### Abstract

The ℓ _{p}-constrained least squares, which is denoted by (Pc), for 0 < p < 1 is addressed. A maximal continuous curve of its critical solutions β(c) for different bounds c forms a critical path which can be constructed with the variational method. The path is a piecewise smooth single-valued function of c, containing non-optimal points such as saddle points and local maxima in general as well as global minima. The path of global minima may coincide with a critical path but may jump from a critical path to another one. The breakpoints of the greedy path (a critical path constructed with a certain greedy selection criterion) coincide with the step-by-step solutions generated by the orthogonal matching pursuit (OMP). A critical point of (Pc) is also a critical point of the ℓ _{p}-penalized least squares (Q _{λ}) which reformulates (P _{c}) with the Lagrangian multiplier. The greedy path is a multi-valued function of λ and is formed by a collection of multiple critical paths of (Q _{λ}).

Original language | English |
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Title of host publication | IEEE International Symposium on Information Theory - Proceedings |

Pages | 2221-2225 |

Number of pages | 5 |

DOIs | |

Publication status | Published - 2012 |

Externally published | Yes |

Event | 2012 IEEE International Symposium on Information Theory, ISIT 2012 - Cambridge, MA, United States Duration: 2012 Jul 1 → 2012 Jul 6 |

### Other

Other | 2012 IEEE International Symposium on Information Theory, ISIT 2012 |
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Country | United States |

City | Cambridge, MA |

Period | 12/7/1 → 12/7/6 |

### Fingerprint

### ASJC Scopus subject areas

- Applied Mathematics
- Modelling and Simulation
- Theoretical Computer Science
- Information Systems

### Cite this

_{p}-constrained least squares (0 < p < 1) and its critical path. In

*IEEE International Symposium on Information Theory - Proceedings*(pp. 2221-2225). [6283848] https://doi.org/10.1109/ISIT.2012.6283848

**ℓ _{p}-constrained least squares (0 < p < 1) and its critical path.** / Yukawa, Masahiro; Amari, Shun Ichi.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

_{p}-constrained least squares (0 < p < 1) and its critical path. in

*IEEE International Symposium on Information Theory - Proceedings.*, 6283848, pp. 2221-2225, 2012 IEEE International Symposium on Information Theory, ISIT 2012, Cambridge, MA, United States, 12/7/1. https://doi.org/10.1109/ISIT.2012.6283848

_{p}-constrained least squares (0 < p < 1) and its critical path. In IEEE International Symposium on Information Theory - Proceedings. 2012. p. 2221-2225. 6283848 https://doi.org/10.1109/ISIT.2012.6283848

}

TY - GEN

T1 - ℓ p-constrained least squares (0 < p < 1) and its critical path

AU - Yukawa, Masahiro

AU - Amari, Shun Ichi

PY - 2012

Y1 - 2012

N2 - The ℓ p-constrained least squares, which is denoted by (Pc), for 0 < p < 1 is addressed. A maximal continuous curve of its critical solutions β(c) for different bounds c forms a critical path which can be constructed with the variational method. The path is a piecewise smooth single-valued function of c, containing non-optimal points such as saddle points and local maxima in general as well as global minima. The path of global minima may coincide with a critical path but may jump from a critical path to another one. The breakpoints of the greedy path (a critical path constructed with a certain greedy selection criterion) coincide with the step-by-step solutions generated by the orthogonal matching pursuit (OMP). A critical point of (Pc) is also a critical point of the ℓ p-penalized least squares (Q λ) which reformulates (P c) with the Lagrangian multiplier. The greedy path is a multi-valued function of λ and is formed by a collection of multiple critical paths of (Q λ).

AB - The ℓ p-constrained least squares, which is denoted by (Pc), for 0 < p < 1 is addressed. A maximal continuous curve of its critical solutions β(c) for different bounds c forms a critical path which can be constructed with the variational method. The path is a piecewise smooth single-valued function of c, containing non-optimal points such as saddle points and local maxima in general as well as global minima. The path of global minima may coincide with a critical path but may jump from a critical path to another one. The breakpoints of the greedy path (a critical path constructed with a certain greedy selection criterion) coincide with the step-by-step solutions generated by the orthogonal matching pursuit (OMP). A critical point of (Pc) is also a critical point of the ℓ p-penalized least squares (Q λ) which reformulates (P c) with the Lagrangian multiplier. The greedy path is a multi-valued function of λ and is formed by a collection of multiple critical paths of (Q λ).

UR - http://www.scopus.com/inward/record.url?scp=84867496879&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84867496879&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2012.6283848

DO - 10.1109/ISIT.2012.6283848

M3 - Conference contribution

AN - SCOPUS:84867496879

SN - 9781467325790

SP - 2221

EP - 2225

BT - IEEE International Symposium on Information Theory - Proceedings

ER -