sparse image representation via combined transforms - Convex ...

2.2. SPARSITY AND COMPRESSION 17
with the largest n amplitudes, then the compression number c(n) is the square root of the
RSS distortion of the signal reconstructed by the coefficients with the n largest amplitudes.
The following result can be found in [47].
Lemma 2.2 For any sequence θ, ifm =1/p − 1/2, the following inequality is true:
c(N) ≤ α p N −m |θ| wl p, N ≥ 1,
where α p is a constant determined only by the value of p.
From the above lemma, a small weak l p norm implies a small compression number.
Rate of Recovery
The rate of recovery comes from statistics, particularly in density estimation. For a sequence
θ, the rate of recovery is defined as
r(ɛ) =
∞∑
min{θi 2 ,ɛ 2 }.
i=1
Lemma 2.3 For any sequence θ, ifr =1− p/2, the following inequality is true:
r(ɛ) ≤ α ′ p|θ| p wl p (ɛ 2 ) r , ɛ > 0,
where α ′ p is a constant.
This implies that a small weak l p norm leads to a small rate of recovery. In some cases
(for example, in density estimation) we choose rate of recovery as a measure of sparsity.
The weak l p norm is therefore a good measure of sparsity too.
Lemma 1 in [46] shows that all these measures are equivalent in an asymptotic sense.
Critical Index
In order to define the critical index of a functional space, we need to introduce some new
notation. A detailed discussion of this can be found in [47]. Suppose Θ is the functional
space that we are considering. (In the transform coding scenario, the functional space Θ
includes all the coefficient vectors.) An infinite-length sequence θ = {θ i : i ∈ N} is in a weak