Many statistical estimation procedures lead to nonconvex optimization problems. Algorithms to solve these problems are often guaranteed to output a stationary point of the optimization problem. Oracle inequalities are an important theoretical instrument to assess the statistical performance of an estimator. Oracle results have focused on the theoretical properties of the uncomputable (global) minimum or maximum. In this paper, a general framework used for convex optimization problems to derive oracle inequalities for stationary points is extended. A main new ingredient of these oracle inequalities is that they are sharp: they show closeness to the best approximation within the model plus a remainder term. We apply this framework to different estimation problems.

Large scale and structurally complex volume datasets from high-resolution 3D imaging devices or computational simulations pose a number of technical challenges for interactive visual analysis. In this paper, we present the first integration of a multiscale volume representation based on tensor approximation within a GPU-accelerated out-of-core multiresolution rendering framework. Specific contributions include (a) a hierarchical brick-tensor decomposition approach for pre-processing large volume data, (b) a GPU accelerated tensor reconstruction implementation exploiting CUDA capabilities, and (c) an effective tensor-specific quantization strategy for reducing data transfer bandwidth and out-of-core memory footprint. Our multiscale representation allows for the extraction, analysis and display of structural features at variable spatial scales, while adaptive level-of-detail rendering methods make it possible to interactively explore large datasets within a constrained memory footprint. The quality and performance of our prototype system is evaluated on large structurally complex datasets, including gigabyte-sized micro-tomographic volumes.