  The Loewner framework is one of the most successful data-driven model order
reduction techniques. If $N$ is the cardinality of a given data set, the
so-called Loewner and shifted Loewner matrices
$\mathbb{L}\in\mathbb{C}^{N\times N}$ and $\mathbb{S}\in\mathbb{C}^{N\times N}$
can be defined by solely relying on information encoded in the considered data
set and they play a crucial role in the computation of the sought rational
model approximation.
  In particular, the singular value decomposition of a linear combination of
$\mathbb{S}$ and $\mathbb{L}$ provides the tools needed to construct accurate
models which fulfill important approximation properties with respect to the
original data set.
  However, for highly-sampled data sets, the dense nature of $\mathbb{L}$ and
$\mathbb{S}$ leads to numerical difficulties, namely the failure to allocate
these matrices in certain memory-limited environments or excessive
computational costs. Even though they do not possess any sparsity pattern, the
Loewner and shifted Loewner matrices are extremely structured and, in this
paper, we show how to fully exploit their Cauchy-like structure to reduce the
cost of computing accurate rational models while avoiding the explicit
allocation of $\mathbb{L}$ and $\mathbb{S}$. In particular, the use of the
\emph{hierarchically semiseparable} format allows us to remarkably lower both
the computational cost and the memory requirements of the Loewner framework
obtaining a novel scheme whose costs scale with $N \log N$.
