  Compression techniques for deep neural network models are becoming very
important for the efficient execution of high-performance deep learning systems
on edge-computing devices. The concept of model compression is also important
for analyzing the generalization error of deep learning, known as the
compression-based error bound. However, there is still huge gap between a
practically effective compression method and its rigorous background of
statistical learning theory. To resolve this issue, we develop a new
theoretical framework for model compression and propose a new pruning method
called {\it spectral pruning} based on this framework. We define the ``degrees
of freedom'' to quantify the intrinsic dimensionality of a model by using the
eigenvalue distribution of the covariance matrix across the internal nodes and
show that the compression ability is essentially controlled by this quantity.
Moreover, we present a sharp generalization error bound of the compressed model
and characterize the bias--variance tradeoff induced by the compression
procedure. We apply our method to several datasets to justify our theoretical
analyses and show the superiority of the the proposed method.
