We study regularization for the deep linear network (DLN) using the entropy formula introduced in arXiv:2509.09088. The equilibria and gradient flow of the free energy on the Riemannian manifold of end-to-end maps of the DLN are characterized for energies that depend symmetrically on the singular values of the end-to-end matrix. The only equilibria are minimizers and the set of minimizers is an orbit of the orthogonal group. In contrast with random matrix theory there is no singular value repulsion. The corresponding gradient flow reduces to a one-dimensional ordinary differential equation whose solution gives explicit relaxation rates toward the minimizers. We also study the concavity of the entropy in the chamber of singular values. The entropy is shown to be strictly concave in the Euclidean geometry on the chamber but not in the Riemannian geometry defined by the DLN metric.

翻译：我们利用arXiv:2509.09088中引入的熵公式研究了深度线性网络的正则化。在深度线性网络端到端映射的黎曼流形上，对于依赖于端到端矩阵奇异值的对称能量函数，我们刻画了自由能的平衡点与梯度流。仅有的平衡点即为极小值点，且极小值点集合构成正交群的一个轨道。与随机矩阵理论不同，此处不存在奇异值排斥现象。相应的梯度流可约化为一维常微分方程，其解给出了向极小值点弛豫的显式速率。我们还研究了熵在奇异值区域内的凹性。结果表明，熵在奇异值区域的欧几里得几何中是严格凹的，但在深度线性网络度量定义的黎曼几何中则非严格凹。