论文标题
无限维汉密尔顿 - 雅各比方程，用于稀疏图的统计推断
Infinite-dimensional Hamilton-Jacobi equations for statistical inference on sparse graphs
论文作者
论文摘要
我们研究了在非阴性措施和单调的非线性上提出的无限二维汉密尔顿 - 雅各比方程的适当性。我们的结果将用于伴侣工作中，以提出一个猜想，并证明有关稀疏制度中分类随机块模型中渐近互信息的部分结果。我们认为的方程式是根据解决方案的gateaux导数来自然说明的，这与以前的衍生物通常为传输类型不同。我们介绍了一个有限维汉密尔顿 - 雅各布方程的近似家族，并使用非线性的单调性来表明不需要规定边界条件即可建立良好的拟合度。然后将无限维汉密尔顿 - 雅各比方程的解决方案定义为这些近似溶液的极限。在凸非线性的特殊环境中，我们还提供了溶液的HOPF极端变异表示。
We study the well-posedness of an infinite-dimensional Hamilton-Jacobi equation posed on the set of non-negative measures and with a monotonic non-linearity. Our results will be used in a companion work to propose a conjecture and prove partial results concerning the asymptotic mutual information in the assortative stochastic block model in the sparse regime. The equation we consider is naturally stated in terms of the Gateaux derivative of the solution, unlike previous works in which the derivative is usually of transport type. We introduce an approximating family of finite-dimensional Hamilton-Jacobi equations, and use the monotonicity of the non-linearity to show that no boundary condition needs to be prescribed to establish well-posedness. The solution to the infinite-dimensional Hamilton-Jacobi equation is then defined as the limit of these approximating solutions. In the special setting of a convex non-linearity, we also provide a Hopf-Lax variational representation of the solution.
