这篇论文用神经网络分析发现仅三个系数就能区分全正矩阵，还给出了几何解释，对矩阵理论和机器学习都有启发。
该论文研究了全正矩阵能否通过特征多项式最高阶系数进行区分。使用神经网络分类器和特征归因方法，发现系数(a_{n-1}, a_{n-2}, a_{n-3})在维度5、10、30上能有效分离全正矩阵与非全正矩阵。分离边界呈非线性，可通过三维系数空间中的马氏椭球自然描述，不同结构族（如正双对角矩阵乘积、Vandermonde矩阵、Cauchy矩阵）的椭球特征不同，且维度越高分离越明显。
Totally Positive Matrices and the Highest-Order Coefficients of the Characteristic Polynomial
We investigate the extent to which totally positive matrices can be distinguished through the highest-order coefficients of their characteristic polynomials. To identify the most informative coefficients, we also employed neural-network classifiers together with feature-attribution methods. Using datasets built from several structured totally positive families, including products of positive bidiagonal matrices, Vandermonde matrices, and Cauchy matrices, we find that the coefficients (a_{n-1}, a_{n-2}, a_{n-3}) already contain strong discriminatory information for separating totally positive from non-totally positive matrices in dimensions 5, 10, and 30. The resulting separation is markedly nonlinear and admits a natural geometric description in the corresponding three-dimensional coefficient space by means of Mahalanobis ellipsoids. These ellipsoids enclose the totally positive samples while excluding most non-totally positive ones. Moreover, different structured totally positive families exhibit distinct ellipsoidal signatures, and the separation between these signatures increases with the dimension. These observations lead us to formulate a conjecture on the geometric separation of structured totally positive families in the space determined by the three highest-order characteristic coefficients.