Reconstructing Latent Orderings by Spectral Clustering

Abstract

Spectral clustering uses a graph Laplacian spectral embedding to enhance the cluster structure of some data sets. When the embedding is one dimensional, it can be used to sort the items (spectral ordering). A number of empirical results also suggests that a multidimensional Laplacian embedding enhances the latent ordering of the data, if any. This also extends to circular orderings, a case where unidimensional embeddings fail. We tackle the task of retrieving linear and circular orderings in a unifying framework, and show how a latent ordering on the data translates into a filamentary structure on the Laplacian embedding. We propose a method to recover it, illustrated with numerical experiments on synthetic data and real DNA sequencing data. The code and experiments are available at https: //github.com/antrec/mdso.

Localization in 1D non-parametric latent space models from pairwise affinities

“…S , x * S ) = O n log(n) . We propose two estimators fulfilling this requirement: (a) a first one, which requires no additional assumptions, but which has a super-polynomial computational complexity; (b) a second one, adapted from [Recanati et al, 2018], which has a polynomial computational complexity, but for which we prove a O n log(n) control only for a class of random geometric graphs.…”

“…For this noiseless version of Example 3, efficient algorithms have been proposed using convex optimization [Fogel et al, 2013], or spectral methods [Atkins et al, 1998]. The exact seriation problem has been solved on toroidal R-matrices in the noiseless case [Recanati et al, 2018], by using a spectral algorithm. A perturbation analysis has also been sketched in [Recanati et al, 2018].…”

“…The exact seriation problem has been solved on toroidal R-matrices in the noiseless case [Recanati et al, 2018], by using a spectral algorithm. A perturbation analysis has also been sketched in [Recanati et al, 2018]. As a byproduct of our analysis, we provide some more explicit recovery bounds in our specific setting with noisy observations.…”

“…x(1) S requires to minimize (18) over Π n 0 , which is an instance of the Quadratic Assignment Problem (QAP), which is known to be NP-hard and hard to approximate [Queyranne, 1986, Sahni andGonzalez, 1976]. A spectral relaxation of the QAP has been shown to be successful for reordering a pre (toroidal) R-matrix [Atkins et al, 1998, Recanati et al, 2018, and hence for solving the noiseless seriation problem for R-matrices. This vanilla spectral algorithm proposed in [Atkins et al, 1998] takes as input any symmetric matrix M ∈ R N ×N and output N points in R 2 .…”

“…, n} as a torus with the corresponding distance d(i, j) = min(|j − i|, |n + i − j|) for any 1 ≤ i, j ≤ n. A toroidal R-matrix is any symmetric matrix B whose entries decrease when moving away from the diagonal with respect to the toroidal distance: B i,j ≥ B i+1,j when d(i, j) < d(i+1, j) and B i,j ≥ B i,j+1 when d(i, j) < d(i, j +1). As in Example 3 above, a pre-toroidal R-matrix is defined as a permutation of a toroidal R-matrix and the statistical seriation model is defined analogously [Recanati et al, 2018]. Again, we can recast this model as a latent space model on X = {1, .…”