Showing papers by "David L. Donoho published in 1992"

Breakdown Properties of Location Estimates Based on Halfspace Depth and Projected Outlyingness

Abstract: We describe multivariate generalizations of the median, trimmed mean and $W$ estimates. The estimates are based on a geometric construction related to "projection pursuit." They are both affine equivariant (coordinate-free) and have high breakdown point. The generalization of the median has a breakdown point of at least $1/(d + 1)$ in dimension $d$ and the breakdown point can be as high as $1/3$ under symmetry. In contrast, various estimators based on rejecting apparent outliers and taking the mean of the remaining observations have breakdown points not larger than $1/(d + 1)$ in dimension $d$.

Maximum entropy and the nearly black object

Abstract: SUMMARY Maximum entropy (ME) inversion is a non-linear inversion technique for inverse problems where the object to be recovered is known to be positive. It has been applied in areas ranging from radio astronomy to various forms of spectroscopy, sometimes with dramatic success. In some cases, ME has attained an order of magnitude finer resolution and/or an order of magnitude smaller noise level than that obtainable by standard linear methods. The dramatic successes all seem to occur in cases where the object to be recovered is 'nearly black': essentially zero in the vast majority of samples. We show that near-blackness is required, both for signal-to-noise enhancements and for superresolution. However, other methods-in particular, minimum /1-norm reconstruction-may exploit near-blackness to an even greater extent.

Superresolution via sparsity constraints

Abstract: Consider the problem of recovering a measure $\mu $ supported on a lattice of span $\Delta $, when measurements are only available concerning the Fourier Transform $\hat \mu (\omega )$ at frequencies $|\omega | \leqslant \Omega $. If $\Omega $ is much smaller than the Nyquist frequency ${\pi / \Delta }$ and the measurements are noisy, then, in general, stable recovery of $\mu $ is impossible. In this paper it is shown that if, in addition, we know that the measure $\mu $ satisfies certain sparsity constraints, then stable recovery is possible. Say that a set has Rayleigh index less than or equal to R if in any interval of length ${{4\pi } / \Omega } \cdot R$ there are at most R elements. Indeed, if the (unknown) support of $\mu $ is known, a priori, to have Rayleigh index at most R, then stable recovery is possible with a stability coefficient that grows at most like $\Delta ^{ - 2R - 1} $ as $\Delta \to 0$. This result validates certain practical efforts, in spectroscopy, seismic prospecting, and astrono...

Signal recovery and the large sieve

Abstract: Inequalities are developed for the fraction of a bandlimited function’s $L_p $ norm that can be concentrated on any set of small “Nyquist density.” Two applications are mentioned. First, that a bandlimited function corrupted by impulsive noise can be reconstructed perfectly, provided the noise is concentrated on a set of Nyquist density $ < 1/\pi $; second, that a wideband signal supported on a set of Nyquist density $ < 1/ \pi$ can be reconstructed stably from noisy data, even when the low-frequency information is completely missing.

Renormalization Exponents and Optimal Pointwise Rates of Convergence

Abstract: Simple renormalization arguments can often be used to calculate optimal rates of convergence for estimating linear functionals from indirect measurements contaminated with white noise. This allows one to quickly identify optimal rates for certain problems of density estimation, nonparametric regression, signal recovery and tomography. Optimal kernels may also be derived from renormalization; we give examples for deconvolution and tomography.