Our primary focus in the book thus far has been on sampling and recovery of signals that lie in a single subspace. Subspace models lead to powerful sampling theorems, resulting in perfect recovery of the signal from its linear and nonlinear samples under very broad conditions. An appealing feature of the recovery algorithms is that they can often be implemented using simple digital and analog filtering, generalizing the Shannon-Nyquist theorem to a much broader set of input classes. The subspace viewpoint also leads to nice geometrical interpretations of classic and new sampling theorems in terms of orthogonal and oblique projections.

To a large extent, the notion that any possible vector in a given subspace is a valid signal has driven the explosion in the dimensionality of the data we must sample and process. Despite the simplicity and geometric appeal of subspace modeling, there are many classes of signals whose structure is not captured well by a single subspace. This is particularly true in applications in which the sampler does not have full knowledge of the received signal. In response to these challenges, there has been a surge of interest in recent years, across many fields, in a variety of low-dimensional signal representations that quantify the notion that the number of degrees of freedom in high-dimensional signals is often quite small compared with their ambient dimension. Low-dimensional modeling has been used extensively in machine learning, parameter estimation, and detection techniques.