The main goal of this paper is to understand which properties of a basis are important for certain direct and inverse theorems in nonlinear approximation. We study greedy approximation with regard to bases with different properties. We consider bases that are tensor products of univariate greedy bases. Some results known for unconditional bases are extended to the case of quasi-greedy bases.
We consider the moving least-square (MLS) method by the coefficient-based regression framework with -regularizer and the sample dependent hypothesis spaces. The data dependent characteristic of the new algorithm provides flexibility and adaptivity for MLS. We carry out a rigorous error analysis by using the stepping stone technique in the error decomposition. The concentration technique with the -empirical covering number is also employed in our study to improve the sample error. We derive the satisfactory learning rate that can be arbitrarily close to the best rate under more natural and much simpler conditions.
We investigate the efficiency of Chebyshev Thresholding Greedy Algorithm (CTGA) for an n-term approximation with respect to general bases in a Banach space. We show that the convergence property of CTGA is better than TGA for non-quasi-greedy bases. Then we determine the exact rate of the Lebesgue constants L ch n for two examples of such bases: the trigonometric system and the summing basis. We also establish the upper estimates for L ch n with respect to general bases in terms of quasi-greedy parameter, democracy parameter and A-property parameter. These estimates do not involve an unconditionality parameter, therefore they are better than those of TGA. In particular, for conditional quasi-greedy bases, a faster convergence rate is obtained.
MSC: 41A25; 41A46; 42A10
We study restricted Monte Carlo integration for anisotropic H枚lder-Nikolskii classes. The results show that with clog 2 n random bits we have the same optimal order for the nth minimal Monte Carlo integration error as with arbitrary random numbers. We also study the computation of integration on anisotropic Sobolev classes in the quantum setting and present the optimal bound of nth minimal query error. The results show that the error bound of quantum algorithms is much smaller than that of deterministic and randomized algorithms.
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