# Algorithms for Piecewise Constant Signal Approximations

## TL;DR

This paper compares algorithms for approximating one-dimensional signals with piecewise constant segments, highlighting the limitations of existing methods and proposing a new optimization approach that performs better on nonsmooth signals.

## Contribution

It reformulates a recent adaptive sampling method, analyzes its limitations, and introduces a particle swarm optimization approach that overcomes these issues.

## Key findings

- The direct optimization approach outperforms the Dar--Bruckstein method on nonsmooth signals.
- The principle of error balancing is not always optimal for l^2 norm data approximation.
- The paper provides a theoretical analysis of existing algorithms and proposes a more flexible alternative.

## Abstract

We consider the problem of finding optimal piecewise constant approximations of one-dimensional signals. These approximations should consist of a specified number of segments (samples) and minimise the mean squared error to the original signal. We formalise this goal as a discrete nonconvex optimisation problem, for which we study two algorithms. First we reformulate a recent adaptive sampling method by Dar and Bruckstein in a compact and transparent way. This allows us to analyse its limitations when it comes to violations of its three key assumptions: signal smoothness, local linearity, and error balancing. As a remedy, we propose a direct optimisation approach which does not rely on any of these assumptions and employs a particle swarm optimisation algorithm. Our experiments show that for nonsmooth signals or low sample numbers, the direct optimisation approach offers substantial qualitative advantages over the Dar--Bruckstein method. As a more general contribution, we disprove the optimality of the principle of error balancing for optimising data in the l^2 norm.

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Source: https://tomesphere.com/paper/1903.01320