  Recent works on over-parameterized neural networks have shown that the
stochasticity in optimizers has the implicit regularization effect of
minimizing the sharpness of the loss function (in particular, the trace of its
Hessian) over the family zero-loss solutions. More explicit forms of flatness
regularization also empirically improve the generalization performance.
However, it remains unclear why and when flatness regularization leads to
better generalization. This work takes the first step toward understanding the
inductive bias of the minimum trace of the Hessian solutions in an important
setting: learning deep linear networks from linear measurements, also known as
\emph{deep matrix factorization}. We show that for all depth greater than one,
with the standard Restricted Isometry Property (RIP) on the measurements,
minimizing the trace of Hessian is approximately equivalent to minimizing the
Schatten 1-norm of the corresponding end-to-end matrix parameters (i.e., the
product of all layer matrices), which in turn leads to better generalization.
We empirically verify our theoretical findings on synthetic datasets.
