In this paper we consider the problem about level difference between an original arrangement
and arrangements reconstructed from partial information of the original arrangement, motivated
from the learning theory where reconstruction of the arrangement from partial information can be
considered as the inference of a network of threshold functions. If the difference of the level for an
arbitrary point in the space between the original arrangement and the reconstructed arrangements
is small, then even if each hyperplane may be different from the original one, the reconstructed
arrangements consistent with partial information can be used as a good approximation of the
original.
We prove that, given the level information of one point per cell for an unknown simple ar-
rangement of finite set of hyperplanes in the d-dimensional space, the difference of the level of an
arbitrary point between the original arrangement and any reconstructed arrangement, which is
consistent with the level information, is (1) less than or equal to d for general d if the dual graph
of the reconstructed arrangement is same as that of the original arrangement, and (2) less than
or equal to 3 for planar case even if the dual graph of the reconstructed arrangement is different
from that of the original arrangement.