Nonlinear System Identification using Volterra series and Tensor network in Polynomial Nonlinear State Space framework

Low-rank approximations (approximations of a matrix by a matrix of lower rank) are widely used in data mining, machine learning and signal processing as a tool for dimensionality reduction, feature extraction, and classification. The optimal solution is well-understood and can be obtained from the truncated singular value decomposition. To make use of the intrinsic properties of the data, two generalizations of the low-rank approximation problem are available: one to affinely structured matrices and one to higher-order tensors. Structured matrix approximations are used in system identification, signal processing and computer algebra, among others. The goal of structured low-rank approximation (SLRA) is to approximate a given structured matrix, e.g., symmetric, Hankel or Sylvester matrix, by a low-rank matrix with the same structure. On the other hand, data are often naturally multi-dimensional (multiway). For example, term-document matrix in text mining or user-item matrix in recommender systems are naturally extended to term-document-time and user-item-time tensors under the realization that topics and user preferences change in time. An important problem then is approximating tensors by ‘low-rank’ tensors, even though the concept of tensor rank is not uniquely defined. In this PhD thesis we will generalize the low-rank approximation problem even further and consider structured tensor low-rank approximations. Such tensors appear in higher order statistics (symmetric tensors), in chemometrics (Hankel-structured tensors), etc. Existing research is mainly focused on imposing structure on the factors of the approximation. In this thesis, we will impose structure on (the elements of) the approximation directly, which is more difficult and computationally more expensive and thus requires a dedicated study. We will develop and implement algorithms and adapt them to particular application domains.