Table 1 Key notation used in the proof
From: Bandlimited graph signal reconstruction by diffusion operator
Symbol | Description |
P w | The graph frequency cut-off operator |
w | The graph cut-off frequency |
S | The downsampling set |
V | The set of vertices of the graph |
E | The set of edges connecting vertices |
\({\mathcal {F}}\) | The graph Fourier transform |
\({{\mathcal {F}}^{- 1}}\) | The inverse graph Fourier transform |
P T | The downsampling operator |
u,v | The vertex |
P d | The diffusion operator |
f | A bandlimited graph signal |
P m | The local-mean diffusion operator |
P b | The global-bias diffusion operator |
K(u) | The local-diffused unsampled vertices set of the sampled vertex u |
d(v) | The degree of the vertex v |
α 1,α 2 | The downsampling parameter |
M | The cardinality of the downsampling vertices set |
N | The cardinality of the graph vertices set |
Y | The cardinality of the eigenvalue less than the graph cut-off frequency |
H K(u) | The maximal distance from sampled vertex u to other vertices within K(u) |
X K(u) | The cardinality of K(u) |
R max | The maximal value of the product of X K(u) and H K(u) in graph |
J(u) | The degree operator of K(u) |
J max | The maximal value J(u) in graph |