Besov Balls Research Articles

Tree Approximation and Optimal Encoding

Tree approximation is a new form of nonlinear approximation which appears naturally in some applications such as image processing and adaptive numerical methods. It is somewhat more restrictive than the usual n-term approximation. We show that the restrictions of tree approximation cost little in terms of rates of approximation. We then use that result to design encoders for compression. These encoders are universal (they apply to general functions) and progressive (increasing accuracy is obtained by sending bit stream increments). We show optimality of the encoders in the sense that they provide upper estimates for the Kolmogorov entropy of Besov balls.

Adaptive estimation in autoregression or -mixing regression via model selection

We study the problem of estimatingsome unknown regression function in a $\beta$-mixing dependent framework. To this end, we consider some collection of models which are finite dimensional spaces. A penalized least-squares estimator (PLSE) is built on a data driven selected model among this collection. We state non asymptotic risk bounds for this PLSE and give several examples where the procedure can be applied (autoregression, regression with arithmetically $\beta$-mixing design points, regression with mixing errors, estimation in additive frameworks, estimation of the order of the autoregression). In addition we show that under a weak moment condition on the errors, our estimator is adaptive in the minimax sense simultaneously over some family of Besov balls.

Adaptive Estimation of the Spectrum of a Stationary Gaussian Sequence

In this paper, we study the problem of nonparametric adaptive estimation of the spectral density f of a stationary Gaussian sequence. For this purpose, we consider a collection of finite-dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on possibly irregular grids or spaces of trigonometric polynomials). We estimate the spectral density by a projection estimator based on the periodogram and constructed on a data-driven choice of linear space from the collection. This data-driven choice is made via the minimization of a penalized projection contrast. The penalty function depends on If In,, but we give results including the estimation of this bound. Moreover, we give extensions to the case of unbounded spectral densities (long-memory processes). In all cases, we state non-asymptotic risk bounds in L2-norm for our estimator, and we show that it is adaptive in the minimax sense over a large class of Besov balls.

Model selection for (auto-)regression with dependent data

In this paper, we study the problem of non parametric estimation of an unknown regression function from dependent data with sub-Gaussian errors. As a particular case, we handle the autoregressive framework. For this purpose, we consider a collection of finite dimensional linear spaces (e.g. linear spaces spanned by wavelets or piecewise polynomials on a possibly irregular grid) and we estimate the regression function by a least-squares estimator built on a data driven selected linear space among the collection. This data driven choice is performed via the minimization of a penalized criterion akin to the Mallows' C p . We state non asymptotic risk bounds for our estimator in some -norm and we show that it is adaptive in the minimax sense over a large class of Besov balls of the form Bα,p,∞ (R) with p ≥ 1.

An Adaptive Compression Algorithm in Besov Spaces

Given a function f on [0,1] and a wavelet-type expansion of f , we introduce a new algorithm providing an approximation $\tilde f of f with a prescribed number D of nonzero coefficients in its expansion. This algorithm depends only on the number of coefficients to be kept and not on any smoothness assumption on f . Nevertheless it provides the optimal rate D-α of approximation with respect to the Lq -norm when f belongs to some Besov space Bαp,∈fty whenever α>(1/p-1/q)+ . These results extend to more general expansions including splines and piecewise polynomials and to multivariate functions. Moreover, this construction allows us to compute easily the metric entropy of Besov balls.

Density estimation by kernel and wavelets methods: Optimality of Besov spaces

This paper is showing that the saturation space of the minimax rate associated to a L p loss and linear estimators is the Besov space B s∞ p . More precisely, it is shown that if a function space included in L p is such that its minimax rate is the usual one s/(1 + 2 s) and if this rate is attained by a sequence of linear estimators, then this space is included in a ball of the space B s∞ p . This implies, for example, that the minimax rates that have been estimated for the Sobolev balls are in fact only a consequence of their inclusions in such Besov balls