### 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 | 2012 IEEE International Symposium on Information Theory Proceedings, ISIT 2012 |

Pages | 2221-2225 |

Number of pages | 5 |

DOIs | |

Publication status | Published - 2012 Oct 22 |

Externally published | Yes |

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

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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### 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 |

### ASJC Scopus subject areas

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

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## Cite this

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

*2012 IEEE International Symposium on Information Theory Proceedings, ISIT 2012*(pp. 2221-2225). [6283848] (IEEE International Symposium on Information Theory - Proceedings). https://doi.org/10.1109/ISIT.2012.6283848