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 is widely used in ecology research. Such networks are often sparse and inherently imperfect in their detection. Existing models mainly focus on interaction recovery, while the induced similarity graphs are much less studied. Moreover, sparsity is often not controlled, and scale is unbalanced, leading to oversparse or poorly rescaled estimates with degrading structural recovery. To address these issues, we propose a framework for structured sparse nonnegative low-rank factorization with detection probability estimation. We impose nonconvex $\ell_{1/2}$ regularization on the latent similarity and connectivity structures to promote sparsity within-group similarity and cross-group connectivity with better relative scale. The resulting optimization problem is nonconvex and nonsmooth. To solve it, we develop an ADMM-based algorithm with adaptive penalization and scale-aware initialization and establish its asymptotic feasibility and KKT stationarity of cluster points under mild regularity conditions. Experiments on synthetic and real-world ecological datasets demonstrate improved recovery of latent factors and similarity/connectivity structure relative to existing baselines.

翻译：从计数数据中恢复潜在结构在网络推断中受到了广泛关注，特别是在需要同时揭示二分网络中的跨组交互和组内相似性模式时，这在生态学研究中应用广泛。此类网络通常具有稀疏性且检测过程天然存在不完全性。现有模型主要聚焦于交互恢复，而对由此衍生的相似性图的研究较少。此外，稀疏性常缺乏有效控制，且尺度不平衡，导致估计结果过度稀疏或重新缩放不当，从而降低结构恢复质量。为解决这些问题，我们提出了一种结合检测概率估计的结构化稀疏非负低秩分解框架。我们对潜在相似性与连通性结构施加非凸$\ell_{1/2}$正则化，以促进组内相似性和跨组连通性的稀疏性，并实现更优的相对尺度。由此产生的优化问题具有非凸非光滑性。为此，我们开发了一种基于ADMM的算法，该算法采用自适应惩罚与尺度感知初始化，并在温和的正则性条件下建立了其渐近可行性以及聚类点的KKT平稳性。在合成数据集和真实生态数据集上的实验表明，与现有基线方法相比，该方法在潜在因子及相似性/连通性结构的恢复方面均有显著提升。