AI 中文总结
研究在高维中不同数据集共享条件依赖结构但边强度各异时估计多个精度矩阵的问题，提出乘性图形套索（Mglasso）方法，通过分解矩阵并优化惩罚对数似然函数来估计，有理论保证且模拟显示效果优于基准方法，还经实际应用验证。
AI 中文摘要
在高维中估计多个精度矩阵存在重大挑战，特别是当不同数据集共享共同的条件依赖结构但具有特定总体的交互强度时。我们通过引入乘性图形套索（Mglasso）来解决这个问题，这是一种在共享稀疏性约束下联合估计多个高斯图形模型中精度矩阵的方法。每个精度矩阵被分解为共享结构矩阵\(\boldsymbol{\Theta}\)（编码共同条件独立图）和特定总体矩阵\(\boldsymbol{\Gamma}_{l}\)（捕获总体间边强度变化）的舒尔 - 哈达玛积。我们优化一个惩罚对数似然函数，利用\(\ell_1\)惩罚来强制共同稀疏性，利用弗罗贝尼乌斯范数惩罚来调节特定总体的变化。使用与梯度下降集成的交替方向乘子法（ADMM）算法有效地进行优化。理论上，我们建立了目标函数的局部严格凸性，并提供了严格的高维一致性保证。广泛的模拟表明，与基准组图形套索（GGL）相比，在较小样本量下具有更好的模型选择一致性。最后，通过实际应用进一步验证了该方法的实用性。
英文摘要
Estimating multiple precision matrices in high-dimension presents significant challenges, particularly when distinct datasets share a common conditional dependency structure but exhibit population-specific interaction strengths. We address this problem by introducing the Multiplicative Graphical Lasso (Mglasso), a method for jointly estimating precision matrices across multiple Gaussian graphical models under a shared sparsity constraint. Each precision matrix is decomposed as a Schur-Hadamard product of a shared structural matrix $\boldsymbolΘ$, which encodes the common conditional independence graph, and a population-specific matrix $\boldsymbolΓ_{l}$, which captures variation in edge strengths across populations. We optimize a penalized log-likelihood that utilizes an $\ell_1$-penalty to enforce common sparsity and a Frobenius norm penalty to regulate population-specific variations. The optimization is efficiently performed using the Alternating Direction Method of Multipliers (ADMM) algorithm integrated with gradient descent. Theoretically, we establish the local strict convexity of the objective function and provide rigorous high-dimensional consistency guarantees, including supremum norm error bounds and exact support recovery under sub-Gaussian tail conditions. Extensive simulations show superior model selection consistency at smaller sample sizes compared to the benchmark Group Graphical Lasso (GGL). Finally, the method's practical utility is further validated through real-world applications.
