Ioana Dumitriu
Minimizing Communication for Eigenproblems and the Singular Value Decomposition.
Tue, 11/16/2010 - 16:01 — SomNambulAuthors: Ioana Dumitriu, Grey Ballard, James Demmel
Subjects: Numerical Analysis
AbstractAlgorithms have two costs: arithmetic and communication. The latter
represents the cost of moving data, either between levels of a memory
hierarchy, or between processors over a network. Communication often dominates
arithmetic and represents a rapidly increasing proportion of the total cost, so
we seek algorithms that minimize communication. In \cite{BDHS10} lower bounds
were presented on the amount of communication required for essentially all
$O(n^3)$-like algorithms for linear algebra, including eigenvalue problems and
the SVD.Sparse regular random graphs: spectral density and eigenvectors.
Thu, 10/29/2009 - 16:01 — SomNambulAuthors: Soumik Pal, Ioana Dumitriu
Subjects: Probability
AbstractWe examine the empirical distribution of the eigenvalues and the eigenvectors
of adjacency matrices of regular random graphs. We find that when the degree
sequence of the graph slowly increases to infinity with the number of vertices,
the empirical spectral distribution converges to the semicircular law.
Moreover, we prove concentration estimates on the number of eigenvalues over
progressively smaller intervals. We also show that, with high probability, all
the eigenvectors are delocalized and none of them, except the first, are
concentrated around lower-dimensional subspaces.Tridiagonal realization of the anti-symmetric Gaussian $\beta$-ensemble.
Wed, 10/28/2009 - 16:01 — SomNambulAuthors: Peter J. Forrester, Ioana Dumitriu
Subjects: Mathematical Physics
AbstractThe Householder reduction of a member of the anti-symmetric Gaussian unitary
ensemble gives an anti-symmetric tridiagonal matrix with all independent
elements. The random variables permit the introduction of a positive parameter
$\beta$, and the eigenvalue probability density function of the corresponding
random matrices can be computed explicitly, as can the distribution of
$\{q_i\}$, the first components of the eigenvectors.
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