Besov Scales Research Articles

Quasi-Linear Wavelet Estimation

The main paradigm of the modern wavelet theory of spatial adaptation formulated by Donoho and Johnstone is that there is a divergence between the linear minimax adaptation theory and the heuristic guiding algorithm development that leads to the necessity of using strongly nonlinear adaptive thresholded methods. On the other hand, it is well known that linear adaptive estimates are the best whenever an estimated function is smooth. Is it possible to suggest a quasi-linear wavelet estimate, by adding to a linear adaptive estimate a minimal number of nonlinear terms on finest scales, that offers advantages of linear adaptive estimates and at the same time matches asymptotic properties of strongly nonlinear procedures like the benchmark SureShrink? The answer is “yes,” and we discuss quasi-linear estimation both theoretically and via a Monte Carlo study. In particular, I show that, asymptotically, a quasi-linear procedure not only matches properties of SureShrink over the Besov scale, but also allows us to relax familiar assumptions and solve a long-standing problem of rate and sharp optimal estimation of monotone functions. For the case of small sample sizes and functions that contain spiky/jumps parts and smooth parts, a quasi-linear estimate performs exceptionally well in terms of visual aesthetic appeal, approximation, and data compression.

Minimax estimation via wavelet shrinkage

We attempt to recover an unknown function from noisy, sampled data. Using orthonormal bases of compactly supported wavelets, we develop a nonlinear method which works in the wavelet domain by simple nonlinear shrinkage of the empirical wavelet coefficients. The shrinkage can be tuned to be nearly minimax over any member of a wide range of Triebel- and Besov-type smoothness constraints and asymptotically mini-max over Besov bodies with $p \leq q$. Linear estimates cannot achieve even the minimax rates over Triebel and Besov classes with $p<2$, so the method can significantly outperform every linear method (e.g., kernel, smoothing spline, sieve in a minimax sense). Variants of our method based on simple threshold nonlinear estimators are nearly minimax. Our method possesses the interpretation of spatial adaptivity; it reconstructs using a kernel which may vary in shape and bandwidth from point to point, depending on the data. Least favorable distributions for certain of the Triebel and Besov scales generate objects with sparse wavelet transforms. Many real objects have similarly sparse transforms, which suggests that these minimax results are relevant for practical problems. Sequels to this paper, which was first drafted in November 1990, discuss practical implementation, spatial adaptation properties, universal near minimaxity and applications to inverse problems.

Atomic decompositions on manifolds with bounded geometry

We give the atomic decomposition of Triebel–Lizorkin and Besov scales on Riemannian manifolds of bounded geometry. In order to do it we modify the moment condition for atoms. 1 Preliminaries A fundamental technique in harmonic analysis is to represent a function or distribution as a linear combination of function of an elementary form. One of the example is the atomic decomposition. The decomposition appeared to be useful for characterizations of various function spaces as well as for the investigation of continuity properties of pseudodifferential operators. The decomposition comes from the theory of homogeneous Hardy spaces Hp, but it works for inhomogeneous function spaces as well. Frazier and Jawerth have got the atomic decomposition theorem for the Triebel-Lizorkin and Besov scales [2, 3]. A smooth (s,p)–atom centered in the ball B(xo, r) is a test function a satisfying the following three condition: supp a ⊂ B(xo, 2r) (1) sup x∈X | (Da)(x) | ≤ r n p , for any α, | α |≤ L, (2) ∫ Rn xa(x) = 0 for any β , | β |≤M, (3) where L,M are positive constant depending on s, p. The inhomogeneous spaces of F s p,q − Bs p,q type are invariant with respect to large class of diffeomorphism, some assumptions about derivatives are needed. But, if we look at the above condition of atoms we see immediately that the first two condition are invariant with respect to the diffeomorphism after the suitable normalization of the constants. But the third condition called the moment condition is not invariant. In the paper we give a new definition of the moment condition, which is invariant with respect to diffeomorphisms . In consequence we get atomic decompositions of the Triebel-Lizorkin and Besov scales on Riemannian manifolds with bounded geometry. 1.1 Function spaces For convenience we recall the definition of the Triebel-Lizorkin and Besov scales on Rn. Let η ∈ S(Rn) (the Schwartz space) be a non-negative function with η(x) = 1 if |x| ≤ 1 and η(x) = 0 if |x| ≥ 3 2 . We put η0 = η, η1 = η( x 2 )− η(x) ηk = η1(2x), k = 2, 3, . . . Then supp ηj ⊂ {x : 2j−1 ≤| x |≤ 32j−1}, j = 1, 2, . . . and ∑∞ j=0 ηj(x) = 1.

Adapting to Unknown Smoothness via Wavelet Shrinkage

Abstract We attempt to recover a function of unknown smoothness from noisy sampled data. We introduce a procedure, SureShrink, that suppresses noise by thresholding the empirical wavelet coefficients. The thresholding is adaptive: A threshold level is assigned to each dyadic resolution level by the principle of minimizing the Stein unbiased estimate of risk (Sure) for threshold estimates. The computational effort of the overall procedure is order N · log(N) as a function of the sample size N. SureShrink is smoothness adaptive: If the unknown function contains jumps, then the reconstruction (essentially) does also; if the unknown function has a smooth piece, then the reconstruction is (essentially) as smooth as the mother wavelet will allow. The procedure is in a sense optimally smoothness adaptive: It is near minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet. We know from a previous paper by the authors that traditional smoothing methods—kernels, splines, and orthogonal series estimates—even with optimal choices of the smoothing parameter, would be unable to perform in a near-minimax way over many spaces in the Besov scale. Examples of SureShrink are given. The advantages of the method are particularly evident when the underlying function has jump discontinuities on a smooth background.

Adapting to Unknown Smoothness via Wavelet Shrinkage

Abstract We attempt to recover a function of unknown smoothness from noisy sampled data. We introduce a procedure, SureShrink, that suppresses noise by thresholding the empirical wavelet coefficients. The thresholding is adaptive: A threshold level is assigned to each dyadic resolution level by the principle of minimizing the Stein unbiased estimate of risk (Sure) for threshold estimates. The computational effort of the overall procedure is order N · log(N) as a function of the sample size N. SureShrink is smoothness adaptive: If the unknown function contains jumps, then the reconstruction (essentially) does also; if the unknown function has a smooth piece, then the reconstruction is (essentially) as smooth as the mother wavelet will allow. The procedure is in a sense optimally smoothness adaptive: It is near minimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet. We know from a previous paper by the authors that traditional smoot...