Algorithm A.3.

function Align $\left({\mathcal{G}}^{(X)},{\mathcal{G}}^{(Y)},t,\ell \right)={\mathrm{\Phi }}_t{}^{(X,Y)}$

Compute and apply an alignment matrix to two diffusion maps.

Input:		Laplacian eigensystems and degrees	${\mathcal{G}}^{(X)},{\mathcal{G}}^{(Y)}:{\mathcal{G}}^{(Z)}=\left\{{\mathrm{\Psi }}^{(Z)},{\mathrm{\Lambda }}^{(Z)},D^{(Z)}\right\}$
		Alignment diffusion	t
		Alignment band count	$\ell$
Output:		Aligned diffusion map	${\mathrm{\Phi }}_t{}^{(X,Y)}$
1:	for Z=X,Y do
2:	${\mathrm{\Psi }}^{(Z)}\leftarrow {\mathrm{\Psi }}^{(Z)}\backslash {\psi }_1^{(Z)};{\mathrm{\Lambda }}^{(Z)}\leftarrow {\mathrm{\Lambda }}^{(Z)}\backslash {\lambda }_1^{(Z)}$
3:	${\mathrm{\Phi }}^{0(Z)}\leftarrow D^{{(Z)}^{1/2}}{\mathrm{\Psi }}^{(Z)}$			▷ Graph Fourier transform
4:	$\widehat{Z}\leftarrow {\mathrm{\Psi }}^{{(Z)}^T}Z$
5:	end for
6:	$w^{(X,Y)}\leftarrow \text{Bandlimiting Weights}\left({\mathrm{\Lambda }}^{(X)},{\mathrm{\Lambda }}^{(Y)},\ell \right)$			▷ Alg. A.4
7:	for i = 2, N1 do
8:	for j = 2,N2 do
9:	$C(i-1,j-1)\leftarrow w_{ij}{}^{(X,Y)}\langle \widehat{X}(i-1,:),\widehat{Y}(j-1,:)\rangle$			▷ Bandlimited correlatios
10:	end for
11:	end for
12:	U, S, V ← SVD(C)
13:	T ← UVT			▷ Orthogonalization Sec. 4.2
14:	${\mathrm{\Phi }}_t^{(X,Y)}\leftarrow \left[\begin{array}{cc}{\mathrm{\Phi }}^{0^{(X)}}& {\mathrm{\Phi }}^{0^{(X)}}\mathbf{T}\\ {\mathrm{\Phi }}^{0^{(Y)}}{\mathbf{T}}^T& {\mathrm{\Phi }}^{0^{(Y)}}\end{array}\right]{\left[\begin{array}{cc}{\mathrm{\Lambda }}^{(X)}& 0\\ 0& {\mathrm{\Lambda }}^{(Y)}\end{array}\right]}^t$
15:	return ${\mathrm{\Phi }}_t{}^{(\overline{X},Y)}$