We establish the fundamental limits of lossless linear analog compression by
considering the recovery of random vectors
x∈Rm from the noiseless linear
measurements
y=Ax with
measurement matrix A∈Rn×m. Specifically,
for a random vector x∈Rm of arbitrary
distribution we show that x can be recovered with
zero error probability from
n>infdimMB(U) linear measurements,
where dimMB(⋅) denotes the lower
modified Minkowski dimension and the infimum is over all sets
U⊆Rm with P[x∈U]=1. This achievability statement holds for Lebesgue almost all measurement
matrices A. We then show that s-rectifiable random vectors---a
stochastic generalization of s-sparse vectors---can be recovered with zero
error probability from n>s linear measurements. From classical compressed
sensing theory we would expect n≥s to be necessary for successful
recovery of x. Surprisingly, certain classes of
s-rectifiable random vectors can be recovered from fewer than s
measurements. Imposing an additional regularity condition on the distribution
of s-rectifiable random vectors x, we do get the
expected converse result of s measurements being necessary. The resulting
class of random vectors appears to be new and will be referred to as
s-analytic random vectors