45,642 research outputs found
Deterministic Sampling and Range Counting in Geometric Data Streams
We present memory-efficient deterministic algorithms for constructing
epsilon-nets and epsilon-approximations of streams of geometric data. Unlike
probabilistic approaches, these deterministic samples provide guaranteed bounds
on their approximation factors. We show how our deterministic samples can be
used to answer approximate online iceberg geometric queries on data streams. We
use these techniques to approximate several robust statistics of geometric data
streams, including Tukey depth, simplicial depth, regression depth, the
Thiel-Sen estimator, and the least median of squares. Our algorithms use only a
polylogarithmic amount of memory, provided the desired approximation factors
are inverse-polylogarithmic. We also include a lower bound for non-iceberg
geometric queries.Comment: 12 pages, 1 figur
Constant-Factor Approximation for TSP with Disks
We revisit the traveling salesman problem with neighborhoods (TSPN) and
present the first constant-ratio approximation for disks in the plane: Given a
set of disks in the plane, a TSP tour whose length is at most times
the optimal can be computed in time that is polynomial in . Our result is
the first constant-ratio approximation for a class of planar convex bodies of
arbitrary size and arbitrary intersections. In order to achieve a
-approximation, we reduce the traveling salesman problem with disks, up
to constant factors, to a minimum weight hitting set problem in a geometric
hypergraph. The connection between TSPN and hitting sets in geometric
hypergraphs, established here, is likely to have future applications.Comment: 14 pages, 3 figure
Residual Minimizing Model Interpolation for Parameterized Nonlinear Dynamical Systems
We present a method for approximating the solution of a parameterized,
nonlinear dynamical system using an affine combination of solutions computed at
other points in the input parameter space. The coefficients of the affine
combination are computed with a nonlinear least squares procedure that
minimizes the residual of the governing equations. The approximation properties
of this residual minimizing scheme are comparable to existing reduced basis and
POD-Galerkin model reduction methods, but its implementation requires only
independent evaluations of the nonlinear forcing function. It is particularly
appropriate when one wishes to approximate the states at a few points in time
without time marching from the initial conditions. We prove some interesting
characteristics of the scheme including an interpolatory property, and we
present heuristics for mitigating the effects of the ill-conditioning and
reducing the overall cost of the method. We apply the method to representative
numerical examples from kinetics - a three state system with one parameter
controlling the stiffness - and conductive heat transfer - a nonlinear
parabolic PDE with a random field model for the thermal conductivity.Comment: 28 pages, 8 figures, 2 table
Robust regression with imprecise data
We consider the problem of regression analysis with imprecise data. By imprecise data we mean imprecise observations of precise quantities in the form of sets of values. In this paper, we explore a recently introduced likelihood-based approach to regression with such data. The approach is very general, since it covers all kinds of imprecise data (i.e. not only intervals) and it is not restricted to linear regression. Its result consists of a set of functions, reflecting the entire uncertainty of the regression problem. Here we study in particular a robust special case of the likelihood-based imprecise regression, which can be interpreted as a generalization of the method of least median of squares. Moreover, we apply it to data from a social survey, and compare it with other approaches to regression with imprecise data. It turns out that the likelihood-based approach is the most generally applicable one and is the only approach accounting for multiple sources of uncertainty at the same time
Sum-of-squares proofs and the quest toward optimal algorithms
In order to obtain the best-known guarantees, algorithms are traditionally
tailored to the particular problem we want to solve. Two recent developments,
the Unique Games Conjecture (UGC) and the Sum-of-Squares (SOS) method,
surprisingly suggest that this tailoring is not necessary and that a single
efficient algorithm could achieve best possible guarantees for a wide range of
different problems.
The Unique Games Conjecture (UGC) is a tantalizing conjecture in
computational complexity, which, if true, will shed light on the complexity of
a great many problems. In particular this conjecture predicts that a single
concrete algorithm provides optimal guarantees among all efficient algorithms
for a large class of computational problems.
The Sum-of-Squares (SOS) method is a general approach for solving systems of
polynomial constraints. This approach is studied in several scientific
disciplines, including real algebraic geometry, proof complexity, control
theory, and mathematical programming, and has found applications in fields as
diverse as quantum information theory, formal verification, game theory and
many others.
We survey some connections that were recently uncovered between the Unique
Games Conjecture and the Sum-of-Squares method. In particular, we discuss new
tools to rigorously bound the running time of the SOS method for obtaining
approximate solutions to hard optimization problems, and how these tools give
the potential for the sum-of-squares method to provide new guarantees for many
problems of interest, and possibly to even refute the UGC.Comment: Survey. To appear in proceedings of ICM 201
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