Envelopes: A new chapter in partial least squares regression

We describe and elaborate on foundations that connect partial least squares regression with recently developed envelope theory and methodology. These foundations explain why PLS regression can work well in high-dimensional regressions where the number of predictors exceeds the number of observations and set it apart from other predictive methodologies. We hope that our foundational perspective will stimulate cross-fertilization between statistics and chemometrics, leading eventually to important methodological advancements.