4,553 research outputs found
Controlled Sequential Monte Carlo
Sequential Monte Carlo methods, also known as particle methods, are a popular
set of techniques for approximating high-dimensional probability distributions
and their normalizing constants. These methods have found numerous applications
in statistics and related fields; e.g. for inference in non-linear non-Gaussian
state space models, and in complex static models. Like many Monte Carlo
sampling schemes, they rely on proposal distributions which crucially impact
their performance. We introduce here a class of controlled sequential Monte
Carlo algorithms, where the proposal distributions are determined by
approximating the solution to an associated optimal control problem using an
iterative scheme. This method builds upon a number of existing algorithms in
econometrics, physics, and statistics for inference in state space models, and
generalizes these methods so as to accommodate complex static models. We
provide a theoretical analysis concerning the fluctuation and stability of this
methodology that also provides insight into the properties of related
algorithms. We demonstrate significant gains over state-of-the-art methods at a
fixed computational complexity on a variety of applications
Optimising the Solovay-Kitaev algorithm
The Solovay-Kitaev algorithm is the standard method used for approximating
arbitrary single-qubit gates for fault-tolerant quantum computation. In this
paper we introduce a technique called "search space expansion", which modifies
the initial stage of the Solovay-Kitaev algorithm, increasing the length of the
possible approximating sequences but without requiring an exhaustive search
over all possible sequences. We show that our technique, combined with a GNAT
geometric tree search outputs gate sequences that are almost an order of
magnitude smaller for the same level of accuracy. This therefore significantly
reduces the error correction requirements for quantum algorithms on encoded
fault-tolerant hardware.Comment: 9 page
Applications of incidence bounds in point covering problems
In the Line Cover problem a set of n points is given and the task is to cover
the points using either the minimum number of lines or at most k lines. In
Curve Cover, a generalization of Line Cover, the task is to cover the points
using curves with d degrees of freedom. Another generalization is the
Hyperplane Cover problem where points in d-dimensional space are to be covered
by hyperplanes. All these problems have kernels of polynomial size, where the
parameter is the minimum number of lines, curves, or hyperplanes needed. First
we give a non-parameterized algorithm for both problems in O*(2^n) (where the
O*(.) notation hides polynomial factors of n) time and polynomial space,
beating a previous exponential-space result. Combining this with incidence
bounds similar to the famous Szemeredi-Trotter bound, we present a Curve Cover
algorithm with running time O*((Ck/log k)^((d-1)k)), where C is some constant.
Our result improves the previous best times O*((k/1.35)^k) for Line Cover
(where d=2), O*(k^(dk)) for general Curve Cover, as well as a few other bounds
for covering points by parabolas or conics. We also present an algorithm for
Hyperplane Cover in R^3 with running time O*((Ck^2/log^(1/5) k)^k), improving
on the previous time of O*((k^2/1.3)^k).Comment: SoCG 201
Primordial non-Gaussianity and the CMB bispectrum
We present a new formalism, together with efficient numerical methods, to
directly calculate the CMB bispectrum today from a given primordial bispectrum
using the full linear radiation transfer functions. Unlike previous analyses
which have assumed simple separable ansatze for the bispectrum, this work
applies to a primordial bispectrum of almost arbitrary functional form, for
which there may have been both horizon-crossing and superhorizon contributions.
We employ adaptive methods on a hierarchical triangular grid and we establish
their accuracy by direct comparison with an exact analytic solution, valid on
large angular scales. We demonstrate that we can calculate the full CMB
bispectrum to greater than 1% precision out to multipoles l<1800 on reasonable
computational timescales. We plot the bispectrum for both the superhorizon
('local') and horizon-crossing ('equilateral') asymptotic limits, illustrating
its oscillatory nature which is analogous to the CMB power spectrum
Experimental evidence for the Volume Conjecture for the simplest hyperbolic non-2-bridge knot
Loosely speaking, the Volume Conjecture states that the limit of the n-th
colored Jones polynomial of a hyperbolic knot, evaluated at the primitive
complex n-th root of unity is a sequence of complex numbers that grows
exponentially. Moreover, the exponential growth rate is proportional to the
hyperbolic volume of the knot.
We provide an efficient formula for the colored Jones function of the
simplest hyperbolic non-2-bridge knot, and using this formula, we provide
numerical evidence for the Hyperbolic Volume Conjecture for the simplest
hyperbolic non-2-bridge knot.Comment: Published by Algebraic and Geometric Topology at
http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-17.abs.htm
A geometric approach to free variable loop equations in discretized theories of 2D gravity
We present a self-contained analysis of theories of discrete 2D gravity
coupled to matter, using geometric methods to derive equations for generating
functions in terms of free (noncommuting) variables. For the class of discrete
gravity theories which correspond to matrix models, our method is a
generalization of the technique of Schwinger-Dyson equations and is closely
related to recent work describing the master field in terms of noncommuting
variables; the important differences are that we derive a single equation for
the generating function using purely graphical arguments, and that the approach
is applicable to a broader class of theories than those described by matrix
models. Several example applications are given here, including theories of
gravity coupled to a single Ising spin (), multiple Ising spins (), a general class of two-matrix models which includes the Ising theory and
its dual, the three-state Potts model, and a dually weighted graph model which
does not admit a simple description in terms of matrix models.Comment: 40 pages, 8 figures, LaTeX; final publication versio
Uniform Time Average Consistency of Monte Carlo Particle Filters
We prove that bootstrap type Monte Carlo particle filters approximate the
optimal nonlinear filter in a time average sense uniformly with respect to the
time horizon when the signal is ergodic and the particle system satisfies a
tightness property. The latter is satisfied without further assumptions when
the signal state space is compact, as well as in the noncompact setting when
the signal is geometrically ergodic and the observations satisfy additional
regularity assumptions.Comment: 21 pages, 1 figur
How quantizable matter gravitates: a practitioner's guide
We present the practical step-by-step procedure for constructing canonical
gravitational dynamics and kinematics directly from any previously specified
quantizable classical matter dynamics, and then illustrate the application of
this recipe by way of two completely worked case studies. Following the same
procedure, any phenomenological proposal for fundamental matter dynamics must
be supplemented with a suitable gravity theory providing the coefficients and
kinematical interpretation of the matter equations, before any of the two
theories can be meaningfully compared to experimental data.Comment: 45 pages, no figure
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