the spectral representation of a graph filter is independent from the graph

Data

subject:: Data Science Methods for Large Scale Graphs parent:: Graph Signals and Graph Signal Processing theme:: math notes

Remark

From the spectral representation of a convolutional graph filter $\hat{y}={\sum }_{k=0}^{K-1}{h}_k{\Lambda }^k\hat{x}$, we can see that this representation is completely defined by the polynomial $h(\lambda )={\sum }_{k=0}^{K-1}{h}_k{\lambda }^k$ in the spectral domain. ie, the filter is independent from the graph

• We can see from above that the filter frequency response only depends on the coefficients for the filter $h_k$ and the eigenvalues $\Lambda$ of the shift operator.

• If we consider another graph $\tilde{G}$ with shift operator $\tilde{S}$ and $\sigma (\tilde{S})=\tilde{\Lambda }$. Then $H(\tilde{S})={\sum }_{k=0}^{K-1}{h}_k{\tilde{\Lambda }}^k$

Proof

By inspection

Mentions

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