Sampling Theorems for Signals from the Union of Finite-Dimensional Linear Subspaces

Description/Abstract

Compressed sensing is an emerging signal acquisition technique that enables signals to be sampled
well below the Nyquist rate, given that the signal has a sparse representation in an orthonormal basis. In
fact, sparsity in an orthonormal basis is only one possible signal model that allows for sampling strategies
below the Nyquist rate. In this paper we consider a more general signal model and assume signals that
live on or close to the union of linear subspaces of low dimension. We present sampling theorems for
this model that are in the same spirit as the Nyquist-Shannon sampling theorem in that they connect the
number of required samples to certain model parameters.
Contrary to the Nyquist-Shannon sampling theorem, which gives a necessary and sufficient condition
for the number of required samples as well as a simple linear algorithm for signal reconstruction, the
model studied here is more complex. We therefore concentrate on two aspects of the signal model, the
existence of one to one maps to lower dimensional observation spaces and the smoothness of the inverse
map. We show that almost all linear maps are one to one when the observation space is at least of the same
dimension as the largest dimension of the convex hull of the union of any two subspaces in the model.
However, we also show that in order for the inverse map to have certain smoothness properties such as
a given finite Lipschitz constant, the required observation dimension necessarily depends logarithmically