  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.
