Laplacian eigenmaps [Belkin and Niyogi, 2002] decompose a function of the weight matrix known as the graph Laplacian to obtain a new representation of the data. Inspired by MDS, we consider a rescaled variation of standard eigenmaps. The Laplacian matrix S of a weighted graph G is defined by S(i,j)={−wijifi≠j,di−wiiifi=j, where di = ∑j wij is the so-called degree of vertex i. In matrix form, S=D−W, where D = diag(d1,…, dn) is a diagonal matrix. The normalized graph Laplacian is a matrix defined as ℒ=D−1/2SD−1/2. Entries in the kernel matrix, hij, measure the similarity, or correlation, between subjects, making it a good candidate for a weight matrix: the larger the entry for a pair (i,j), the stronger the connection between the subjects within the pair. For a weight matrix, 0 indicates an unconnected pair. Negative values in H correspond to negative correlations, hence motivating the choice of 0 for these pairs. We define the weights as wij={hijifhij≥0,0otherwise.
