Deterministic wavelet thresholding for maximum-error metrics

Abstract

Several studies have demonstrated the effectiveness of the wavelet, decomposition as a tool for reducing large amounts of data down to compact, wavelet synopses that can be used to obtain fast, accurate approximate answers to user queries. While conventional wavelet synopses are based on greedily minimizing the overall root-mean-squared (i.e., L2-norm) error in the data approximation, recent work has demonstrated that such synopses can suffer from important problems, including severe bias and wide variance in the quality of the data reconstruction, and lack of non-trivial guarantees for individual approximate answers. As a result, probabilistic thresholding schemes have been recently proposed as a means of building wavelet synopses that try to probabilistically control other approximation-error metrics, such as the maximum relative error in data-value reconstruction, which is arguably the most important for approximate query answers and meaningful error guarantees.One of the main open problems posed by this earlier work is whether it is possible to design efficient deterministic wavelet-thresholding algorithms for minimizing non-L2 error metrics that are relevant to approximate query processing systems, such as maximum relative or maximum absolute error. Obviously, such algorithms can guarantee better wavelet synopses and avoid the pitfalls of probabilistic techniques (e.g., "bad" coin-flip sequences) leading to poor solutions. In this paper, we address this problem and propose novel, computationally efficient schemes for deterministic wavelet thresholding with the objective of optimizing maximum-error metrics. We introduce an optimal low polynomial-time algorithm for one-dimensional wavelet thresholding--our algorithm is based on a new Dynamic-Programming (DP) formulation, and can be employed to minimize the maximum relative or absolute error in the data reconstruction. Unfortunately, directly extending our one-dimensional DP algorithm to multi-dimensional wavelets results in a super-exponential increase in time complexity with the data dimensionality. Thus, we also introduce novel, polynomial-time approximation schemes (with tunable approximation guarantees for the target maximum-error metric) for deterministic wavelet thresholding in multiple dimensions.