This question may be slightly related to this question on length of the representation of a number in a certain basis.
Introduction / Background In image and video coding, particularly the quantization step, one usually wants to get to store numbers with the least number of required symbols. One way to avoid coding a number altogether is to try to somehow achieve a sparse signal. This has been done a lot ( and in many different fields as well ) during the last 10 years. has been increasingly popular due to availability of fast algorithms to solve optimization problems on this form $$\min \left\{\sum_k|\cdot|_n + |\cdot |_1\right\}$$ I.e. 1-norm regularized optimization problems which often produce sparse results.
However if we would not want the entire number to be sparse, but their representation in some basis to have least expected number of digits per symbol. Is there some way we can apply the previously known methods (preferrably linear or convex optimization) to achieve this?
Own work (actually it's more of an idea than work):
Maybe write the number as a sum (linear combination) and then some sequence of increasingly weighted $l_1$ norms on the individual terms?