  First-principles calculations of electron interactions in materials have seen
rapid progress in recent years, with electron-phonon (e-ph) interactions being
a prime example. However, these techniques use large matrices encoding the
interactions on dense momentum grids, which reduces computational efficiency
and obscures interpretability. For e-ph interactions, existing interpolation
techniques leverage locality in real space, but the high dimensionality of the
data remains a bottleneck to balance cost and accuracy. Here we show an
efficient way to compress e-ph interactions based on singular value
decomposition (SVD), a widely used matrix / image compression technique.
Leveraging (un)constrained SVD methods, we accurately predict material
properties related to e-ph interactions - including charge mobility, spin
relaxation times, band renormalization, and superconducting critical
temperature - while using only a small fraction (1-2%) of the interaction data.
These findings unveil the hidden low-dimensional nature of e-ph interactions.
Furthermore, they accelerate state-of-the-art first-principles e-ph
calculations by about two orders of magnitudes without sacrificing accuracy.
Our Pareto-optimal parametrization of e-ph interactions can be readily
generalized to electron-electron and electron-defect interactions, as well as
to other couplings, advancing quantitative studies of condensed matter.
