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Partial Least Squares (PLS) methods have been heavily exploited to analysethe association between two blocs of data. These powerful approaches can beapplied to data sets where the number of variables is greater than the numberof observations and in presence of high collinearity between variables.Different sparse versions of PLS have been developed to integrate multiple datasets while simultaneously selecting the contributing variables. Sparsemodelling is a key factor in obtaining better estimators and identifyingassociations between multiple data sets. The cornerstone of the sparsityversion of PLS methods is the link between the SVD of a matrix (constructedfrom deflated versions of the original matrices of data) and least squaresminimisation in linear regression. We present here an accurate description ofthe most popular PLS methods, alongside their mathematical proofs. A unifiedalgorithm is proposed to perform all four types of PLS including theirregularised versions. Various approaches to decrease the computation time areoffered, and we show how the whole procedure can be scalable to big data sets.

A Unified Parallel Algorithm for Regularized Group PLS Scalable to Big Data