  Finding interesting symmetrical topological structures in high-dimensional
systems is an important problem in statistical machine learning. Limited amount
of available high-dimensional data and its sensitivity to noise pose
computational challenges to find symmetry. Our paper presents a new method to
find local symmetries in a low-dimensional 2-D grid structure which is embedded
in high-dimensional structure. To compute the symmetry in a grid structure, we
introduce three legal grid moves (i) Commutation (ii) Cyclic Permutation (iii)
Stabilization on sets of local grid squares, grid blocks. The three grid moves
are legal transformations as they preserve the statistical distribution of
hamming distances in each grid block. We propose and coin the term of grid
symmetry of data on the 2-D data grid as the invariance of statistical
distributions of hamming distance are preserved after a sequence of grid moves.
We have computed and analyzed the grid symmetry of data on multivariate
Gaussian distributions and Gamma distributions with noise.
