A scalable projective scaling algorithm for lp loss with convex penalizations

Hongbo Zhou, Qiang Cheng

Research output: Contribution to journalArticlepeer-review

3 Scopus citations


This paper presents an accurate, efficient, and scalable algorithm for minimizing a special family of convex functions, which have a lp loss function as an additive component. For this problem, well-known learning algorithms often have well-established results on accuracy and efficiency, but there exists rarely any report on explicit linear scalability with respect to the problem size. The proposed approach starts with developing a second-order learning procedure with iterative descent for general convex penalization functions, and then builds efficient algorithms for a restricted family of functions, which satisfy the Karmarkar's projective scaling condition. Under this condition, a light weight, scalable message passing algorithm (MPA) is further developed by constructing a series of simpler equivalent problems. The proposed MPA is intrinsically scalable because it only involves matrix-vector multiplication and avoids matrix inversion operations. The MPA is proven to be globally convergent for convex formulations; for nonconvex situations, it converges to a stationary point. The accuracy, efficiency, scalability, and applicability of the proposed method are verified through extensive experiments on sparse signal recovery, face image classification, and over-complete dictionary learning problems.

Original languageEnglish
Article number6808493
Pages (from-to)265-276
Number of pages12
JournalIEEE Transactions on Neural Networks and Learning Systems
Issue number2
StatePublished - 1 Feb 2015


  • Convex function
  • Karmarkar's projective scaling condition
  • l loss function
  • message passing algorithm (MPA)
  • minimization-majorization (MM)
  • nonconvex
  • scalability


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