论文标题
用于推理基质张量产品的汉密尔顿-Jacobi方程
Hamilton-Jacobi equations for inference of matrix tensor products
论文作者
论文摘要
我们研究了与有限级矩阵张量产品的推理问题相关的自由能的高维极限。通常，我们通过独特的解决方案将限制到某个汉密尔顿 - 雅各比方程。在对方程式中非线性的其他假设下，该方程式是由模型明确确定的，我们用解决方案确定限制。考虑了两个解决方案的概念，弱解决方案和粘度解决方案，每个解决方案都有其自身的优势，需要不同的治疗方法。为了具体，我们将结果应用于I.I.D.的模型。条目和对称相互作用。特别是，对于一阶甚至订购张量产品，我们确定了限制并获得收敛率的估计；对于其他奇数，可以获得上限。
We study the high-dimensional limit of the free energy associated with the inference problem of finite-rank matrix tensor products. In general, we bound the limit from above by the unique solution to a certain Hamilton-Jacobi equation. Under additional assumptions on the nonlinearity in the equation which is determined explicitly by the model, we identify the limit with the solution. Two notions of solutions, weak solutions and viscosity solutions, are considered, each of which has its own advantages and requires different treatments. For concreteness, we apply our results to a model with i.i.d. entries and symmetric interactions. In particular, for the first order and even order tensor products, we identify the limit and obtain estimates on convergence rates; for other odd orders, upper bounds are obtained.
