  We study the implicit bias of gradient flow (i.e., gradient descent with
infinitesimal step size) on linear neural network training. We propose a tensor
formulation of neural networks that includes fully-connected, diagonal, and
convolutional networks as special cases, and investigate the linear version of
the formulation called linear tensor networks. With this formulation, we can
characterize the convergence direction of the network parameters as singular
vectors of a tensor defined by the network. For $L$-layer linear tensor
networks that are orthogonally decomposable, we show that gradient flow on
separable classification finds a stationary point of the $\ell_{2/L}$
max-margin problem in a "transformed" input space defined by the network. For
underdetermined regression, we prove that gradient flow finds a global minimum
which minimizes a norm-like function that interpolates between weighted
$\ell_1$ and $\ell_2$ norms in the transformed input space. Our theorems
subsume existing results in the literature while removing standard convergence
assumptions. We also provide experiments that corroborate our analysis.
