Relaxation Kernel, Spectral Dissipation, and Global Convergence of Blahut--Arimoto Dynamics

arXiv:2604.25106v3 Announce Type: replace Abstract: We develop a spectral theory for continuous- and discrete-time Blahut--Arimoto (BA) dynamics, centered on the relaxation kernel $ \G = \E_p[K^*_X \otimes K^*_X] $. Five main results are established. (i) Along the continuous-time BA flow, the free energy satisfies the exact $ \chi^2 $-dissipation identity $ \dot F_\beta = -\D(q) $, where $ \D(q)=\chi^2(\T q \| q) $ is the Pearson $ \chi^2 $-divergence. (ii) The operator $ \G $ admits a threefold identity: it is simultaneously the Gram matrix of the equilibrium Gibbs kernels, the linearised generator of the BA vector field, and the Fisher--Rao Hessian of the free energy at the fixed point. (iii) For the discrete iteration, the one-step Lyapunov dissipation decomposes spectrally as $ \Delta\mathcal{L}^{(2)} = \sum_i c_i^2\, d(\lambda_i) $, where $ d(\lambda) = -\lambda + \tfrac{3}{2}\lambda^2 - \tfrac{1}{2}\lambda^3 $. This reveals a double bottleneck at $ \lambda\approx 0 $ and $ \lambda\approx 1 $, with optimal dissipation near $ \lambda\approx 0.423 $. (iv) Global convergence follows a two-stage mechanism: $ \chi^2 $-dissipation drives finite-time entry into a local neighbourhood, after which the spectral gap $ \lam = \lambda_{\min}(\G|_T) $ governs exponential contraction. (v) The KL convergence factor is explicit: $ \KL(q^*\|q_{n+1}) \le (1-\lam)^2\,\KL(q^*\|q_n) + O(\|v_n\|_*^3) $, with per-iteration improvement $ \gamma = \lam(2-\lam) $. For Gaussian sources, $ \lam = 1/(2\beta\sigma^2) $ and the Jacobian is diagonalised by Hermite polynomials. The spectral formula complements Hayashi's global convergence theory with a constructive, computable local rate.