A Solution for the Greedy Approximation of a Step Function with a Waveform Dictionary

TL;DR

This paper develops a greedy algorithm for approximating step functions using waveform and wavelet dictionaries, providing closed-form solutions and practical methods for signals with discontinuities.

Contribution

It introduces a novel approach combining matching pursuit with wavelet dictionaries for efficient step function approximation, including closed-form solutions and iterative methods.

Findings

Closed-form solution for initial optimization with wavelet dictionary

Practical iterative approximation method for real-valued sequences

Effective approximation demonstrated on simulated and real data with discontinuities

Abstract

In this paper we consider a step function characterized by an arbitrary sequence of real-valued scalars and approximate it with a matching pursuit (MP) algorithm. We utilize a waveform dictionary with rectangular window functions as part of this algorithm. We show that the waveform dictionary is not necessary when all of the scalars are either non positive or non negative and the parameters of a wavelet dictionary on an integer lattice yields a closed-form solution for the initial optimization problem as part of the MP. Additionally, for any real-valued scalar sequence, we provide a solution with a related wavelet dictionary at each iteration of the algorithm. This allows for practical calculation of the approximating function, which we use to provide examples on simulated and real univariate time series data that display discontinuities in its underlying structure where the step function can be thought of as a sample from a signal of interest.