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This paper is a survey of recent results on some problems of supervised learning in the setting formulated by Cucker and Smale. Supervised learning, or learning-from-examples, refers to a process that builds on the base of available data of inputs and outputs , i = 1,...,m, a function that best represents the relation between the inputs x ∈ X and the corresponding outputs y ∈ Y. The goal is to find an estimator on the base of given data that approximates well the regression function of...
We study nonlinear m-term approximation in a Banach space with regard to a basis. It is known that in the case of a greedy basis (like the Haar basis in , 1 < p < ∞) a greedy type algorithm realizes nearly best m-term approximation for any individual function. In this paper we generalize this result in two directions. First, instead of a greedy algorithm we consider a weak greedy algorithm. Second, we study in detail unconditional nongreedy bases (like the multivariate Haar basis in ,...
We study the following nonlinear method of approximation by trigonometric polynomials. For a periodic function f we take as an approximant a trigonometric polynomial of the form , where is a set of cardinality m containing the indices of the m largest (in absolute value) Fourier coefficients f̂(k) of the function f. Note that Gₘ(f) gives the best m-term approximant in the L₂-norm, and therefore, for each f ∈ L₂, ||f-Gₘ(f)||₂ → 0 as m → ∞. It is known from previous results that in the case of...
We consider convergence of thresholding type approximations with regard to general complete minimal systems eₙ in a quasi-Banach space X. Thresholding approximations are defined as follows. Let eₙ* ⊂ X* be the conjugate (dual) system to eₙ; then define for ε > 0 and x ∈ X the thresholding approximations as , where . We study a generalized version of that we call the weak thresholding approximation. We modify the in the following way. For ε > 0, t ∈ (0,1) we set and consider the weak...
We study Lebesgue-type inequalities for greedy approximation with respect to quasi-greedy bases. We mostly concentrate on the spaces. The novelty of the paper is in obtaining better Lebesgue-type inequalities under extra assumptions on a quasi-greedy basis than known Lebesgue-type inequalities for quasi-greedy bases. We consider uniformly bounded quasi-greedy bases of , 1 < p < ∞, and prove that for such bases an extra multiplier in the Lebesgue-type inequality can be taken as C(p)ln(m+1)....
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