Graphical continuous Lyapunov models offer a new perspective on modeling causally interpretable dependence structure in multivariate data by treating each independent observation as a one-time cross-sectional snapshot of a temporal process. Specifically, the models assume the observations to be cross-sections of independent multivariate Ornstein-Uhlenbeck processes in equilibrium. The Gaussian equilibrium exists under a stability assumption on the drift matrix, and the equilibrium covariance matrix is determined by the continuous Lyapunov equation. Each graphical continuous Lyapunov model assumes the drift matrix to be sparse with a support determined by a directed graph. A natural approach to model selection in this setting is to use an $\ell_1$-regularization approach that seeks to find a sparse approximate solution to the Lyapunov equation when given a sample covariance matrix. We study the model selection properties of the resulting lasso technique by applying the primal-dual witness technique for support recovery. Our analysis uses special spectral properties of the Hessian of the considered loss function in order to arrive at a consistency result. While the lasso technique is able to recover useful structure, our results also demonstrate that the relevant irrepresentability condition may be violated in subtle ways, preventing perfect recovery even in seemingly favorable settings.
翻译:图形连续的 Lyapunov 模型为在多变量数据中模拟因果解释依赖结构提供了一个新的视角,将每个独立观测作为时间过程的一次性跨部门快照。 具体地说, 模型假设这些观测是平衡中独立的多变量 Ornstein- Uhlenbeck 进程的交叉部分。 高斯平衡存在于漂流矩阵的稳定性假设之下, 平衡共变矩阵由连续的 Lyapunov 方程式决定。 每个图形连续的 Lyapunov 模型都假设漂移矩阵在由定向图表确定的支持下会消失。 在此环境下, 模型选择的自然方法是使用$\ ell_ 1$- 常规化方法, 在给样本共变式矩阵时, 寻求为Lyapunov 等方程式找到稀少的近似解决方案。 我们通过应用原始证人技术来支持恢复, 研究由此产生的拉索技术的模型选择属性。 我们的分析使用了所考虑的损失函数赫萨德的特殊光谱特性, 以便得出一致性的结果。 lassoso 技术在真实的精确性环境下, 也能够证明我们无法恢复的精确性状况。