Approximation algorithms for wavelet transform coding of data streams

TL;DR

This paper introduces new approximation algorithms for wavelet transform coding of data streams that minimize various p-norm distances, including adaptive quantization, with provable guarantees and polynomial time schemes.

Contribution

It presents the first algorithms for general p-norm minimization in wavelet coding, including a polynomial time scheme for Haar basis and methods for adaptive quantization.

Findings

Algorithms work in one-pass sublinear-space data stream model

Universal representation guarantees approximation under all p-norms

First algorithms for bit-budget adaptive quantization

Abstract

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.