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We study a simple \"horizon model\" for the problem of recovering an image from noisy data; in this model the image has an edge with α - Holder regularity. Adopting the viewpoint of computational harmonic analysis, we develop an overcomplete collection of atoms called wedgelets, dyadically organized indicator functions with a variety of locations, scales and orientations. The wedgelet representation provides nearly optimal representations of objects in the horizon model, as measured by minimax description length. We show how to rapidly compute a wedgelet approximation to noisy data by finding a special edgelet-decorated recursive partition which minimizes a complexity-penalized sum of squares. This estimate, using sufficient subpixel resolution, achieves nearly the minimax mean-squared error in the horizon model. In fact, the method is adaptive in the sense that it achieves nearly the minimax risk for any value of the unknown degree of regularity of the horizon, 1 ≤ α ≤ 2. Wedgelet analysis and denoising may be used successfully outside the horizon model. We study images modelled as indicators of star-shaped sets with smooth boundaries and show that complexity-penalized wedgelet partitioning achieves nearly the minimax risk in that setting also.

Variable selection is important for developing accurate and interpretable prediction models. While classical and penalized methods are widely used, few simulation studies provide meaningful comparisons. This study compares their predictive performance and model complexity in low-dimensional data. Three classical methods (best subset selection, backward elimination, and forward selection) and four penalized methods (nonnegative garrote (NNG), lasso, adaptive lasso (ALASSO), and relaxed lasso (RLASSO)) were compared. Tuning parameters were selected using cross-validation (CV), Akaike information criterion (AIC), and Bayesian information criterion (BIC). Classical methods performed similarly and produced worse predictions than penalized methods in limited-information scenarios (small samples, high correlation, and low signal-to-noise ratio (SNR)), but performed comparably or better in sufficient-information scenarios (large samples, low correlation, and high SNR). Lasso was superior under limited information but was less effective in sufficient-information scenarios. NNG, ALASSO, and RLASSO outperformed lasso in sufficient-information scenarios, with no clear winner among them. AIC and CV produced similar results and outperformed BIC, except in sufficient-information settings, where BIC performed better. Our findings suggest that no single method consistently outperforms others, as performance depends on the amount of information in the data. Lasso is preferred in limited-information settings, whereas classical methods are more suitable in sufficient-information settings, as they also tend to select simpler models.

We use the concepts of stochastic complexity, description length, and model selection to develop data-based methods for choosing smoothing parameters in nonparametric density estimation. In the case of histogram estimators, we derive a simple, exact formula for stochastic complexity when the prior distribution of cell probabilities is uniform over the class of all possible choices. The formula depends only on the data and the smoothing parameter, which is readily chosen according to the criterion of minimum stochastic complexity. Approaches based on stochastic complexity and description length are shown to be asymptotically equivalent in certain circumstances. They produce a degree of smoothing which is almost optimal from the viewpoint of minimizing L∞, or supremum, distance, but which smooths a little more than is optimal in the sense of minimizing Lr distance for any finite value of r.