DocumentCode :
44939
Title :
Can Critical-Point Paths Under {\\ell }_{{p}} -Regularization (0< p< 1) Reach the Sparsest Least Squares S
Author :
Kwangjin Jeong ; Yukawa, Masahiro ; Amari, Shun-Ichi
Author_Institution :
Dept. of Electron. & Electr. Eng., Keio Univ., Yokohama, Japan
Volume :
60
Issue :
5
fYear :
2014
fDate :
May-14
Firstpage :
2960
Lastpage :
2968
Abstract :
The solution path of the least square problem under ℓp-regularization (0 <; p <; 1) is studied, where the Lagrangian multiplier λ due to the constraint is the parameter of the path. It is first proven that the least square solution of an unconstrained overdetermined linear system is connected with the origin, under a mild condition, by a continuous path of critical points of an ℓp-regularized squared error function. Based on this fact, it is proven that every sparsest least square solution of an underdetermined system is connected with the origin by a critical-point path. The existence theorem holds more generally for any least square solution whose support has its associated submatrix of the fat sensing matrix be full column rank. This is a sufficient condition for the existence, and allows to reduce the underdetermined problem to an overdetermined one with the off-support variable(s) nullified. A necessary condition is that the gradient of the ℓp regularizer with respect to the support variables lies in the row space of the submatrix (which is not necessarily full column rank).
Keywords :
critical points; least squares approximations; optimisation; ℓp-regularization; ℓp-regularized squared error function; Lagrangian multiplier; critical-point paths; sensing matrix; sparsest least square solution; unconstrained overdetermined linear system; Eigenvalues and eigenfunctions; Joining processes; Linear systems; Optimization; Sensors; Silicon; Vectors; $ell_{p}$ -norm regularization $(0
fLanguage :
English
Journal_Title :
Information Theory, IEEE Transactions on
Publisher :
ieee
ISSN :
0018-9448
Type :
jour
DOI :
10.1109/TIT.2014.2312723
Filename :
6776531
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