In addition to subsampling, goals of competitive object learning
are the minimization of the expected quantization error and
entropy maximization. A finite set of 3D scan points is
subsambled to the set
. Error minimization is done with respect to the following
function:

with the set of samples and the Voronoi set of unit , i.e., and . Entropy maximization guarantees inherent robustness. The failure of reference vectors, i.e., missing 3D points, affects only a limited fraction of the data. Interpreting the generation of an input signal and the subsequent mapping onto the nearest sample in as a random experiment which assigns a value to the random variable , then maximizing the entropy

is equivalent to equiprobable samples. The following neural gas algorithm learns and subsamples 3D points clouds [7]:

- i.).
- Initialize the set to contain vectors, randomly from the input set. Set .
- ii.).
- Generate at random an input element , i.e., select a point from .
- iii.).
- Order all elements of according to their distance to , i.e., find the sequence of indices such that is the reference vector closest to , is the reference vector second closest to , etc., , is the reference vector such that vectors exists that are closer to than . denotes the number associated with .
- iv.).
- Adapt the reference vectors according to

with the following time dependencies:

- v.).
- Increase the time parameter .