View full-text article in PMC J Mach Learn Res. Author manuscript; available in PMC: 2019 May 22. Published in final edited form as: J Mach Learn Res. 2015;16:3367–3402. Search in PMC Search in PubMed View in NLM Catalog Add to search Copyright and License information PMC Copyright notice Algorithm 2.1. Rank-Restricted Soft SVD1.InitializeA=UDwhereUm×ris a randomly chosen matrix with orthonormal columnsandD=Ir,ther×ridentity matrix.2.GivenA,solve forB:minimizeB‖X−ABT‖FT+λ‖B‖F2.(15)This is a multiresponse ridge regression, with solutionB˜T=(D2+λI)−1DUTX.(16)This is simply matrix multiplication followed by coordinate-wise shrinkage.3.Compute the SVD ofB˜D=V˜D˜2RT,and letV←V˜,D←D˜,andB=VD.4.GivenB,solve forA:minimizeA‖X−ABT‖FT+λ‖A‖F2.(17)This is also a multiresponse ridge regression, with solutionA˜=XVD(D2+λI)−1.(18)Again matrix multiplication followed by coordinate-wise shrinkage.5.Compute the SVD ofA˜D=U˜D˜2RT,and letU←U˜,D←D˜,andA=UD.6.Repeat steps(2)−(5)until convergence ofABT.7.ComputeM=XV,andthenit'sSVD:M=UDσRT.Then outputU,V←VRandSλ(Dσ)=diag[(σ1−λ)+,…,(σr−λ)+].