23 research outputs found
Spectral Clustering of Graphs with the Bethe Hessian
Spectral clustering is a standard approach to label nodes on a graph by
studying the (largest or lowest) eigenvalues of a symmetric real matrix such as
e.g. the adjacency or the Laplacian. Recently, it has been argued that using
instead a more complicated, non-symmetric and higher dimensional operator,
related to the non-backtracking walk on the graph, leads to improved
performance in detecting clusters, and even to optimal performance for the
stochastic block model. Here, we propose to use instead a simpler object, a
symmetric real matrix known as the Bethe Hessian operator, or deformed
Laplacian. We show that this approach combines the performances of the
non-backtracking operator, thus detecting clusters all the way down to the
theoretical limit in the stochastic block model, with the computational,
theoretical and memory advantages of real symmetric matrices.Comment: 8 pages, 2 figure
Spectral density of the non-backtracking operator
The non-backtracking operator was recently shown to provide a significant
improvement when used for spectral clustering of sparse networks. In this paper
we analyze its spectral density on large random sparse graphs using a mapping
to the correlation functions of a certain interacting quantum disordered system
on the graph. On sparse, tree-like graphs, this can be solved efficiently by
the cavity method and a belief propagation algorithm. We show that there exists
a paramagnetic phase, leading to zero spectral density, that is stable outside
a circle of radius , where is the leading eigenvalue of the
non-backtracking operator. We observe a second-order phase transition at the
edge of this circle, between a zero and a non-zero spectral density. That fact
that this phase transition is absent in the spectral density of other matrices
commonly used for spectral clustering provides a physical justification of the
performances of the non-backtracking operator in spectral clustering.Comment: 6 pages, 6 figures, submitted to EP
Matrix Completion from Fewer Entries: Spectral Detectability and Rank Estimation
The completion of low rank matrices from few entries is a task with many
practical applications. We consider here two aspects of this problem:
detectability, i.e. the ability to estimate the rank reliably from the
fewest possible random entries, and performance in achieving small
reconstruction error. We propose a spectral algorithm for these two tasks
called MaCBetH (for Matrix Completion with the Bethe Hessian). The rank is
estimated as the number of negative eigenvalues of the Bethe Hessian matrix,
and the corresponding eigenvectors are used as initial condition for the
minimization of the discrepancy between the estimated matrix and the revealed
entries. We analyze the performance in a random matrix setting using results
from the statistical mechanics of the Hopfield neural network, and show in
particular that MaCBetH efficiently detects the rank of a large
matrix from entries, where is a constant close to .
We also evaluate the corresponding root-mean-square error empirically and show
that MaCBetH compares favorably to other existing approaches.Comment: NIPS Conference 201
Clustering from Sparse Pairwise Measurements
We consider the problem of grouping items into clusters based on few random
pairwise comparisons between the items. We introduce three closely related
algorithms for this task: a belief propagation algorithm approximating the
Bayes optimal solution, and two spectral algorithms based on the
non-backtracking and Bethe Hessian operators. For the case of two symmetric
clusters, we conjecture that these algorithms are asymptotically optimal in
that they detect the clusters as soon as it is information theoretically
possible to do so. We substantiate this claim for one of the spectral
approaches we introduce
Case Report Convergence Insufficiency/Divergence Insufficiency Convergence Excess/Divergence Excess: Some Facts and Fictions
Great discrepancies are often encountered between the distance fixation and the near-fixation esodeviations and exodeviations. They are all attributed to either anomalies of the AC/A ratio or anomalies of the fusional convergence or divergence amplitudes. We report a case with pseudoconvergence insufficiency and another one with pseudoaccommodative convergence excess. In both cases, conv./div. excess and insufficiency were erroneously attributed to anomalies of the AC/A ratio or to anomalies of the fusional amplitudes. Our purpose is to show that numerous factors, other than anomalies in the AC/A ratio or anomalies in the fusional conv. or divergence amplitudes, can contaminate either the distance or the near deviations. This results in significant discrepancies between the distance and the near deviations despite a normal AC/A ratio and normal fusional amplitudes, leading to erroneous diagnoses and inappropriate treatment models