ABSTRACT
The previous chapter covered background materials on linear subspace learning. From this chapter on, we shall proceed to multiple dimensions with tensorlevel computational thinking. Multilinear algebra is the foundation of multilinear subspace learning (MSL). Thus, we first review the basic notations and operations in multilinear algebra, as well as popular tensor decompositions. In the presentation, we include some discussions of the second-order case (for matrix data) as well, which can be understood in the context of linear algebra. Next, we introduce the important concept of multilinear projections for direct mapping of tensors to a lower-dimensional representation, as shown in Figure 3.1. They include elementary multilinear projection (EMP), tensor-to-vector projection (TVP), and tensor-to-tensor projection (TTP), which project an input tensor to a scalar, a vector, and a tensor, respectively. Their relationships are analyzed in detail subsequently. Finally, we extend commonly used vector-based scatter measures to tensors and scalars for optimality criterion construction in MSL.
