  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.
