Entropy and domination for quasi-Hitchin representations

What the math gives to ML

The paper develops a constructive comparison between complex matrix products and positive real envelopes using weighted planar networks: replacing every parameter by its modulus turns path sums into sums of absolute path weights, yielding entrywise and minor-wise domination through the triangle inequality. The strict version shows that non-real phases can cause genuine cancellation in selected minors, while the positive envelope preserves all corresponding path contributions. This suggests phase-aware stability certificates and regularizers for complex-valued recurrent, state-space, or deep linear modules: maintain a nonnegative envelope whose growth bounds the complex computation, and explicitly control the gap between the two. The construction is most transferable when the layer is factored into sparse triangular or network-like matrices, where path contributions can be computed efficiently.