Algorithm A.5.

function MultiAlignment $(\mathcal{X},\mathcal{K},t,\ell )={{\mathrm{\Phi }}_t}^{(1,\dots ,n)}$

Align the diffusion maps of multiple datasets.

Input:		Data sets	$\mathcal{X}=\left\{{\mathcal{X}}^{(1)},\dots ,{\mathcal{X}}^{(n)}\right\}$
		Kernel parameters	$\mathcal{K}=\left\{{\mathcal{K}}^{(X)},\dots ,{\mathcal{K}}^{(n)}\right\}$
		Alignment diffusion time	t
		Alignment band count	$\ell$
Output:		Unified diffusion map	Φt(1,...,n)
1:	for i=1,n do
2:	${\mathcal{G}}^{(i)}\leftarrow \left\{W^{(i)},{ℒ}^{(i)},D^{(i)}\right\}\leftarrow \text{GaussKernelGraph}\left({\mathcal{X}}^{(i)},{\mathcal{K}}^{(i)}\right)$			▷ Alg. A.2
3:	${\mathrm{\Psi }}^{(i)},{\mathrm{\Lambda }}^{(i)}\leftarrow \mathrm{SVD}\left(I-{ℒ}^{(i)}\right)$			▷ Graph Fourier basis
4:	${\mathrm{\Psi }}^{(i)}\leftarrow {\mathrm{\Psi }}^{(i)}\backslash {\psi }_1^{(i)};{\mathrm{\Lambda }}^{(i)}\leftarrow {\mathrm{\Lambda }}^{(i)}\backslash {\lambda }_1^{(i)}$
5:	${\widehat{\mathcal{X}}}^{(i)}\leftarrow {\mathrm{\Psi }}^{{(i)}^T}{\mathcal{X}}^{(i)}$			▷ Graph Fourier transform
6:	${{\mathrm{\Phi }}_0}^{(i)}\leftarrow D^{{(i)}^{1/2}}{\mathrm{\Psi }}^{(i)}$
7:	end for
8:	for i = 1, n do
9:	${\mathrm{\Phi }}_t{(i,i)}^{(1,\dots ,n)}\leftarrow {{\mathrm{\Phi }}_0}^{(i)}{\mathrm{\Lambda }}^{t^{(i)}}$
10:	for j = i + 1, n do
11:	w(i, j) ← BandlimitingWeights(Λ(i), Λ(j), $\ell$)			▷ Alg. A.4
12:	for $\ell$ = 2, Ni do
13:	for k = 2, Nj do.
14:	$C^{(i,j)}(\ell -1,k-1)\leftarrow w^{(i,j)}\left({{\lambda }_{\ell }}^{(i)},{{\lambda }_k}^{(j)}\right)\langle {\widehat{X}}^{(i)}(\ell -1,:),{\widehat{X}}^{(j)}(k-1,:)\rangle$			▷ Bandlimited correlation
15:	end for
16:	end for
17:	U(i,j),S(i,j),V(i,j) ← C(i,j)
18:	$T^{(i\to j)}\leftarrow U^{(i,j)}{V}^{{(i,j)}^T}:T^{(j\to i)}\leftarrow V^{(i,j)}{U}^{{(i,j)}^T}$
19:	${\mathrm{\Phi }}_t{(i,j)}^{(1,\dots ,n)}\leftarrow {{\mathrm{\Phi }}_0}^{(i)}{T}^{(i\to j)}{\mathrm{\Lambda }}^{{(j)}^t}$
20:	${\mathrm{\Phi }}_t{(j,i)}^{(1,\dots ,n)}\leftarrow {{\mathrm{\Phi }}_0}^{(j)}{T}^{(j\to i)}{\mathrm{\Lambda }}^{{(i)}^t}$
21:	end for
22:	end for
23:	return Φt(1,...,n)