410 research outputs found
Sparse Subspace Clustering: Algorithm, Theory, and Applications
In many real-world problems, we are dealing with collections of
high-dimensional data, such as images, videos, text and web documents, DNA
microarray data, and more. Often, high-dimensional data lie close to
low-dimensional structures corresponding to several classes or categories the
data belongs to. In this paper, we propose and study an algorithm, called
Sparse Subspace Clustering (SSC), to cluster data points that lie in a union of
low-dimensional subspaces. The key idea is that, among infinitely many possible
representations of a data point in terms of other points, a sparse
representation corresponds to selecting a few points from the same subspace.
This motivates solving a sparse optimization program whose solution is used in
a spectral clustering framework to infer the clustering of data into subspaces.
Since solving the sparse optimization program is in general NP-hard, we
consider a convex relaxation and show that, under appropriate conditions on the
arrangement of subspaces and the distribution of data, the proposed
minimization program succeeds in recovering the desired sparse representations.
The proposed algorithm can be solved efficiently and can handle data points
near the intersections of subspaces. Another key advantage of the proposed
algorithm with respect to the state of the art is that it can deal with data
nuisances, such as noise, sparse outlying entries, and missing entries,
directly by incorporating the model of the data into the sparse optimization
program. We demonstrate the effectiveness of the proposed algorithm through
experiments on synthetic data as well as the two real-world problems of motion
segmentation and face clustering
The anatomy of soft approaches
This paper is an inquiry into the nature and characteristics of the so-called soft approaches. As point of departure, two classical references on soft approaches are critically discussed. Six well-known soft approaches are selected for further study and characterisation applying a multi-dimensional framework. In addition, the limitations of such a framework are discussed
Riemannian consensus for manifolds with bounded curvature
Consensus algorithms are popular distributed algorithms for computing aggregate quantities, such as averages, in ad-hoc wireless networks. However, existing algorithms mostly address the case where the measurements lie in Euclidean space. In this work we propose Riemannian consensus, a natural extension of existing averaging consensus algorithms to the case of Riemannian manifolds. Unlike previous generalizations, our algorithm is intrinsic and, in principle, can be applied to any complete Riemannian manifold. We give sufficient convergence conditions on Riemannian manifolds with bounded curvature and we analyze the differences with respect to the Euclidean case. We test the proposed algorithms on synthetic data sampled from the space of rotations, the sphere and the Grassmann manifold.This work was supported by the grant NSF CNS-0834470. Recommended by Associate Editor L. Schenato. (CNS-0834470 - NSF
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