Approximation algorithms for wavelet transform coding of data streams

This paper addresses the problem of finding a B-term wavelet representation of a given discrete function $f \in \real^n$ whose distance from f is minimized. The problem is well understood when we seek to minimize the Euclidean distance between f and its representation. The first known algorithms for finding provably approximate representations minimizing general $\ell_p$ distances (including $\ell_\infty$) under a wide variety of compactly supported wavelet bases are presented in this paper. For the Haar basis, a polynomial time approximation scheme is demonstrated. These algorithms are applicable in the one-pass sublinear-space data stream model of computation. They generalize naturally to multiple dimensions and weighted norms. A universal representation that provides a provable approximation guarantee under all p-norms simultaneously; and the first approximation algorithms for bit-budget versions of the problem, known as adaptive quantization, are also presented. Further, it is shown that the algorithms presented here can be used to select a basis from a tree-structured dictionary of bases and find a B-term representation of the given function that provably approximates its best dictionary-basis representation.Comment: Added a universal representation that provides a provable approximation guarantee under all p-norms simultaneousl

Optimal workload-based weighted wavelet synopses

In recent years wavelets were shown to be effective data synopses. We are concerned with the problem of finding efficiently wavelet synopses for massive data sets, in situations where information about query workload is available. We present linear time, I/O optimal algorithms for building optimal workload-based wavelet synopses for point queries. The synopses are based on a novel construction of weighted inner-products and use weighted wavelets that are adapted to those products. The synopses are optimal in the sense that the subset of retained coefficients is the best possible for the bases in use with respect to either the mean-squared absolute or relative errors. For the latter, this is the first optimal wavelet synopsis even for the regular, non-workload-based case. Experimental results demonstrate the advantage obtained by the new optimal wavelet synopses, as well as the robustness of the synopses to deviations in the actual query workload

Optimal workload-based weighted wavelet synopses

Abstract. In recent years wavelets were shown to be effective data synopses. We are concerned with the problem of finding efficiently wavelet synopses for massive data sets, in situations where information about query workload is available. We present linear time, I/O optimal algorithms for building optimal workload-based wavelet synopses for point queries. The synopses are based on a novel construction of weighted inner-products and use weighted wavelets that are adapted to those products. The synopses are optimal in the sense that the subset of retained coefficients is the best possible for the bases in use with respect to either the mean-squared absolute or relative errors. For the latter, this is the first optimal wavelet synopsis even for the regular, non-workload-based case. Experimental results demonstrate the advantage obtained by the new optimal wavelet synopses, as well as the robustness of the synopses to deviations in the actual query workload.