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High-dimensional linear regression is important in many scientific fields.This article considers discrete measured data of underlying smooth latentprocesses, as is often obtained from chemical or biological systems.Interpretation in high dimensions is challenging because the nullspace and itsinterplay with regularization shapes regression coefficients. The data'snullspace contains all coefficients that satisfy $\mathbf{Xw}=\mathbf{0}$, thusallowing very different coefficients to yield identical predictions. Wedeveloped an optimization formulation to compare regression coefficients andcoefficients obtained by physical engineering knowledge to understand whichpart of the coefficient differences are close to the nullspace. This nullspacemethod is tested on a synthetic example and lithium-ion battery data. The casestudies show that regularization and z-scoring are design choices that, ifchosen corresponding to prior physical knowledge, lead to interpretableregression results. Otherwise, the combination of the nullspace andregularization hinders interpretability and can make it impossible to obtainregression coefficients close to the true coefficients when there is a trueunderlying linear model. Furthermore, we demonstrate that regression methodsthat do not produce coefficients orthogonal to the nullspace, such as fusedlasso, can improve interpretability. In conclusion, the insights gained fromthe nullspace perspective help to make informed design choices for buildingregression models on high-dimensional data and reasoning about potentialunderlying linear models, which are important for system optimization andimproving scientific understanding.

Interpretation of High-Dimensional Linear Regression: Effects of Nullspace and Regularization Demonstrated on Battery Data