String theory mathematics and matrix data analysis

Abstract

Inspired by matrix techniques in quantum field theory and string theory, we review Permutation Invariant Gaussian Matrix Models (PIGMM), which replace the continuous symmetries of traditional Random Matrix Theory, for $ N \times N$ matrices, with finite permutation symmetry, $S_N$. This symmetry-driven approach reduces highly multivariate $N^2$-variable matrix data analysis problems to a rich but tractable space of parameters. The representation theory of $S_N$ brings a highly correlated quadratic matrix action to a near-diagonal form with $13$ parameters. The invariant observables are parameterised by graphs and their expectation values are computed with Wick contractions implemented algorithmically. We review the successful application of PIGMM for data reduction and anomaly detection in computational linguistics, statistical finance and neural network weights. We conclude with a brief discussion of potential future applications to matrix data analysis tasks that exploit hadronization algorithms and the modular structure of collider-physics data.