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

Masahiro Yukawa, Shun Ichi Amari

Research output: Chapter in Book/Report/Conference proceedingConference contribution

3 Citations (Scopus)


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 languageEnglish
Title of host publication2012 IEEE International Symposium on Information Theory Proceedings, ISIT 2012
Number of pages5
Publication statusPublished - 2012 Oct 22
Externally publishedYes
Event2012 IEEE International Symposium on Information Theory, ISIT 2012 - Cambridge, MA, United States
Duration: 2012 Jul 12012 Jul 6

Publication series

NameIEEE International Symposium on Information Theory - Proceedings


Other2012 IEEE International Symposium on Information Theory, ISIT 2012
Country/TerritoryUnited States
CityCambridge, MA

ASJC Scopus subject areas

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


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