approximation of heaviside functions using convolutional graph filters

Data

Note

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

Proof

As an illustration, we want to find coefficients $h_{k}$ such that $\sum_{k = 0}^{K - 1} h_{k} λ^{k} = \tilde{f} (λ)$ ,

\tilde{f} (λ) = \frac{1}{1 + e^{- α (λ - c)}}

Here,

\begin{aligned} \tilde{f} (0) & = \frac{1}{1 + e^{α c}} \\ {\tilde{f}}^{'} (λ) = \frac{1}{(1 + e^{- α (λ - c)})^{2}} e^{- α (λ - c)} + α ⟹ {\tilde{f}}^{'} (0) & = \frac{α e^{α c}}{(1 + e^{- α c})^{2}} \\ {\tilde{f}}^{″} (λ) = \frac{- α^{2} e^{- α (λ - c)}}{(1 + e^{- α (λ - c)})^{2}} + \frac{2 α e^{- 2 α (λ - c)} + α}{(1 + e^{- α (λ - c)})^{3}} \\ ⟹ {\tilde{f}}^{″} (0) & = \frac{α^{2} e^{α c}}{(1 + e^{- α c})^{2}} (\frac{2 e^{α c}}{(1 + e^{- α c})} - 1) \\ ⋮ \end{aligned}

Then $h_{0} = \tilde{f} (0), h_{1} = {\tilde{f}}^{'} (0), h_{2} = \frac{{\tilde{f}}^{″} (0)}{2}$ etc.

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2025-02-03 graphs lecture 4