  A number of recent studies indicate that the charge conduction of the
LaAlO$_3$/SrTiO$_3$ interface at low temperature is confined to filaments which
are linked to structural domain walls in the SrTiO$_3$ with drastic
consequences for example for the temperature dependence of local transport
properties. We demonstrate that as a consequences of this current carrying
filaments on the nano-scale the magnetotransport properties of the interface
are highly anisotropic. Our magnetoresistance measurements reveal that the
magnetoresistance in different nanostructures ($<500nm$) is random in magnitude
and sign, respectively. Warming up nanostructures above the structural phase
transition temperature (105K) results in the significant change in MR. Even a
sign change of the magnetoresistance is possible. The results suggest that
domain walls that are differently oriented with respect to the surface exhibit
different respective magnetoresistance and the total magnetoresistance is a
result of a random domain wall pattern formed during the structural phase
transition in the SrTiO$_3$ at cool down.
