We consider the problem of learning the underlying graph of a sparse Ising model with $p$ nodes from $n$ i.i.d. samples. The most recent and best performing approaches combine an empirical loss (the logistic regression loss or the interaction screening loss) with a regularizer (an L1 penalty or an L1 constraint). This results in a convex problem that can be solved separately for each node of the graph. In this work, we leverage the cardinality constraint L0 norm, which is known to properly induce sparsity, and further combine it with an L2 norm to better model the non-zero coefficients. We show that our proposed estimators achieve an improved sample complexity, both (a) theoretically -- by reaching new state-of-the-art upper bounds for recovery guarantees -- and (b) empirically -- by showing sharper phase transitions between poor and full recovery for graph topologies studied in the literature -- when compared to their L1-based counterparts.
翻译:我们考虑了从一美元(i.d)样本中以美元节点来学习稀疏的Ising模型的基本图解的问题。最新和最有效果的方法是将经验损失(后勤回归损失或互动筛选损失)与正态器(L1罚款或L1限制)结合起来。这导致一个可单独解决的曲线问题,对于图表的每个节点来说都是如此。在这项工作中,我们利用已知能适当诱导散的基点限制L0规范,并进一步将它与L2规范结合起来,以便更好地模拟非零系数。我们表明,我们提议的估算器在理论上(a)通过达到回收保证的新的最先进的高度界限 -- -- 和(b)在经验上 -- -- 通过显示在文献中研究的图表表象的贫弱和完全恢复之间的更敏锐的阶段过渡 -- -- 与其基于L1的对应方相比,实现了更高的样本复杂性。