## 5 Application I: Estimation of the Norms of Localized Graph Spectral Filter Dictionary Atoms
We plot both sets of spectral graph filters on the spectrum of the Stanford bunny graph with \(N=2503\) vertices; however, the design only depends on the spectral range \([0,\lambda_{\text{max}}]\), so the filters will look the same (except stretched) for all graphs considered, e.g., in Figure 5.  
where each \(\mathbf{x}^{(j)}\) is a realization of the random vector \(\mathbf{x}\). In the numerical experiments, we compare the estimates resulting from Chebyshev polynomial approximation to those resulting from spectrum-adapted weighted least squares polynomial approximation. As a computational aside, in the process of estimating the spectral density via KPM in (6), we have already computed \(\tilde{T}_{k}(\mathbf{L})\mathbf{x}^{(j)}\) for each \(k=0,1,\ldots,K\) and each \(j\), where \(\tilde{T}_{k}\) are the Chebyshev polynomials shifted to the interval \([0,\lambda_{\max}]\). From these quantities, we can easily compute the \(p_{l,K}(\mathbf{L})\mathbf{x}^{(j)}\) vectors in (10) for different values of \(l\) (different filters). See [41] (Sec. III.B.1) for details. In the top row of Figure 7, we demonstrate the estimation of the norms of the atoms of a spectral graph wavelet dictionary generated by localizing the four filters in Figure 6b to each of the vertices of the bunny graph. Figure 7a shows the exact norms of the \(NL=2503\cdot 4=10012\) dictionary atoms. In Figure 7b, we plot the estimated norms of the atoms, generated via degree \(K=8\) spectrum-adapted weighted least squares polynomial approximation with \(J=50\) random vectors in (10), against the actual atom norms. The ratios of the estimated atom norms to the actual norms are shown in Figure 7c. We show the mean of the relative error \(\left|\frac{v_{i,j}}{||\boldsymbol{\varphi}_{i,j}||_{2}}-1\right|\) across all of these atoms as a single point in Figure 7d, and also repeat this experiment with different values of \(K\) and \(J\) and different classes of approximating polynomials, as well as for different graphs in Figure 7e-f. On all three graphs and at both values of \(J\), for low degrees \(K\), the estimates generated from the spectrum-adapted polynomial least squares method have lower mean relative error than those based on Chebyshev polynomial approximation. While these examples are on small to medium graphs for comparison to the exact atom norms, the method scales efficiently to dictionaries designed for large, sparse graphs. Figure 7: Estimation of the norms of the atoms of spectral graph wavelet dictionaries. (**a**) Exact atom norms, \(\{||\boldsymbol{\varphi}_{i,j}||_{2}\}\), colored by the indices of the generating filters (shown in Figure 6), each of which is localized to every vertex on the bunny graph. (**b**) Comparison of the estimated norms, \(\{v_{i,j}\}\), to the corresponding exact norms (shown in black). The atoms are sorted in descending order of exact norm to aid visual comparison. (**c**) The ratios of the estimated norms to the exact atom norms. (**d–f**) The mean relative errors across all atoms for dictionaries generated by the same set of filters on three different graphs, with varying polynomial approximation methods, polynomial degrees \(K\), and numbers of random vectors \(J\).