1,734 research outputs found
Almost-Euclidean subspaces of via tensor products: a simple approach to randomness reduction
It has been known since 1970's that the N-dimensional -space contains
nearly Euclidean subspaces whose dimension is . However, proofs of
existence of such subspaces were probabilistic, hence non-constructive, which
made the results not-quite-suitable for subsequently discovered applications to
high-dimensional nearest neighbor search, error-correcting codes over the
reals, compressive sensing and other computational problems. In this paper we
present a "low-tech" scheme which, for any , allows to exhibit nearly
Euclidean -dimensional subspaces of while using only
random bits. Our results extend and complement (particularly) recent work
by Guruswami-Lee-Wigderson. Characteristic features of our approach include (1)
simplicity (we use only tensor products) and (2) yielding "almost Euclidean"
subspaces with arbitrarily small distortions.Comment: 11 pages; title change, abstract and references added, other minor
change
Dynamically Triangulating Lorentzian Quantum Gravity
Fruitful ideas on how to quantize gravity are few and far between. In this
paper, we give a complete description of a recently introduced non-perturbative
gravitational path integral whose continuum limit has already been investigated
extensively in d less than 4, with promising results. It is based on a
simplicial regularization of Lorentzian space-times and, most importantly,
possesses a well-defined, non-perturbative Wick rotation. We present a detailed
analysis of the geometric and mathematical properties of the discretized model
in d=3,4. This includes a derivation of Lorentzian simplicial manifold
constraints, the gravitational actions and their Wick rotation. We define a
transfer matrix for the system and show that it leads to a well-defined
self-adjoint Hamiltonian. In view of numerical simulations, we also suggest
sets of Lorentzian Monte Carlo moves. We demonstrate that certain pathological
phases found previously in Euclidean models of dynamical triangulations cannot
be realized in the Lorentzian case.Comment: 41 pages, 14 figure
Convergence of the critical finite-range contact process to super-Brownian motion above the upper critical dimension: I. The higher-point functions
We consider the critical spread-out contact process in Z^d with d\ge1, whose
infection range is denoted by L\ge1. In this paper, we investigate the r-point
function \tau_{\vec t}^{(r)}(\vec x) for r\ge3, which is the probability that,
for all i=1,...,r-1, the individual located at x_i\in Z^d is infected at time
t_i by the individual at the origin o\in Z^d at time 0. Together with the
results of the 2-point function in [van der Hofstad and Sakai, Electron. J.
Probab. 9 (2004), 710-769; arXiv:math/0402049], on which our proofs crucially
rely, we prove that the r-point functions converge to the moment measures of
the canonical measure of super-Brownian motion above the upper-critical
dimension 4. We also prove partial results for d\le4 in a local mean-field
setting.Comment: 75 pages, 12 figure
Handling convexity-like constraints in variational problems
We provide a general framework to construct finite dimensional approximations
of the space of convex functions, which also applies to the space of c-convex
functions and to the space of support functions of convex bodies. We give
estimates of the distance between the approximation space and the admissible
set. This framework applies to the approximation of convex functions by
piecewise linear functions on a mesh of the domain and by other
finite-dimensional spaces such as tensor-product splines. We show how these
discretizations are well suited for the numerical solution of problems of
calculus of variations under convexity constraints. Our implementation relies
on proximal algorithms, and can be easily parallelized, thus making it
applicable to large scale problems in dimension two and three. We illustrate
the versatility and the efficiency of our approach on the numerical solution of
three problems in calculus of variation : 3D denoising, the principal agent
problem, and optimization within the class of convex bodies.Comment: 23 page
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