This paper establishes risk convergence and asymptotic weight matrix
alignment --- a form of implicit regularization --- of gradient flow and
gradient descent when applied to deep linear networks on linearly separable
data. In more detail, for gradient flow applied to strictly decreasing loss
functions (with similar results for gradient descent with particular decreasing
step sizes): (i) the risk converges to 0; (ii) the normalized i-th weight
matrix asymptotically equals its rank-1 approximation $u_iv_i^{\top}$; (iii)
these rank-1 matrices are aligned across layers, meaning
$|v_{i+1}^{\top}u_i|\to1$. In the case of the logistic loss (binary cross
entropy), more can be said: the linear function induced by the network --- the
product of its weight matrices --- converges to the same direction as the
maximum margin solution. This last property was identified in prior work, but
only under assumptions on gradient descent which here are implied by the
alignment phenomenon.

本篇论文研究了在线性可分数据上应用于深度线性网络的梯度流和梯度下降的风险收敛和渐进权重矩阵对齐 —— 一种隐式正则化方法，详细说明了在套用于严格递减损失函数时（梯度下降的递减步长也是如此）：(i) 风险趋近于 0；（ii）标准化的第 i 个权重矩阵渐进等于其秩 - 1 逼近；（iii）这些秩 - 1 矩阵在层之间对齐，即 |vi+1^Tv_i| -> 1。特别地，在逻辑损失（二元交叉熵）的情况下，还可以说更多的结论：网络激发的线性函数 —— 其权重矩阵的乘积 —— 趋向于与最大边际解同方向。这种性质在之前的工作中已被证明，但仅基于对梯度下降的假设，这里的对齐现象可以证明这些假设。