论文标题
用于结构化动态协方差恢复的非convex框架
A Nonconvex Framework for Structured Dynamic Covariance Recovery
论文作者
论文摘要
我们为高维数据提供了一个灵活但可解释的模型，该模型具有随时间变化的二阶统计数据，动机并应用于功能性神经成像数据。在神经科学文献的推动下，我们将协方差分解为稀疏的空间和光滑的时间成分。虽然这种分解导致简约和域的解释性，但最终的估计问题是非凸。为此，我们设计了一个两阶段优化方案，具有经过精心量身定制的光谱初始化，并结合了迭代精制的交替投影梯度下降。我们证明了线性收敛速率，直至提议的下降方案的非平凡统计误差，并为估计量建立样本复杂性保证。我们进一步量化了多元高斯病例的统计误差。使用模拟和真实脑成像数据的经验结果表明，我们的方法表现优于现有基准。
We propose a flexible yet interpretable model for high-dimensional data with time-varying second order statistics, motivated and applied to functional neuroimaging data. Motivated by the neuroscience literature, we factorize the covariances into sparse spatial and smooth temporal components. While this factorization results in both parsimony and domain interpretability, the resulting estimation problem is nonconvex. To this end, we design a two-stage optimization scheme with a carefully tailored spectral initialization, combined with iteratively refined alternating projected gradient descent. We prove a linear convergence rate up to a nontrivial statistical error for the proposed descent scheme and establish sample complexity guarantees for the estimator. We further quantify the statistical error for the multivariate Gaussian case. Empirical results using simulated and real brain imaging data illustrate that our approach outperforms existing baselines.
