848 research outputs found
Tail bounds for all eigenvalues of a sum of random matrices
This work introduces the minimax Laplace transform method, a modification of
the cumulant-based matrix Laplace transform method developed in "User-friendly
tail bounds for sums of random matrices" (arXiv:1004.4389v6) that yields both
upper and lower bounds on each eigenvalue of a sum of random self-adjoint
matrices. This machinery is used to derive eigenvalue analogues of the
classical Chernoff, Bennett, and Bernstein bounds.
Two examples demonstrate the efficacy of the minimax Laplace transform. The
first concerns the effects of column sparsification on the spectrum of a matrix
with orthonormal rows. Here, the behavior of the singular values can be
described in terms of coherence-like quantities. The second example addresses
the question of relative accuracy in the estimation of eigenvalues of the
covariance matrix of a random process. Standard results on the convergence of
sample covariance matrices provide bounds on the number of samples needed to
obtain relative accuracy in the spectral norm, but these results only guarantee
relative accuracy in the estimate of the maximum eigenvalue. The minimax
Laplace transform argument establishes that if the lowest eigenvalues decay
sufficiently fast, on the order of (K^2*r*log(p))/eps^2 samples, where K is the
condition number of an optimal rank-r approximation to C, are sufficient to
ensure that the dominant r eigenvalues of the covariance matrix of a N(0, C)
random vector are estimated to within a factor of 1+-eps with high probability.Comment: 20 pages, 1 figure, see also arXiv:1004.4389v
Robert MacPherson and arithmetic groups
We survey contributions of Robert MacPherson to the theory of arithmetic
groups. There are two main areas we discuss: (i) explicit reduction theory for
Siegel modular threefolds, and (ii) constructions of compactifications of
locally symmetric spaces. The former is joint work with Mark McConnell, the
latter with Lizhen Ji.Comment: Dedicated to Robert MacPherson on the occasion of his 60th birthda
The complexity and geometry of numerically solving polynomial systems
These pages contain a short overview on the state of the art of efficient
numerical analysis methods that solve systems of multivariate polynomial
equations. We focus on the work of Steve Smale who initiated this research
framework, and on the collaboration between Stephen Smale and Michael Shub,
which set the foundations of this approach to polynomial system--solving,
culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo,
Peter Buergisser and Felipe Cucker
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