rank of a PSD matrix (see, e.g., [@david1994algorithms; @fazel2003log; @mohan2012iterative]). Also, a similar logarithmic penalty has been adopted to approximate the pseudo $\ell_0$ norm for discrete sparse recovery [@rao1999affine; @chartrand2008iteratively; @candes2008enhancing]. From the perspective of LRMR, the atomic $\ell_0$ norm minimizes the number of nonzero eigenvalues of $T\sbra{\m{u}}$ while the atomic norm minimizes the sum of the eigenvalues. In contrast, the new metric $\cM^{\epsilon}\sbra{\m{Y}}$ puts penalty on $\sum_{k=1}^N\ln\abs{\lambda_k+\epsilon}$, where $\lbra{\lambda_k}_{k=1}^N$ denotes the eigenvalues. We plot the function $h(\lambda)=\ln\abs{\lambda+\epsilon}$ with different $\epsilon$’s in Fig. \[Fig:sparsity\] together with the constant function (except at $\lambda=0$) and the identity function corresponding to the $\ell_0$ and $\ell_1$ norms respectively, where $h(\lambda)$ is translated and scaled for better illustration without altering its sparsity-enhancing property. Intuitively, $h(\lambda)$ gets close to the $\ell_1$ norm for large $\epsilon$ while it approaches the $\ell_0$ norm as $\epsilon\rightarrow0$. Therefore, we expect that the new metric $\cM^{\epsilon}\sbra{\m{Y}}$ bridges $\atomn{\m{Y}}$ and $\norm{\m{Y}}_{\cA,0}$ when $\epsilon$ varies from $+\infty$ to $0$. Formally, we have the following results.

![Illustration of the sparsity-promoting property of $\cM^{\epsilon}\sbra{\cdot}$ with respect to $\epsilon$. The plotted curves include the $\ell_0$ and $\ell_1$ norms corresponding to $\norm{\cdot}_{\cA,0}$ and $\norm{\cdot}_{\cA}$ respectively, and $\ln\abs{\lambda+\epsilon}$ corresponding to $\cM^{\epsilon}\sbra{\cdot}$ with $\epsilon=10, 1, 0.1, 10^{-3}$ and $10^{-6}$. 