Compressed Sensing aims to capture attributes of k-sparse signals using
very few measurements. In the standard Compressed Sensing paradigm, the
\m\times \n measurement matrix \A is required to act as a near isometry on
the set of all k-sparse signals (Restricted Isometry Property or RIP).
Although it is known that certain probabilistic processes generate \m \times
\n matrices that satisfy RIP with high probability, there is no practical
algorithm for verifying whether a given sensing matrix \A has this property,
crucial for the feasibility of the standard recovery algorithms. In contrast
this paper provides simple criteria that guarantee that a deterministic sensing
matrix satisfying these criteria acts as a near isometry on an overwhelming
majority of k-sparse signals; in particular, most such signals have a unique
representation in the measurement domain. Probability still plays a critical
role, but it enters the signal model rather than the construction of the
sensing matrix. We require the columns of the sensing matrix to form a group
under pointwise multiplication. The construction allows recovery methods for
which the expected performance is sub-linear in \n, and only quadratic in
\m; the focus on expected performance is more typical of mainstream signal
processing than the worst-case analysis that prevails in standard Compressed
Sensing. Our framework encompasses many families of deterministic sensing
matrices, including those formed from discrete chirps, Delsarte-Goethals codes,
and extended BCH codes.Comment: 16 Pages, 2 figures, to appear in IEEE Journal of Selected Topics in
Signal Processing, the special issue on Compressed Sensin