The name is absent

where a is a positive constant.

Note that the zero norm,∣∣ ∙ ∣∣o, in Equation 2.3 measures the number of non-
zero elements of the vector. This objective function is not convex which means
solving the optimization problem in Equation 2.3 is difficult. Instead, Danaho
et al. showed [10,15,24] that one can use the following optimization problem to
approximate the sparse vector X.

min∣∣X∣∣ι + α∣∣e∣∣₂
such that Y = UX + e.

They proved that, for a Gaussian measurement matrix, an «-sparse vector can
be retrieved via ^ɪ-norm optimization if

_S<C ^K ■
^<Ulog(X/K)’

where C is a constant. Moreover, for a general measurement matrix U, Restricted
Isometry Property (IRP) should be satisfied [15].

Most of the real world vectors have an approximately sparse representation.
A vector Xv ×ι is called approximately «-sparse if it has s large elements and
N — s very small elements. It is also shown that the optimization problem in
Equation 2.4 can be used to recover approximately sparse vectors that fie in weak
l_p ball of radius r [15]. i.e.,

I I              — 1

k∣(i) ≤ ri P, 1 ≤ i ≤ N

21

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