7,561 research outputs found
Constant Factor Approximation for Balanced Cut in the PIE model
We propose and study a new semi-random semi-adversarial model for Balanced
Cut, a planted model with permutation-invariant random edges (PIE). Our model
is much more general than planted models considered previously. Consider a set
of vertices V partitioned into two clusters and of equal size. Let
be an arbitrary graph on with no edges between and . Let
be a set of edges sampled from an arbitrary permutation-invariant
distribution (a distribution that is invariant under permutation of vertices in
and in ). Then we say that is a graph with
permutation-invariant random edges.
We present an approximation algorithm for the Balanced Cut problem that finds
a balanced cut of cost in this model.
In the regime when , this is a
constant factor approximation with respect to the cost of the planted cut.Comment: Full version of the paper at the 46th ACM Symposium on the Theory of
Computing (STOC 2014). 32 page
Correction. Brownian models of open processing networks: canonical representation of workload
Due to a printing error the above mentioned article [Annals of Applied
Probability 10 (2000) 75--103, doi:10.1214/aoap/1019737665] had numerous
equations appearing incorrectly in the print version of this paper. The entire
article follows as it should have appeared. IMS apologizes to the author and
the readers for this error. A recent paper by Harrison and Van Mieghem
explained in general mathematical terms how one forms an ``equivalent workload
formulation'' of a Brownian network model. Denoting by the state vector
of the original Brownian network, one has a lower dimensional state descriptor
in the equivalent workload formulation, where can be chosen as
any basis matrix for a particular linear space. This paper considers Brownian
models for a very general class of open processing networks, and in that
context develops a more extensive interpretation of the equivalent workload
formulation, thus extending earlier work by Laws on alternate routing problems.
A linear program called the static planning problem is introduced to articulate
the notion of ``heavy traffic'' for a general open network, and the dual of
that linear program is used to define a canonical choice of the basis matrix
. To be specific, rows of the canonical are alternative basic optimal
solutions of the dual linear program. If the network data satisfy a natural
monotonicity condition, the canonical matrix is shown to be nonnegative,
and another natural condition is identified which ensures that admits a
factorization related to the notion of resource pooling.Comment: Published at http://dx.doi.org/10.1214/105051606000000583 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Early stopping for statistical inverse problems via truncated SVD estimation
We consider truncated SVD (or spectral cut-off, projection) estimators for a
prototypical statistical inverse problem in dimension . Since calculating
the singular value decomposition (SVD) only for the largest singular values is
much less costly than the full SVD, our aim is to select a data-driven
truncation level only based on the knowledge of
the first singular values and vectors. We analyse in detail
whether sequential {\it early stopping} rules of this type can preserve
statistical optimality. Information-constrained lower bounds and matching upper
bounds for a residual based stopping rule are provided, which give a clear
picture in which situation optimal sequential adaptation is feasible. Finally,
a hybrid two-step approach is proposed which allows for classical oracle
inequalities while considerably reducing numerical complexity.Comment: slightly modified version. arXiv admin note: text overlap with
arXiv:1606.0770
Three Puzzles on Mathematics, Computation, and Games
In this lecture I will talk about three mathematical puzzles involving
mathematics and computation that have preoccupied me over the years. The first
puzzle is to understand the amazing success of the simplex algorithm for linear
programming. The second puzzle is about errors made when votes are counted
during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure
Consistency Thresholds for the Planted Bisection Model
The planted bisection model is a random graph model in which the nodes are
divided into two equal-sized communities and then edges are added randomly in a
way that depends on the community membership. We establish necessary and
sufficient conditions for the asymptotic recoverability of the planted
bisection in this model. When the bisection is asymptotically recoverable, we
give an efficient algorithm that successfully recovers it. We also show that
the planted bisection is recoverable asymptotically if and only if with high
probability every node belongs to the same community as the majority of its
neighbors.
Our algorithm for finding the planted bisection runs in time almost linear in
the number of edges. It has three stages: spectral clustering to compute an
initial guess, a "replica" stage to get almost every vertex correct, and then
some simple local moves to finish the job. An independent work by Abbe,
Bandeira, and Hall establishes similar (slightly weaker) results but only in
the case of logarithmic average degree.Comment: latest version contains an erratum, addressing an error pointed out
by Jan van Waai
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