The four vector norms that play signicant roles in the compressed sensing framework are the $\iota_0$ , the $\iota _1$ , $\iota_2$, and $\iota_\infty$ norms, denoted by $\|x\|_0$, $\|x\|_1$, $\|x\|_2$ and $\|x\|_\infty$ respectively.
Given a vector $x\in R^m$.
$\|x\|_0$ is equal to the number of the non-zero elements in the vector $x$.
$\|x\|_1=\sum_{i=1}^{m}|x_i|$. $\|x\|_2=\sqrt{x_1^2+x_2^2+...+x_m^2}$. $\|x\|_\infty=\max_i |x_i|$. The vector norm $\|x\|_p$ for $p=1, 2, 3, ...$ is defined as $\|x\|_p=(\sum_{i=1}^m |x_i|^p )^\frac{1}{p}.$ The $p$-norm of vector $x$ is implemented as Norm[$x$, $p$], with the 2-norm being returned by Norm[$x$][1].
These norms have natural generalizations to matrices, inheriting many appealing properties from
the vector case. In particular, there is a parallel duality structure.
For two rectangular matrices $X\in R ^{m \times n}$, $X\in R ^{m \times n}$ .
Define the inner product as $<X,Y>:= \sum_{i=1}^m \sum_{j=1}^n X_{ij}Y_{ij}= \sqrt{Tr(X^T Y)}.$
The norm associated with this inner product is called the Frobenius (or Hilbert-Schmidt) norm
$\|X\|_F$ . The Frobenius norm is also equal to the Euclidean, or $\iota_2$, norm of the vector of singular
The operator norm (or induced 2-norm) of a matrix is equal to its largest singular value (i.e., the $\iota_1$ norm of the singular values):
$\|X\|:= \delta_ 1(X).$
The nuclear norm of a matrix is equal to the sum of its singular values, i.e.,
$\|X\|_*:= \sum_{i=1}^r {\delta_i(X)}$,
and is alternatively known by several other names including the Schatten 1-norm, the Ky Fan r-norm, and the trace class norm. Since the singular values are all positive, the nuclear norm is equal to the $\iota_1$ norm of the vector of singular values. These three norms are related by the following inequalities which hold for any matrix $X$ of rank at most $r$:
[2]Recht, Benjamin, Maryam Fazel, and Pablo A. Parrilo. "Guaranteed minimum-rank solutions of linear matrix equations via nuclear norm minimization." SIAM review 52, no. 3 (2010): 471-501.