We compute the high-dimensional limit of the free energy associated with a multi-layer generalized linear model. Under certain technical assumptions, we identify the limit in terms of a variational formula. The approach is to first show that the limit is a solution to a Hamilton-Jacobi equation whose initial condition is related to the limiting free energy of a model with one fewer layer. Then, we conclude by an iteration.
翻译:我们计算了与多层通用线性模型相关的自由能源的高维限值。根据某些技术假设,我们确定变式公式的限值。我们的方法是首先显示这一限值是汉密尔顿-贾科比等式的解决方案,该等式的初始条件与限制一个少一层的模型的免费能源有关。然后,我们通过迭代来得出结论。