Multilinear algebra in machine learning and signal processing
Résumé
We will discuss how numerical multilinear algebra arises in both discriminative and generative models in machine learning: tensors in multilinear regression models (generalization of vector space models), symmetric tensors in independent component analysis, and nonnegative tensors in graphical models (ie. Bayesian networks). We will also introduce a multilinear spectral theory and show how the eigenvalues of symmetric tensors may be used to obtain basic results in Spectral Hypergraph Theory. We will also discuss how the problem of blind source separation in signal processing may be solved either with the decomposition a data tensor into a sum of rank-one terms in the presence of sufficient diversity, or with the help of high-order statistics, namely cumulant tensors. In the latter case, the decomposition of the data cumulant, a symmetric tensor, into a linear combination of rank-one symmetric tensors can be used to extract factors when physical diversity does not allow direct and efficient storage of the data.