This paper uncovers an exact $χ^2$ dissipation identity for the Blahut--Arimoto (BA) flow and establishes its fundamental information-geometric structure. While prior works have analyzed BA convergence via the monotone decrease of the Kullback--Leibler divergence, we establish that the continuous-time flow $\partial_τq = \tilde{q}_β[q] - q$ is fundamentally governed by the Pearson $χ^2$ divergence. Specifically, we prove the governing identity $\frac{d}{dτ} Φ_β= -χ^2(\tilde{q}_β\| q)$, elevating $χ^2$ from a mere local approximation to the global non-asymptotic driver of the dynamics. We establish a \emph{unified dissipation identity} revealing that the $χ^2$ and the symmetrised Jeffreys divergence $D_J$ differ only by a cubic residual $\mathcal{N}$, which vanishes \emph{exactly} for the BA flow. This algebraic cancellation identifies the Jacobian $K_q = D\tilde{q}_β[q]$ and its complement $\mathcal{A}_q = I - K_q$ as the Fisher--Rao Hessian of the free energy $Φ_β$. The spectral gap of $\mathcal{A}_{q^*}$ is proved to determine both the local exponential convergence rate (a universal value of $2$ in artificial time) and the stability of the attraction basin. For Gaussian sources, the theory yields three concrete consequences: (i)~the reproduction variance obeys an exact closed ODE; (ii)~the Gaussian distribution emerges as the unique dynamical attractor; and (iii)~in high dimensions, convergence is bottlenecked by the maximum source variance -- \emph{spectral stiffness}. Significantly, this framework provides new dissipative interpretations for fundamental problems including MIMO channel capacity, Wyner--Ziv coding, and the information bottleneck. These results establish the BA flow as a canonical dissipative relaxation and provide a geometric foundation for the dynamical theory of information-theoretic optimisation.

翻译：本文揭示了Blahut—Arimoto(BA)流的精确χ²耗散恒等式，并建立了其基本的信息几何结构。尽管先前的工作通过Kullback—Leibler散度的单调递减分析了BA收敛性，我们证明连续时间流$\partial_τq = \tilde{q}_β[q] - q$从根本上由Pearson χ²散度主导。具体而言，我们证明主导恒等式$\frac{d}{dτ} Φ_β= -χ^2(\tilde{q}_β\| q)$，将χ²从单纯的局部近似提升为动力学的全局非渐近驱动力。我们建立了一个\emph{统一耗散恒等式}，揭示χ²与对称化Jeffreys散度$D_J$仅相差一个三次残差$\mathcal{N}$，该残差对于BA流\emph{精确}消失。这种代数相消将雅可比矩阵$K_q = D\tilde{q}_β[q]$及其补$\mathcal{A}_q = I - K_q$识别为自由能$Φ_β$的Fisher—Rao Hessian矩阵。证明$\mathcal{A}_{q^*}$的谱隙决定了局部指数收敛速率（人工时间中的普适值$2$）和吸引域的稳定性。对于高斯源，该理论产生三个具体结论：(i) 重构方差服从精确封闭常微分方程；(ii) 高斯分布作为唯一动力吸引子出现；(iii) 在高维情形下，收敛受限于最大源方差——\emph{谱刚性}。值得注意的是，该框架为MIMO信道容量、Wyner—Ziv编码和信息瓶颈等基本问题提供了新的耗散解释。这些结果将BA流确立为典范耗散松弛过程，并为信息论优化的动力学理论提供了几何基础。