Showing papers by "David L. Donoho published in 1990"

Minimax Risk Over Hyperrectangles, and Implications

Abstract: Consider estimating the mean of a standard Gaussian shift when that mean is known to lie in an orthosymmetric quadratically convex set in $l_2$. Such sets include ellipsoids, hyperrectangles and $l_p$-bodies with $p > 2$. The minimax risk among linear estimates is within 25% of the minimax risk among all estimates. The minimax risk among truncated series estimates is within a factor 4.44 of the minimax risk. This implies that the difficulty of estimation--a statistical quantity--is measured fairly precisely by the $n$-widths--a geometric quantity. If the set is not quadratically convex, as in the case of $l_p$-bodies with $p < 2$, things change appreciably. Minimax linear estimators may be out-performed arbitrarily by nonlinear estimates. The (ordinary, Kolmogorov) $n$-widths still determine the difficulty of linear estimation, but the difficulty of nonlinear estimation is tied to the (inner, Bernstein) $n$-widths, which can be far smaller. Essential use is made of a new heuristic: that the difficulty of the hardest rectangular subproblem is equal to the difficulty of the full problem.

Does the maximum entropy method improve sensitivity

Abstract: Maximum entropy reconstruction has been used in several fields to produce visually striking reconstructions of positive objects (images, densities, spectra) from noisy, indirect measurements. In magnetic resonance spectroscopy, this technique is notable for its apparent noise suppression and its avoidance of the artifacts that affect discrete Fourier transform spectra of short (zero-extended) data records. In the general case where the length of the reconstructed spectrum exceeds that of the data record or where a convolution kernel is incorporated in the reconstruction, no known analytical solution to the reconstruction problem exists. Consequently, knowledge of the properties of maximum entropy reconstruction has been mainly anecdotal, based on a small selection of published reconstructions. However, in the limiting case where the lengths of the reconstructed spectrum and the data record are the same and a convolution kernel is not applied, the problem can be solved analytically. The solution has a simple structure that helps explain several commonly observed features of maximum entropy reconstructions--for example, the biases in the recovered intensities and the fact that noise near the baseline is more successfully suppressed than is noise superimposed on broad features in the spectrum. The solution also shows that the noise suppression offered by maximum entropy reconstruction could (in this special case) be equally well obtained by a "cosmetic" device: simply displaying the conventional Fourier transform reconstruction using a certain nonlinear plotting scale for the vertical (y) coordinate.