TY - GEN

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

AU - Yukawa, Masahiro

AU - Amari, Shun Ichi

PY - 2012/10/22

Y1 - 2012/10/22

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).

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U2 - 10.1109/ISIT.2012.6283848

DO - 10.1109/ISIT.2012.6283848

M3 - Conference contribution

AN - SCOPUS:84867496879

SN - 9781467325790

T3 - IEEE International Symposium on Information Theory - Proceedings

SP - 2221

EP - 2225

BT - 2012 IEEE International Symposium on Information Theory Proceedings, ISIT 2012

T2 - 2012 IEEE International Symposium on Information Theory, ISIT 2012

Y2 - 1 July 2012 through 6 July 2012

ER -