Hamilton-Jacobi equations for inference of matrix tensor products
Hamilton-Jacobi equations for inference of matrix tensor products
复制标题
用于推理矩阵张量积的 Hamilton-Jacobi 方程
DOI：
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发表时间：
2020
影响因子：
1.5
通讯作者：
J. Xia
中科院分区：
文献类型：
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作者：
Hong;J. Xia
本文研究了有限秩矩阵张量积推论问题中自由能的高维极限。一般来说，我们通过对某一Hamilton-Jacobi方程的唯一解来约束上述极限。在模型明确确定的方程非线性的附加假设下，我们用解来确定极限。考虑了两种解的概念，即弱解和粘性解，它们各有优点，需要不同的处理方法。为了具体起见，我们将我们的结果应用到一个模型，i.i.d.条目和对称交互。特别是，对于一阶和偶数阶张量积，我们确定了极限，并获得了收敛速度的估计;对于其他奇数阶，得到了上界。
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.
DOI：
10.1109/allerton.2019.8919733
发表时间：
2019-09
期刊：
2019 57th Annual Allerton Conference on Communication, Control, and Computing (Allerton)
影响因子：
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作者：
Vaishakhi Mayya;G. Reeves
通讯作者：
Vaishakhi Mayya;G. Reeves
DOI：
10.1109/isit.2019.8849594
发表时间：
2019-07
期刊：
2019 IEEE International Symposium on Information Theory (ISIT)
影响因子：
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作者：
G. Reeves;Vaishakhi Mayya;A. Volfovsky
通讯作者：
G. Reeves;Vaishakhi Mayya;A. Volfovsky