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.

This study proposes the method of quantitative analysis on discussion in workshop for proper operation of discussion. It also shows the characteristics and the roles of discussion in workshop in planning process. There is a perception gap about policy issues of the region between planners who set up a workshop, and participants. The objectives of this study are to make clear its gap quantitatively, and to establish a management solution for minimizing them. This study provides quantitative analysis of a content of discussion to make participant's concern clear. The efficacy of this method is also discussed.