2d:null 2e:["$","title",null,{"children":"Spectral Convolutions Equation | Ameya Daigavane | Observable"}] 33:T8fd,spec_figure = html`

Start with the original features.

${tex.block`\quad {${main_col} h^{(0)}} = x`}

Then iterate, for ${tex`k = 1, 2, \ldots `} upto ${tex` K`}:

${tex.block` \begin{aligned} {\hat{h}^{(k - 1)}} &= U_m^T {${main_col} h^{(k - 1)}} \\ \\ {\hat{g}^{(k)}} &= {${param_col} \hat{w}^{(k)}} \odot {${main_col} \hat{h}^{(k - 1)}} \\ \\ {g^{(k)}} &= U_m {\hat{g}^{(k)}} \\ \\ {${main_col} h^{(k)}} &= \sigma \left({g^{(k)}} \right) \end{aligned} `}

Convert previous feature ${tex`${main_col} h^{(k - 1)}`} to its spectral representation ${tex`\hat{h}^{(k - 1)}`}.

Convolve with filter weights ${tex`{${param_col} \hat{w}^{(k)}}`} in the spectral domain to get ${tex`{\hat{g}^{(k)}}.`}

${tex`\odot`} represents element-wise multiplication.

Convert ${tex`\hat{g}^{(k)}`} back to its natural representation ${tex`{g}^{(k)}`}.

Apply a non-linearity ${tex`\sigma`} to ${tex`{g}^{(k)}`} to get ${tex`${main_col} {h}^{(k)}`}.

Color Codes:

Computed node embeddings.
Learnable parameters.

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\n \n ${tex.block`\n \\quad h^{(k)} = \\begin{bmatrix} h_1^{(k)} \\\\ \\vdots \\\\ h_n^{(k)} \\end{bmatrix}\n `}\n \n\n

\n for each ${tex`k = 0, 1, 2, \\ldots `} upto ${tex` K`}.\n

\n\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":7,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"style = html`\n \n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":343,"value":"import {main_col, main_col_rgb, param_col, param_col_rgb, spectral_col, natural_col} from \"@ameyasd/gnn-styles\";","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W34","stdlib":"1","titling":"auto"}} 32:"$31:body"