1,659 research outputs found
On the Convergence Rate of Decomposable Submodular Function Minimization
Submodular functions describe a variety of discrete problems in machine
learning, signal processing, and computer vision. However, minimizing
submodular functions poses a number of algorithmic challenges. Recent work
introduced an easy-to-use, parallelizable algorithm for minimizing submodular
functions that decompose as the sum of "simple" submodular functions.
Empirically, this algorithm performs extremely well, but no theoretical
analysis was given. In this paper, we show that the algorithm converges
linearly, and we provide upper and lower bounds on the rate of convergence. Our
proof relies on the geometry of submodular polyhedra and draws on results from
spectral graph theory.Comment: 17 pages, 3 figure
Exponentially convergent data assimilation algorithm for Navier-Stokes equations
The paper presents a new state estimation algorithm for a bilinear equation
representing the Fourier- Galerkin (FG) approximation of the Navier-Stokes (NS)
equations on a torus in R2. This state equation is subject to uncertain but
bounded noise in the input (Kolmogorov forcing) and initial conditions, and its
output is incomplete and contains bounded noise. The algorithm designs a
time-dependent gain such that the estimation error converges to zero
exponentially. The sufficient condition for the existence of the gain are
formulated in the form of algebraic Riccati equations. To demonstrate the
results we apply the proposed algorithm to the reconstruction a chaotic fluid
flow from incomplete and noisy data
A unified framework for solving a general class of conditional and robust set-membership estimation problems
In this paper we present a unified framework for solving a general class of
problems arising in the context of set-membership estimation/identification
theory. More precisely, the paper aims at providing an original approach for
the computation of optimal conditional and robust projection estimates in a
nonlinear estimation setting where the operator relating the data and the
parameter to be estimated is assumed to be a generic multivariate polynomial
function and the uncertainties affecting the data are assumed to belong to
semialgebraic sets. By noticing that the computation of both the conditional
and the robust projection optimal estimators requires the solution to min-max
optimization problems that share the same structure, we propose a unified
two-stage approach based on semidefinite-relaxation techniques for solving such
estimation problems. The key idea of the proposed procedure is to recognize
that the optimal functional of the inner optimization problems can be
approximated to any desired precision by a multivariate polynomial function by
suitably exploiting recently proposed results in the field of parametric
optimization. Two simulation examples are reported to show the effectiveness of
the proposed approach.Comment: Accpeted for publication in the IEEE Transactions on Automatic
Control (2014
Matrix Scaling and Balancing via Box Constrained Newton's Method and Interior Point Methods
In this paper, we study matrix scaling and balancing, which are fundamental
problems in scientific computing, with a long line of work on them that dates
back to the 1960s. We provide algorithms for both these problems that, ignoring
logarithmic factors involving the dimension of the input matrix and the size of
its entries, both run in time where is the amount of error we are willing to
tolerate. Here, represents the ratio between the largest and the
smallest entries of the optimal scalings. This implies that our algorithms run
in nearly-linear time whenever is quasi-polynomial, which includes, in
particular, the case of strictly positive matrices. We complement our results
by providing a separate algorithm that uses an interior-point method and runs
in time .
In order to establish these results, we develop a new second-order
optimization framework that enables us to treat both problems in a unified and
principled manner. This framework identifies a certain generalization of linear
system solving that we can use to efficiently minimize a broad class of
functions, which we call second-order robust. We then show that in the context
of the specific functions capturing matrix scaling and balancing, we can
leverage and generalize the work on Laplacian system solving to make the
algorithms obtained via this framework very efficient.Comment: To appear in FOCS 201
Volumetric Spanners: an Efficient Exploration Basis for Learning
Numerous machine learning problems require an exploration basis - a mechanism
to explore the action space. We define a novel geometric notion of exploration
basis with low variance, called volumetric spanners, and give efficient
algorithms to construct such a basis.
We show how efficient volumetric spanners give rise to the first efficient
and optimal regret algorithm for bandit linear optimization over general convex
sets. Previously such results were known only for specific convex sets, or
under special conditions such as the existence of an efficient self-concordant
barrier for the underlying set
- …