Relation between PLS and OLS regression in terms of the eigenvalue distribution of the regressor covariance matrix

Partial least squares (PLS) is a dimensionality reduction technique introduced in the field of chemometrics and successfully employed in many other areas. The PLS components are obtained by maximizing the covariance between linear combinations of the regressors and of the target variables. In this work, we focus on its application to scalar regression problems. PLS regression consists in finding the least squares predictor that is a linear combination of a subset of the PLS components. Alternatively, PLS regression can be formulated as a least squares problem restricted to a Krylov subspace. This equivalent formulation is employed to analyze the distance between ${\hat{\boldsymbol\beta}\;}_{\mathrm{PLS}}^{\scriptscriptstyle {(L)}}$, the PLS estimator of the vector of coefficients of the linear regression model based on $L$ PLS components, and $\hat{\boldsymbol \beta}_{\mathrm{OLS}}$, the one obtained by ordinary least squares (OLS), as a function of $L$. Specifically, ${\hat{\boldsymbol\beta}\;}_{\mathrm{PLS}}^{\scriptscriptstyle {(L)}}$ is the vector of coefficients in the aforementioned Krylov subspace that is closest to $\hat{\boldsymbol \beta}_{\mathrm{OLS}}$ in terms of the Mahalanobis distance with respect to the covariance matrix of the OLS estimate. We provide a bound on this distance that depends only on the distribution of the eigenvalues of the regressor covariance matrix. Numerical examples on synthetic and real-world data are used to illustrate how the distance between ${\hat{\boldsymbol\beta}\;}_{\mathrm{PLS}}^{\scriptscriptstyle {(L)}}$ and $\hat{\boldsymbol \beta}_{\mathrm{OLS}}$ depends on the number of clusters in which the eigenvalues of the regressor covariance matrix are grouped.