Geodesic Quantum $f$-Divergences

What the math gives to ML

The paper supplies a continuous, noncommutative interpolation between standard Petz and maximal quantum f-divergences through an affine-invariant geodesic of relative modular operators. This is transferable to neural models whose predictions or embeddings are positive-definite matrices, including covariance networks, density-matrix classifiers, and matrix-valued representation learners. The key asset is that operator-convex generators preserve data processing along completely positive trace-preserving layers, while the geodesic parameter t provides a tunable tradeoff between conventional and maximal matrix discrepancies. A practical first use is a geodesic divergence auxiliary loss with differentiable eigendecomposition or Lanczos evaluation, combined with an ordinary task loss to prevent representation collapse.