Despite the large research effort devoted to learning dependencies between time series, the state of the art still faces a major limitation: existing methods learn partial correlations but fail to discriminate across distinct frequency bands. Motivated by many applications in which this differentiation is pivotal, we overcome this limitation by learning a block-sparse, frequency-dependent, partial correlation graph, in which layers correspond to different frequency bands, and partial correlations can occur over just a few layers. To this aim, we formulate and solve two nonconvex learning problems: the first has a closed-form solution and is suitable when there is prior knowledge about the number of partial correlations; the second hinges on an iterative solution based on successive convex approximation, and is effective for the general case where no prior knowledge is available. Numerical results on synthetic data show that the proposed methods outperform the current state of the art. Finally, the analysis of financial time series confirms that partial correlations exist only within a few frequency bands, underscoring how our methods enable the gaining of valuable insights that would be undetected without discriminating along the frequency domain.

翻译：尽管已有大量研究致力于学习时间序列之间的依赖关系，当前最先进的方法仍面临一个主要局限：现有方法虽能学习偏相关，但无法区分不同频带。受众多需要这种差异化的应用驱动，我们通过学习一种块稀疏、频率依赖的偏相关图来克服这一局限——该图的各层对应不同频带，且偏相关可能仅出现在少数层中。为此，我们提出并求解了两个非凸学习问题：第一个具有闭式解，适用于已知偏相关数量的场景；第二个基于逐次凸逼近的迭代解法，在无先验知识的通用情况下效果显著。合成数据的数值结果表明，所提方法优于现有最先进技术。最后，对金融时间序列的分析证实了偏相关仅存在于少数频带内，凸显了我们的方法如何在频率域进行区分前就能发现原本无法察觉的重要洞察。