论文标题
汉密尔顿 - 雅各比方程非对称矩阵推断
Hamilton-Jacobi equations for nonsymmetric matrix inference
论文作者
论文摘要
我们研究了与排名一的非对称基质相关的自由能的高维极限。矩阵表示为两个向量的外产物，不一定是独立的。仅假定两个向量的分布具有缩放的界支持。我们将自由能和解决方案之间的差异绑定到合适的汉密尔顿 - 雅各比方程，这两个量要简单得多：这种自由能的浓度速率以及在解耦系统中简单的自由能的收敛速率。为了证明这种方法的多功能性，我们将结果应用于I.I.D.案例和球形案例。通过插入两个简单数量的估计值，我们确定限制并获得收敛速率。
We study the high-dimensional limit of the free energy associated with the inference problem of a rank-one nonsymmetric matrix. The matrix is expressed as the outer product of two vectors, not necessarily independent. The distributions of the two vectors are only assumed to have scaled bounded supports. We bound the difference between the free energy and the solution to a suitable Hamilton-Jacobi equation in terms of two much simpler quantities: concentration rate of this free energy, and the convergence rate of a simpler free energy in a decoupled system. To demonstrate the versatility of this approach, we apply our result to the i.i.d. case and the spherical case. By plugging in estimates of the two simpler quantities, we identify the limits and obtain convergence rates.
