10 research outputs found
Spectrum of non-Hermitian heavy tailed random matrices
Let (X_{jk})_{j,k>=1} be i.i.d. complex random variables such that |X_{jk}|
is in the domain of attraction of an alpha-stable law, with 0< alpha <2. Our
main result is a heavy tailed counterpart of Girko's circular law. Namely,
under some additional smoothness assumptions on the law of X_{jk}, we prove
that there exists a deterministic sequence a_n ~ n^{1/alpha} and a probability
measure mu_alpha on C depending only on alpha such that with probability one,
the empirical distribution of the eigenvalues of the rescaled matrix a_n^{-1}
(X_{jk})_{1<=j,k<=n} converges weakly to mu_alpha as n tends to infinity. Our
approach combines Aldous & Steele's objective method with Girko's Hermitization
using logarithmic potentials. The underlying limiting object is defined on a
bipartized version of Aldous' Poisson Weighted Infinite Tree. Recursive
relations on the tree provide some properties of mu_alpha. In contrast with the
Hermitian case, we find that mu_alpha is not heavy tailed.Comment: Expanded version of a paper published in Communications in
Mathematical Physics 307, 513-560 (2011
Efficient Second-Order Shape-Constrained Function Fitting
We give an algorithm to compute a one-dimensional shape-constrained function
that best fits given data in weighted- norm. We give a single
algorithm that works for a variety of commonly studied shape constraints
including monotonicity, Lipschitz-continuity and convexity, and more generally,
any shape constraint expressible by bounds on first- and/or second-order
differences. Our algorithm computes an approximation with additive error
in time, where
captures the range of input values. We also give a simple greedy algorithm that
runs in time for the special case of unweighted convex
regression. These are the first (near-)linear-time algorithms for
second-order-constrained function fitting. To achieve these results, we use a
novel geometric interpretation of the underlying dynamic programming problem.
We further show that a generalization of the corresponding problems to directed
acyclic graphs (DAGs) is as difficult as linear programming.Comment: accepted for WADS 2019; (v2 fixes various typos