Recovering latent structure from count data has received considerable attention in network inference, particularly when one seeks both cross-group interactions and within-group similarity patterns in bipartite networks, which are widely used in ecology research. Such networks are often sparse and prone to imperfect detection. Existing models mainly focus on interaction recovery, while the induced similarity graphs are much less studied. Moreover, existing models for under-counted data place sparsity, if at all, on the detection model rather than on these graphs. To address these issues, we propose a nonnegative low-rank factorization framework that jointly estimates detection probabilities and promotes sparse within- and cross-group graphs through $\ell_{1/2}$ regularization. For the resulting nonconvex, nonsmooth problem, we develop an ADMM-based algorithm with adaptive penalization, and establish asymptotic feasibility and KKT stationarity of cluster points under stated assumptions. We also give sufficient conditions for exact recovery of within- and cross-group graphs. On synthetic data with up to 20% missing entries, the method recovers the graphs and detection probabilities with competitive accuracy. Across three ecological datasets, it matches the baseline held-out predictions while estimating detection probabilities and within- and cross-group graphs.

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