3 research outputs found
On the convergence of orthogonalization-free conjugate gradient method for extreme eigenvalues of Hermitian matrices: a Riemannian optimization interpretation
In many applications, it is desired to obtain extreme eigenvalues and
eigenvectors of large Hermitian matrices by efficient and compact algorithms.
In particular, orthogonalization-free methods are preferred for large-scale
problems for finding eigenspaces of extreme eigenvalues without explicitly
computing orthogonal vectors in each iteration. For the top eigenvalues,
the simplest orthogonalization-free method is to find the best rank-
approximation to a positive semi-definite Hermitian matrix by algorithms
solving the unconstrained Burer-Monteiro formulation. We show that the
nonlinear conjugate gradient method for the unconstrained Burer-Monteiro
formulation is equivalent to a Riemannian conjugate gradient method on a
quotient manifold with the Bures-Wasserstein metric, thus its global
convergence to a stationary point can be proven. Numerical tests suggest that
it is efficient for computing the largest eigenvalues for large-scale
matrices if the largest eigenvalues are nearly distributed uniformly
A Distributed Block Chebyshev-Davidson Algorithm for Parallel Spectral Clustering
We develop a distributed Block Chebyshev-Davidson algorithm to solve
large-scale leading eigenvalue problems for spectral analysis in spectral
clustering. First, the efficiency of the Chebyshev-Davidson algorithm relies on
the prior knowledge of the eigenvalue spectrum, which could be expensive to
estimate. This issue can be lessened by the analytic spectrum estimation of the
Laplacian or normalized Laplacian matrices in spectral clustering, making the
proposed algorithm very efficient for spectral clustering. Second, to make the
proposed algorithm capable of analyzing big data, a distributed and parallel
version has been developed with attractive scalability. The speedup by parallel
computing is approximately equivalent to , where denotes the
number of processes. {Numerical results will be provided to demonstrate its
efficiency in spectral clustering and scalability advantage over existing
eigensolvers used for spectral clustering in parallel computing environments.