$\ln\abs{\lambda+\epsilon}$ is translated and scaled such that it equals 0 and 1 at $\lambda=0$ and 1 respectively for better illustration.[]{data-label="Fig:sparsity"}](Fig_sparsity.pdf){width="3in"}

Let $\epsilon\rightarrow+\infty$. Then, i.e., they are equivalent infinitesimals. \[thm:epsilontoinf\]

See Appendix \[sec:append:epsilontoinf\].

Let $r=\norm{\m{Y}}_{\cA,0}$ and $\epsilon\rightarrow0$. Then, we have the following results:

1.  If $r\leq N-1$, then i.e., they are equivalent infinities. Otherwise, $\cM^{\epsilon}\sbra{\m{Y}}$ is a positive constant depending only on $\m{Y}$;

2.  Let $\m{u}_{\epsilon}^*$ be the (global) optimizer of $\m{u}$ to the optimization problem in (\[formu:nonconvexrelax\]). Then, the smallest $N-r$ eigenvalues of $T\sbra{\m{u}_{\epsilon}^*}$ are either zero or approach zero as fast as $\epsilon$;

3.  For any cluster point of $\m{u}_{\epsilon}^*$ at $\epsilon=0$, denoted by $\m{u}_0^*$, there exists an atomic decomposition $\m{Y}=\sum_{k=1}^r\m{a}\sbra{f_k}\m{s}_k$ such that $T\sbra{\m{u}_0^*}=\sum_{k=1}^r\twon{\m{s}_k}^2\m{a}\sbra{f_k}\m{a}\sbra{f_k}^H$.[^3]

\[thm:epsilontozero\]

See Appendix \[sec:append:epsilontozero\].

Note that the term $\ln\frac{1}{\epsilon}$ in that becomes unbounded as $\epsilon\rightarrow0$ is not problematic in the optimization problem to introduce, since the objective function $\cM^{\epsilon}\sbra{\m{Y}}$ can be re-scaled by $\sbra{\ln\frac{1}{\epsilon}}^{-1}$ for any $\epsilon>0$ without altering the optimizer. By similar arguments we see that $\epsilon^{-\frac{1}{2}}$ in is not problematic as well.

Theorem \[thm:epsilontoinf\] shows that the new metric $\cM^{\epsilon}\sbra{\m{Y}}$ plays the same role as $\atomn{\m{Y}}$ as $\epsilon\rightarrow+\infty$, while Theorem \[thm:epsilontozero\] states that it approaches $\norm{\m{Y}}_{\cA,0}$ as $\epsilon\rightarrow0$. Consequently, it bridges $\atomn{\m{Y}}$ and 