Our purpose in this paper is to provide a general approach to model selection via penalization for Gaussian regression and to develop our point of view about this subject. The advantage and importance of model selection come from the fact that it provides a suitable approach to many different types of problems, starting from model selection per se (among a family of parametric models, which one is more suitable for the data at hand), which includes for instance variable selection in regression models,...

We prove asymptotic formulas for the behavior of Gelfand and Kolmogorov numbers of Sobolev embeddings between Besov and Triebel-Lizorkin spaces of radial distributions. Our method works also for Weyl numbers.

For a precompact subset K of a Hilbert space we prove the following inequalities: $n^{1/2}cₙ\left(cov\left(K\right)\right)\le c_K(1+{\sum }_{k=1}^{ⁿ}{k}^{-1/2}{e}_k\left(K\right))$, n ∈ ℕ, and $k^{1/2}{c}_{k+n}\left(cov\left(K\right)\right)\le c[lo{g}^{1/2}(n+1)\epsilon ₙ\left(K\right)+{\sum }_{j=n+1}^{\infty }{\epsilon }_j\left(K\right)/\left(jlo{g}^{1/2}(j+1)\right)]$, k,n ∈ ℕ, where cₙ(cov(K)) is the nth Gelfand number of the absolutely convex hull of K and ${\epsilon }_k\left(K\right)$ and $e_k\left(K\right)$ denote the kth entropy and kth dyadic entropy number of K, respectively. The inequalities are, essentially, a reformulation of the corresponding inequalities given in [CKP] which yield asymptotically optimal estimates of the Gelfand numbers cₙ(cov(K)) provided that the entropy numbers εₙ(K) are slowly...

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 $L_p\left([0,1]\right)$, 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 ${}^d=\times ...\times$ in $L_p\left({[0,1]}^d\right)$,...

*This research was supported by the National Science Foundation Grant DMS 0200187 and by ONR Grant N00014-96-1-1003This paper is a survey which also contains some new results on the nonlinear approximation with regard to a basis or, more generally, with regard to a minimal system. Approximation takes place in a Banach or in a quasi-Banach space. The last decade was very successful in studying nonlinear approximation. This was motivated by numerous applications. Nonlinear approximation is important...