On the optimality of the proper orthogonal decomposition and balanced truncation

The proper orthogonal decomposition (POD), also known as Karhunen-Loeve decomposition or principal component analysis, and balanced truncation, are shown to be optimal in the sense of distance minimizations in spaces of Hilbert-Schmidt or trace-class 2 integral operators. Both POD and balanced truncation are shown to be optimal approximations by finite rank operators in the Hilbert-Schmidt norm. Optimality of balanced truncation seems to have been overlooked in the literature, and in fact, it is commonly thought to be non-optimal in any sense. The role of POD and balanced truncation in minimizing different n-widths of specific compact operators is discussed. The n-widths quantify inherent and representation errors due to lack of data or inaccurate measurements and loss of information.