Scientific Computing Seminar

We will discuss generalizations of factor analysis and principal component analysis to multiway data. Just as matrix-based methods are often used in the analysis of 2-way data arrays (modelled as matrices), tensor-based methods may be used to effectively analyze k-way data arrays (modelled as order-k tensors). Unfortunately, the properties of tensors of order 3 or higher is markedly different from those of tensors of order 2 (ie. matrices) and thus straight-forward generalizations of matrix techniques to tensors often do not work. We will show that an "Eckart-Young Theorem"-like result for tensors is non-existent for tensors of order 3 or higher, regardless of the choice of norm. In fact, the problem of finding an optimal low-rank approximation to an arbitrary tensor is ill-posed in every sense of the word -- nonexistent, nonunique and unstable. The ill-posedness of the problem leads to serious ill-conditioning in practice when one tries to compute a near optimal low-rank tensor approximation. We will show how the problem may be remedied by a natural redefinition of rank and with the use of the hyperdeterminant -- a generalization of the matrix determinant to k-way arrays, first introduced by Cayley 160 years ago but remained largely unknown until recently. We will give examples of applications.