  We provide a function space characterization of the inductive bias resulting
from minimizing the $\ell_2$ norm of the weights in multi-channel convolutional
neural networks with linear activations and empirically test our resulting
hypothesis on ReLU networks trained using gradient descent. We define an
induced regularizer in the function space as the minimum $\ell_2$ norm of
weights of a network required to realize a function. For two layer linear
convolutional networks with $C$ output channels and kernel size $K$, we show
the following: (a) If the inputs to the network are single channeled, the
induced regularizer for any $K$ is independent of the number of output channels
$C$. Furthermore, we derive the regularizer is a norm given by a semidefinite
program (SDP). (b) In contrast, for multi-channel inputs, multiple output
channels can be necessary to merely realize all matrix-valued linear functions
and thus the inductive bias does depend on $C$. However, for sufficiently large
$C$, the induced regularizer is again given by an SDP that is independent of
$C$. In particular, the induced regularizer for $K=1$ and $K=D$ (input
dimension) is given in closed form as the nuclear norm and the $\ell_{2,1}$
group-sparse norm, respectively, of the Fourier coefficients of the linear
predictor. We investigate the broader applicability of our theoretical results
to implicit regularization from gradient descent on linear and ReLU networks
through experiments on MNIST and CIFAR-10 datasets.
