, by Vector Quantization, a dictionary D = {x1 , . . . , xp } with p m Step : Manifold Learning on the dictionary Laplacian Eigenmaps Manifold Learning searches Φ such that 1 2 ij Φ(xi ) − Φ(xj ) 2 KD(i, j) = Tr(ΦT LΦ) with ΦT DDΦ = I. Compute the similarity matrix KD between vectors xi ∈ D with KD(i, j) = k(xi , xj ) = exp − x i −x j 2 2 σ2 with σ = max (x i ,x j )∈D xi − xj 2 2 Compute the degree diagonal matrix DD of KD Solution is obtained with the eigen-decomposition of the normalized Laplacian L = I − D−1 2 D KDD−1 2 D as L = ΦDΠDΦT D with eigenvectors ΦD = [Φ1 D , · · · , Φp D ] and eigenvalues ΠD = diag[λ1, · · · , λp] O. L´ ezoray Multivariate approaches for graph signal morphological processing / 8