approximation of heaviside functions using convolutional graph filters

Data

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

Note

We can approximate heaviside functions using the logistic function as a proxy for our target function

analytic, and recall that we can represent any analytic function with convolutional graph filters.

Logistic functions are

As an illustration, we want to find coefficients $h_k$ such that ${\sum }_{k=0}^{K-1}{h}_k{\lambda }^k=\tilde{f}(\lambda )$, $\tilde{f}(\lambda )=\frac{1}{1+e^{-\alpha (\lambda -c)}}$ Here,

\tilde{f}(0) &= \frac{1}{1+e^{\alpha c}} \\ \tilde{f}'(\lambda) = \frac{1}{(1+e^{-\alpha(\lambda-c)})^2}e^{-\alpha(\lambda-c)}+\alpha \implies \tilde{f}'(0) &= \frac{\alpha e^{\alpha c}}{(1+e^{-\alpha c})^2} \\ \tilde{f}''(\lambda)=\frac{-\alpha^2e^{-\alpha(\lambda-c)}}{(1+e^{-\alpha(\lambda-c)})^2} + \frac{2\alpha e^{-2\alpha(\lambda-c)} +\alpha}{(1+e^{-\alpha(\lambda-c)})^3} \\ \implies \tilde{f}''(0) &= \frac{\alpha^2e^{\alpha c}}{(1+e^{-\alpha c})^2}\left(\frac{2e^{\alpha c}}{(1+e^{-\alpha c})} -1\right) \\ \vdots \end{aligned}$$ Then $h_{0}=\tilde{f} (0), h_{1}=\tilde{f}'(0), h_{2}=\frac{\tilde{f}''(0)}{2}$ etc.

Mentions

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