  Heavy-tailed distributions have been studied in statistics, random matrix
theory, physics, and econometrics as models of correlated systems, among other
domains. Further, heavy-tail distributed eigenvalues of the covariance matrix
of the weight matrices in neural networks have been shown to empirically
correlate with test set accuracy in several works (e.g. arXiv:1901.08276), but
a formal relationship between heavy-tail distributed parameters and
generalization bounds was yet to be demonstrated. In this work, the compression
framework of arXiv:1802.05296 is utilized to show that matrices with heavy-tail
distributed matrix elements can be compressed, resulting in networks with
sparse weight matrices. Since the parameter count has been reduced to a sum of
the non-zero elements of sparse matrices, the compression framework allows us
to bound the generalization gap of the resulting compressed network with a
non-vacuous generalization bound. Further, the action of these matrices on a
vector is discussed, and how they may relate to compression and resilient
classification is analyzed.
