  In this work, we explore modewise Johnson-Lindenstrauss embeddings (JLEs) as
a tool to reduce the computational cost and memory requirements of nuclear
many-body methods. JLEs are randomized projections of high-dimensional data
tensors onto low-dimensional subspaces that preserve key structural features.
Such embeddings allow for the oblivious and incremental compression of large
tensors, e.g., the nuclear Hamiltonian, into significantly smaller random
sketches that still allow for the accurate calculation of ground-state energies
and other observables. Their oblivious character makes it possible to compress
a tensor without knowing in advance exactly what observables one might want to
approximate at a later time. This opens the door for the use of tensors that
are much too large to store in memory, e.g., complete two-plus three-nucleon
Hamiltonians in large, symmetry-unrestricted bases. Such compressed
Hamiltonians can be stored and used later on with relative ease.
  As a first step, we analyze the JLE's impact on the second-order Many-Body
Perturbation Theory (MBPT) corrections for nuclear ground-state observables.
Numerical experiments for a wide range of closed-shell nuclei, model spaces and
state-of-the-art nuclear interactions demonstrate the validity and potential of
the proposed approach: We can compress nuclear Hamiltonians hundred- to
thousand-fold while only incurring mean relative errors of 1\% or less in
ground-state observables. Importantly, we show that JLEs capture the relevant
physical information contained in the highly structured Hamiltonian tensor
despite their random characteristics. In addition to the significant storage
savings, the achieved compressions imply multiple order-of magnitude reductions
in computational effort when the compressed Hamiltonians are used in
higher-order MBPT or nonperturbative many-body methods.
