  It has been shown that the effectiveness of graph convolutional network (GCN)
for recommendation is attributed to the spectral graph filtering. Most
GCN-based methods consist of a graph filter or followed by a low-rank mapping
optimized based on supervised training. However, we show two limitations
suppressing the power of graph filtering: (1) Lack of generality. Due to the
varied noise distribution, graph filters fail to denoise sparse data where
noise is scattered across all frequencies, while supervised training results in
worse performance on dense data where noise is concentrated in middle
frequencies that can be removed by graph filters without training. (2) Lack of
expressive power. We theoretically show that linear GCN (LGCN) that is
effective on collaborative filtering (CF) cannot generate arbitrary embeddings,
implying the possibility that optimal data representation might be unreachable.
  To tackle the first limitation, we show close relation between noise
distribution and the sharpness of spectrum where a sharper spectral
distribution is more desirable causing data noise to be separable from
important features without training. Based on this observation, we propose a
generalized graph normalization G^2N to adjust the sharpness of spectral
distribution in order to redistribute data noise to assure that it can be
removed by graph filtering without training. As for the second limitation, we
propose an individualized graph filter (IGF) adapting to the different
confidence levels of the user preference that interactions can reflect, which
is proved to be able to generate arbitrary embeddings. By simplifying LGCN, we
further propose a simplified graph filtering (SGFCF) which only requires the
top-K singular values for recommendation. Finally, experimental results on four
datasets with different density settings demonstrate the effectiveness and
efficiency of our proposed methods.
