  This paper investigates some theoretical properties of the Partial Least
Square (PLS) method. We focus our attention on the single component case, that
provides a useful framework to understand the underlying mechanism. We provide
a non-asymptotic upper bound on the quadratic loss in prediction with high
probability in a high dimensional regression context. The bound is attained
thanks to a preliminary test on the first PLS component. In a second time, we
extend these results to the sparse partial least squares (sPLS) approach. In
particular, we exhibit upper bounds similar to those obtained with the lasso
algorithm, up to an additional restricted eigenvalue constraint on the design
matrix.
