### Abstract

This paper presents an accurate, efficient, and scalable algorithm for minimizing a special family of convex functions, which have a l_{p} 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 language | English |
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Article number | 6808493 |

Pages (from-to) | 265-276 |

Number of pages | 12 |

Journal | IEEE Transactions on Neural Networks and Learning Systems |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - 1 Feb 2015 |

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### Keywords

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

### Cite this

_{p}loss with convex penalizations.

*IEEE Transactions on Neural Networks and Learning Systems*,

*26*(2), 265-276. [6808493]. https://doi.org/10.1109/TNNLS.2014.2314129