Searching for Topological Symmetry in Data Haystack
This addresses computational challenges in statistical machine learning for identifying symmetries in noisy, high-dimensional data, but it appears incremental as it focuses on a specific grid-based approach.
The paper tackles the problem of finding symmetrical topological structures in high-dimensional data by introducing a new method to detect local symmetries in a 2-D grid embedded in high dimensions, using three legal grid moves that preserve statistical distributions of hamming distances, and it computes grid symmetry for multivariate Gaussian and Gamma distributions with noise.
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.