Multiplicative Updates for the Lasso

Multiplicative updates have proven useful for non-negativity constrained optimization. Presently, we demonstrate how multiplicative updates also can be used for unconstrained optimization. This is for instance useful when estimating the least absolute shrinkage and selection operator (LASSO) i.e. least squares minimization with L₁-norm regularization, since the multiplicative updates (MU) can efficiently exploit the structure of the problem traditionally solved using quadratic programming (QP). We derive two algorithms based on MU for the LASSO and compare the performance to Matlabs standard QP solver as well as the basis pursuit denoising algorithm (BP) which can be obtained from www.sparselab.stanford.edu. The algorithms were tested on three benchmark bio-informatic datasets: A small scale data set where the number of observations is larger than the number of variables estimated (M < J) and two large scale microarray data sets (M Gt J). For small scale data the two MU algorithms, QP and BP give identical results while the time used is more or less of the same order. However, for large scale problems QP is unstable and slow. Both algorithms based on MU on the other hand are stable and faster but not as efficient as the BP algorithm and converge slowly for small regularizations. The benefit of the present MU algorithms is that they are easy to implement, they bridge multiplicative updates to unconstrained optimization and the updates derived monotonically decrease the cost-function thus does not need any objective function evaluation. Finally, both MU are potentially useful for a wide range of other models such as the elastic net or the fused LASSO. The Matlab implementations of the LASSO based on MU can be downloaded.