although $L_1$ norm based penalty leads to computationally sound solutions to the original issue, it also introduces modeling bias into the penalized maximum likelihood estimation. As illustrated in the figure, when the underlying true values of the matrix entries are sufficiently large, the corresponding $L_1$ norm based regularization performs linearly increased penalty to those entries, which thus results in a severe bias w.r.t. the maximum likelihood estimation [@FanASA]. In contrast, $L_0$ norm based penalty avoids such issue by constraining the penalties of non-zero elements to be constant. It has been proved in [@TibLASSO] that the $L_1$ norm penalty discovers the underlined sparse structure when some suitable assumptions are satisfied. However, in general cases, the quality of the solutions is not clear.

Iterative proportional scaling from the Bethe reference point {#sec:iip}
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Bethe reference point and optimal $1$-link correction {#sec:onelink}
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As observed in the previous section, when using the Bethe approximation to find an approximate solution to the IIP, the consistency check should then be that either the factor graph be sparse, nearly a tree, either the coupling are small. There are then two distinct ways of using the Bethe approximation:

-   the direct way, where the form of the 