data Analysis on Matrix Manifolds: (with Manopt)

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We want to start with few remarks predating considerably the emerging of the idea for writing this book and our collaboration in general. They are related to the first authors own experience which made him explore matrix manifolds in data analysis problems.

NTT was in an early stage of his research career, after his Ph.D. was completed on a completely different topic. He was assigned to study factor analysis (FA) and do some programming for a particular software product. While working on FA, NTT realized that the most interesting part for him is the FA interpretation and the so-called rotation methods (see Section 4.5). NTT recognized that the main goal of FA is to produce simple for interpretation results, which is achieved by orthogonal rotation of some initial, usually difficult to interpret, FA solution (known as factor loadings matrix). However, how we can define what is simple for interpretation results? The problem is really tough, especially if you try to capture its meaning in a single mathematical expression/formula. Because of that, a huge number of different formulas were proposed each of them claiming to approximate in some sense the idea for simple for interpretation results. In FA, these formulas are called simple structure criteria. They are supposed to measure/quantify the simplicity of a certain FA solution. Let be such a criterion. Then, if is simpler than , and if is even simpler than , then we should have , or depending on the sign of