We study the convergence of gradient flows related to learning deep linear
neural networks (where the activation function is the identity map) from data.
In this case, the composition of the network layers amounts to simply
multiplying the weight matrices of all layers together, resulting in an
overparameterized problem. The gradient flow with respect to these factors can
be re-interpreted as a Riemannian gradient flow on the manifold of rank-$r$
matrices endowed with a suitable Riemannian metric. We show that the flow
always converges to a critical point of the underlying functional. Moreover, we
establish that, for almost all initializations, the flow converges to a global
minimum on the manifold of rank $k$ matrices for some $k\leq r$.

本文研究了与从数据中学习深度线性神经网络（其中激活函数为恒等映射）相关的梯度流的收敛性，结果表明梯度流总是收敛于潜在函数的临界点。

学习深度线性神经网络：黎曼梯度流与全局极小值的收敛

Learning deep linear neural networks: Riemannian gradient flows and  convergence to global minimizers

We analyze algorithms for approximating a function $f(x) = \Phi x$ mapping
$\Re^d$ to $\Re^d$ using deep linear neural networks, i.e. that learn a
function $h$ parameterized by matrices $\Theta_1,...,\Theta_L$ and defined by
$h(x) = \Theta_L \Theta_{L-1} ... \Theta_1 x$. We focus on algorithms that
learn through gradient descent on the population quadratic loss in the case
that the distribution over the inputs is isotropic.
We provide polynomial bounds on the number of iterations for gradient descent
to approximate the least squares matrix $\Phi$, in the case where the initial
hypothesis $\Theta_1 = ... = \Theta_L = I$ has excess loss bounded by a small
enough constant. On the other hand, we show that gradient descent fails to
converge for $\Phi$ whose distance from the identity is a larger constant, and
we show that some forms of regularization toward the identity in each layer do
not help.
If $\Phi$ is symmetric positive definite, we show that an algorithm that
initializes $\Theta_i = I$ learns an $\epsilon$-approximation of $f$ using a
number of updates polynomial in $L$, the condition number of $\Phi$, and
$\log(d/\epsilon)$. In contrast, we show that if the least squares matrix
$\Phi$ is symmetric and has a negative eigenvalue, then all members of a class
of algorithms that perform gradient descent with identity initialization, and
optionally regularize toward the identity in each layer, fail to converge.
We analyze an algorithm for the case that $\Phi$ satisfies $u^{\top} \Phi u >
0$ for all $u$, but may not be symmetric. This algorithm uses two regularizers:
one that maintains the invariant $u^{\top} \Theta_L \Theta_{L-1} ... \Theta_1 u
> 0$ for all $u$, and another that "balances" $\Theta_1, ..., \Theta_L$ so that
they have the same singular values.

本文介绍了使用梯度下降算法在深度线性神经网络中学习的近似函数的算法，并分析了在输入分布为各向同性时，使用多项式约束的梯度下降算法能多快地逼近最小二乘矩阵，以及针对不同情况下的正则化方法如何对收敛时效产生影响。此外，文章还探讨了对称正定矩阵条件下的算法，以及对于某些不对称矩阵的处理方式。