The $θ$-Symmetric SRG with Applications to Stability of Cactus Dynamic Networks

What the math gives to ML

The paper introduces a phase-parameterized scaled relative graph (theta-SRG) that encloses matrix spectra while retaining gain and phase information, and uses chord and arc closure rules to certify nonsingularity of sums and products. Its key transferable asset is a computable phase-sector certificate: the angular spread of a matrix around phase theta equals a one-dimensional norm minimization. In neural networks, this can be applied to Jacobians of residual blocks and feedback-like compositions to obtain phase-aware invertibility and long-horizon propagation margins that are less conservative than spectral-radius or zero-phase bounds. The most practical transfers are a Jacobian regularizer and a phase-aware controller for branch gains or optimizer dynamics.