Hai ZHANG, Zong Ben XU, Yao WANG, Xiang Yu CHANG, Yong LIANG
We derive a sharp nonasymptotic bound of parameter estimation of the L1/2 regularization. The bound shows that the solutions of the L1/2 regularization can achieve a loss within logarithmic factor of an ideal mean squared error and therefore underlies the feasibility and effectiveness of the L1/2 regularization. Interestingly, when applied to compressive sensing, the L1/2 regularization scheme has exhibited a very promising capability of completed recovery from a much less sampling information. As compared with the Lp (0 < p < 1) penalty, it is appeared that the L1/2 penalty can always yield the most sparse solution among all the Lp penalty when 1/2 ≤ p< 1, and when 0 < p < 1/2, the Lp penalty exhibits the similar properties as the L1/2 penalty. This suggests that the L1/2 regularization scheme can be accepted as the best and therefore the representative of all the Lp (0 < p < 1) regularization schemes.
with variable exponents α(·) and p(·), and prove the boundedness of multilinear Calderón-Zygmund singular operators on the product of these spaces.
