Sigma Flows for Image and Data Labeling and Learning Structured Prediction

The paper’s Theorem 4.19 proves that, under the low-frequency support assumption a_in(t)=0 for n not in N(t), the L2-norm of the state diverges along the σ-φ-flow by establishing a strictly positive time derivative of the Lyapunov functional Φ and exploiting the bound φ ≤ 0; the key steps are the weighted testing with (g_ε)ϑ, the identity (4.63d), and the spectral bound leading to dΦ/dt ≥ Σ_{n∈N} ((c2+ε)λ_n + ε m^2) ||a_n||^2 > 0, from which divergence of the θ-coordinate L2-norm follows (Theorem 4.19 and its proof: definition of N in (4.85), inequality (4.94)–(4.98)) . By contrast, the model’s argument tests the PDE against ϑ without the (g_ε)-weight and invokes the matrix bound (4.30) as a direct lower bound for the Christoffel term; that step does not follow from (4.30), which concerns B(θ) = g + (1/2) d g · θ appearing only after the weighted manipulation used in the paper . The model also leaves the time-derivative of the volume form uncontrolled and assumes uniform equivalence of time-dependent L2 norms without an upper bound on h_t, whereas the paper avoids such issues by using a functional independent of h_t’s derivatives and a weighted test identity analogous to (4.61b) . Hence the paper’s proof is correct under its stated assumptions, while the model’s proof is flawed in key steps, even though it reaches the same high-level conclusion.