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

We develop a spectral theory for continuous- and discrete-time Blahut–Arimoto (BA) dynamics, centered on the relaxation kernel $\text{G}={\text{E}}_p[K_X^{\ast }\otimes K_X^{\ast }]$. Five main results are established. (i) Along the continuous-time BA flow, the free energy satisfies the exact ${\chi }^2$-dissipation identity ${\dot{F}}_{\beta }=-\text{D}(q)$, where $\text{D}(q)={\chi }^2(\text{T}q\mathrm{\parallel }q)$ is the Pearson ${\chi }^2$-divergence. (ii) The operator $\text{G}$ 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 $\mathrm{\Delta }{\mathcal{L}}^{(2)}={\sum }_i{c}_i^{2}d({\lambda }_i)$, where $d(\lambda )=-\lambda +\frac{3}{2}{\lambda }^2-\frac{1}{2}{\lambda }^3$. 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 $\text{lam}={\lambda }_{\min }(\text{G}{\mathrm{\mid }}_T)$ governs exponential contraction. (v) The KL convergence factor is explicit: $\text{KL}(q^{\ast }\mathrm{\parallel }{q}_{n+1})\le (1-\text{lam}{)}^2\text{KL}(q^{\ast }\mathrm{\parallel }{q}_n)+O(\mathrm{\parallel }{v}_n{\mathrm{\parallel }}_{\ast }^3)$, with per-iteration improvement $\gamma =\text{lam}(2-\text{lam})$. For Gaussian sources, $\text{lam}=1\mathrm{/}(2\beta {\sigma }^2)$ and the Jacobian is diagonalised by Hermite polynomials. The spectral formula complements Hayashi’s global convergence theory with a constructive, computable local rate.