  In this paper, we consider parameter recovery for non-overlapping
convolutional neural networks (CNNs) with multiple kernels. We show that when
the inputs follow Gaussian distribution and the sample size is sufficiently
large, the squared loss of such CNNs is $\mathit{~locally~strongly~convex}$ in
a basin of attraction near the global optima for most popular activation
functions, like ReLU, Leaky ReLU, Squared ReLU, Sigmoid and Tanh. The required
sample complexity is proportional to the dimension of the input and polynomial
in the number of kernels and a condition number of the parameters. We also show
that tensor methods are able to initialize the parameters to the local strong
convex region. Hence, for most smooth activations, gradient descent following
tensor initialization is guaranteed to converge to the global optimal with time
that is linear in input dimension, logarithmic in precision and polynomial in
other factors. To the best of our knowledge, this is the first work that
provides recovery guarantees for CNNs with multiple kernels under polynomial
sample and computational complexities.
