  The goal of model compression is to reduce the size of a large neural network
while retaining a comparable performance. As a result, computation and memory
costs in resource-limited applications may be significantly reduced by dropping
redundant weights, neurons, or layers. There have been many model compression
algorithms proposed that provide impressive empirical success. However, a
theoretical understanding of model compression is still limited. One problem is
understanding if a network is more compressible than another of the same
structure. Another problem is quantifying how much one can prune a network with
theoretically guaranteed accuracy degradation. In this work, we propose to use
the sparsity-sensitive $\ell_q$-norm ($0<q<1$) to characterize compressibility
and provide a relationship between soft sparsity of the weights in the network
and the degree of compression with a controlled accuracy degradation bound. We
also develop adaptive algorithms for pruning each neuron in the network
informed by our theory. Numerical studies demonstrate the promising performance
of the proposed methods compared with standard pruning algorithms.
