### 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 |
---|---|

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 |

### Fingerprint

### 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

}

_{p}loss with convex penalizations',

*IEEE Transactions on Neural Networks and Learning Systems*, vol. 26, no. 2, 6808493, pp. 265-276. https://doi.org/10.1109/TNNLS.2014.2314129

**A scalable projective scaling algorithm for l _{p} loss with convex penalizations.** / Zhou, Hongbo; Cheng, Qiang.

Research output: Contribution to journal › Article

TY - JOUR

T1 - A scalable projective scaling algorithm for lp loss with convex penalizations

AU - Zhou, Hongbo

AU - Cheng, Qiang

PY - 2015/2/1

Y1 - 2015/2/1

N2 - 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.

AB - 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.

KW - Convex function

KW - Karmarkar's projective scaling condition

KW - l loss function

KW - message passing algorithm (MPA)

KW - minimization-majorization (MM)

KW - nonconvex

KW - scalability

UR - http://www.scopus.com/inward/record.url?scp=84921454442&partnerID=8YFLogxK

U2 - 10.1109/TNNLS.2014.2314129

DO - 10.1109/TNNLS.2014.2314129

M3 - Article

C2 - 25608289

AN - SCOPUS:84921454442

VL - 26

SP - 265

EP - 276

JO - IEEE Transactions on Neural Networks and Learning Systems

JF - IEEE Transactions on Neural Networks and Learning Systems

SN - 2162-237X

IS - 2

M1 - 6808493

ER -

_{p}loss with convex penalizations. IEEE Transactions on Neural Networks and Learning Systems. 2015 Feb 1;26(2):265-276. 6808493. https://doi.org/10.1109/TNNLS.2014.2314129