  This paper determines to within a single measurement the minimum number of
measurements required to successfully reconstruct a signal drawn from a
Gaussian mixture model in the low-noise regime. The method is to develop upper
and lower bounds that are a function of the maximum dimension of the linear
subspaces spanned by the Gaussian mixture components. The method not only
reveals the existence or absence of a minimum mean-squared error (MMSE) error
floor (phase transition) but also provides insight into the MMSE decay via
multivariate generalizations of the MMSE dimension and the MMSE power offset,
which are a function of the interaction between the geometrical properties of
the kernel and the Gaussian mixture. These results apply not only to standard
linear random Gaussian measurements but also to linear kernels that minimize
the MMSE. It is shown that optimal kernels do not change the number of
measurements associated with the MMSE phase transition, rather they affect the
sensed power required to achieve a target MMSE in the low-noise regime.
Overall, our bounds are tighter and sharper than standard bounds on the minimum
number of measurements needed to recover sparse signals associated with a union
of subspaces model, as they are not asymptotic in the signal dimension or
signal sparsity.
