Geometric turbulence: a geodesic-regression crisis indicator for equity covariance dynamics, with evidence from African markets

What the math gives to ML

The paper supplies a principled way to process symmetric positive-definite (SPD) matrices without leaving the SPD cone, replacing Euclidean operations on covariance entries by intrinsic operations. This transfers naturally to neural modules that predict covariance, attention kernels, uncertainty matrices, or graph diffusion operators: regress in a matrix-log coordinate system and map predictions back with the matrix exponential. The affine-invariant metric additionally provides a scale- and coordinate-aware discrepancy that can serve as a loss or monitoring signal. The most practical first experiment is a log-Euclidean temporal covariance-prediction head, comparing its validity, calibration, and forecasting error against an unconstrained vectorized matrix head.