In this work we present a computationally efficient linear optimization approach for estimating the cross--power spectrum of an hidden multivariate stochastic process from that of another observed process. Sparsity in the resulting estimator of the cross--power is induced through $\ell_1$ regularization and the Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is used for computing such an estimator. With respect to a standard implementation, we prove that a proper initialization step is sufficient to guarantee the required symmetric and antisymmetric properties of the involved quantities. Further, we show how structural properties of the forward operator can be exploited within the FISTA update in order to make our approach adequate also for large--scale problems such as those arising in context of brain functional connectivity. The effectiveness of the proposed approach is shown in a practical scenario where we aim at quantifying the statistical relationships between brain regions in the context of non-invasive electromagnetic field recordings. Our results show that our method provide results with an higher specificity that classical approaches based on a two--step procedure where first the hidden process describing the brain activity is estimated through a linear optimization step and then the cortical cross--power spectrum is computed from the estimated time--series.

翻译：本文提出一种计算高效的线性优化方法，用于从观测过程的交叉功率谱估计隐藏多元随机过程的交叉功率谱。通过$\ell_1$正则化诱导交叉功率估计量的稀疏性，并采用快速迭代收缩阈值算法（FISTA）进行计算。相较于标准实现，我们证明适当的初始化步骤足以保证相关量所需的对称与反对称性质。进一步，我们展示了如何在FISTA更新中利用前向算子的结构特性，使该方法能够适用于大规模问题，例如脑功能连接领域产生的问题。本方法的有效性在实践场景中得到验证：我们旨在量化非侵入性电磁场记录背景下脑区间的统计关系。结果表明，与传统基于两步流程的方法相比，本方法具有更高的特异性——传统方法首先通过线性优化步骤估计描述脑活动的隐藏过程，再从估计的时间序列计算皮层交叉功率谱。