Searching for Topological Symmetry in Data Haystack

TL;DR

This paper introduces a novel method for detecting local topological symmetries in high-dimensional data by analyzing invariances in a 2-D grid structure using specific legal grid moves, applicable to various distributions.

Contribution

The paper presents a new approach to identify local symmetries in high-dimensional data through grid moves that preserve statistical properties, enabling symmetry detection in noisy data.

Findings

Grid symmetry can be computed for Gaussian and Gamma distributions.

Legal grid moves preserve the distribution of Hamming distances.

Method effectively detects local symmetries in noisy high-dimensional data.

Abstract

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.