Tensor decompositions:Principles and application to food sciences

Tensors of order d may be seen as arrays of entries indexed by d indices. They naturally appear as data, and arrays in applications such as chemistry, food science, forensics, environmental analysis and many other fields. Extracting and visualizing the underlying features from tensors is an important source separation problem. This chapter first describes an important class of data mining methods for tensors, namely low-rank tensor approximations (CPD, Tucker3) in the case of order d=3. In such a case, striking differences already exist compared to low-rank approximations of matrices, which are tensors of order d=2. Constrained decompositions and coupled decompositions, which are important variants of tensor decompositions, are also discussed in detail, along with practical learning algorithms. Finally, tensor decompositions are illustrated as a tool for source separation in food sciences. In particular fluorescence spectroscopy, electrophoresis in gel, or chromatography especially coupled with mass spectrometry, are techniques where tensor decompositions are known to be useful. Some of the many other source separation problems that may be tackled with tensor decompositions are briefly discussed in the concluding remarks.