  The hierarchical matrix ($\mathcal{H}^{2}$-matrix) formalism provides a way
to reinterpret the Fast Multipole Method and related fast summation schemes in
linear algebraic terms. The idea is to tessellate a matrix into blocks in such
as way that each block is either small or of numerically low rank; this enables
the storage of the matrix and the application of it to a vector in linear or
close to linear complexity. A key motivation for the reformulation is to extend
the range of dense matrices that can be represented. Additionally,
$\mathcal{H}^{2}$-matrices in principle also extend the range of operations
that can be executed to include matrix inversion and factorization. While such
algorithms can be highly efficient for certain specialized formats (such as
HBS/HSS matrices based on ``weak admissibility''), inversion algorithms for
general $\mathcal{H}^{2}$-matrices tend to be based on nested recursions and
recompressions, making them challenging to implement efficiently. An exception
is the \textit{strong recursive skeletonization (SRS)} algorithm by Minden, Ho,
Damle, and Ying, which involves a simpler algorithmic flow. However, SRS
greatly increases the number of blocks of the matrix that need to be stored
explicitly, leading to high memory requirements. This manuscript presents the
\textit{randomized strong recursive skeletonization (RSRS)} algorithm, which is
a reformulation of SRS that incorporates the randomized SVD (RSVD) to
simultaneously compress and factorize an $\mathcal{H}^{2}$-matrix. RSRS is a
``black box'' algorithm that interacts with the matrix to be compressed only
via its action on vectors; this extends the range of the SRS algorithm (which
relied on the ``proxy source'' compression technique) to include dense matrices
that arise in sparse direct solvers.
