31 research outputs found
Sparse Power Factorization: Balancing peakiness and sample complexity
In many applications, one is faced with an inverse problem, where the known
signal depends in a bilinear way on two unknown input vectors. Often at least
one of the input vectors is assumed to be sparse, i.e., to have only few
non-zero entries. Sparse Power Factorization (SPF), proposed by Lee, Wu, and
Bresler, aims to tackle this problem. They have established recovery guarantees
for a somewhat restrictive class of signals under the assumption that the
measurements are random. We generalize these recovery guarantees to a
significantly enlarged and more realistic signal class at the expense of a
moderately increased number of measurements.Comment: 18 page
Upper and lower bounds for the Lipschitz constant of random neural networks
Empirical studies have widely demonstrated that neural networks are highly
sensitive to small, adversarial perturbations of the input. The worst-case
robustness against these so-called adversarial examples can be quantified by
the Lipschitz constant of the neural network. In this paper, we study upper and
lower bounds for the Lipschitz constant of random ReLU neural networks.
Specifically, we assume that the weights and biases follow a generalization of
the He initialization, where general symmetric distributions for the biases are
permitted. For shallow neural networks, we characterize the Lipschitz constant
up to an absolute numerical constant. For deep networks with fixed depth and
sufficiently large width, our established upper bound is larger than the lower
bound by a factor that is logarithmic in the width