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备受关注，尤其是在生态学研究中广泛使用的二部网络中同时寻求跨组交互与组内相似性模式时。此类网络通常具有稀疏性且检测过程本身存在固有缺陷。现有模型主要聚焦于交互恢复，而对诱导相似性图的研究相对不足。此外，稀疏性常缺乏控制且尺度失衡，导致估计结果过度稀疏或缩放不当，进而削弱结构恢复效果。针对这些问题，我们提出了一种融合检测概率估计的结构化稀疏非负低秩分解框架。通过对潜在相似性与连通性结构施加非凸$\ell_{1/2}$正则化，我们能够在更优相对尺度下增强组内相似性与跨组连通性的稀疏性。该优化问题具有非凸非光滑特性。为求解此问题，我们开发了基于ADMM的自适应惩罚算法，并采用尺度感知初始化策略，在温和正则性条件下证明了其渐近可行性及聚类点的KKT平稳性。在合成数据集与现实生态数据集上的实验表明，相较现有基准方法，本方法在潜在因子及相似性/连通性结构恢复方面均有显著提升。