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The Loewner framework is one of the most successful data-driven model orderreduction techniques. If $N$ is the cardinality of a given data set, theso-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 dataset and they play a crucial role in the computation of the sought rationalmodel approximation.  In particular, the singular value decomposition of a linear combination of$\mathbb{S}$ and $\mathbb{L}$ provides the tools needed to construct accuratemodels which fulfill important approximation properties with respect to theoriginal 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 allocatethese matrices in certain memory-limited environments or excessivecomputational costs. Even though they do not possess any sparsity pattern, theLoewner and shifted Loewner matrices are extremely structured and, in thispaper, we show how to fully exploit their Cauchy-like structure to reduce thecost of computing accurate rational models while avoiding the explicitallocation of $\mathbb{L}$ and $\mathbb{S}$. In particular, the use of the\emph{hierarchically semiseparable} format allows us to remarkably lower boththe computational cost and the memory requirements of the Loewner frameworkobtaining a novel scheme whose costs scale with $N \log N$.

An efficient, memory-saving approach for the Loewner framework