  High-dimensional linear regression is important in many scientific fields.
This article considers discrete measured data of underlying smooth latent
processes, as is often obtained from chemical or biological systems.
Interpretation in high dimensions is challenging because the nullspace and its
interplay with regularization shapes regression coefficients. The data's
nullspace contains all coefficients that satisfy $\mathbf{Xw}=\mathbf{0}$, thus
allowing very different coefficients to yield identical predictions. We
developed an optimization formulation to compare regression coefficients and
coefficients obtained by physical engineering knowledge to understand which
part of the coefficient differences are close to the nullspace. This nullspace
method is tested on a synthetic example and lithium-ion battery data. The case
studies show that regularization and z-scoring are design choices that, if
chosen corresponding to prior physical knowledge, lead to interpretable
regression results. Otherwise, the combination of the nullspace and
regularization hinders interpretability and can make it impossible to obtain
regression coefficients close to the true coefficients when there is a true
underlying linear model. Furthermore, we demonstrate that regression methods
that do not produce coefficients orthogonal to the nullspace, such as fused
lasso, can improve interpretability. In conclusion, the insights gained from
the nullspace perspective help to make informed design choices for building
regression models on high-dimensional data and reasoning about potential
underlying linear models, which are important for system optimization and
improving scientific understanding.
