15,563 research outputs found
Combinatorial limitations of average-radius list-decoding
We study certain combinatorial aspects of list-decoding, motivated by the
exponential gap between the known upper bound (of ) and lower
bound (of ) for the list-size needed to decode up to
radius with rate away from capacity, i.e., 1-\h(p)-\gamma (here
and ). Our main result is the following:
We prove that in any binary code of rate
1-\h(p)-\gamma, there must exist a set of
codewords such that the average distance of the
points in from their centroid is at most . In other words,
there must exist codewords with low "average
radius." The standard notion of list-decoding corresponds to working with the
maximum distance of a collection of codewords from a center instead of average
distance. The average-radius form is in itself quite natural and is implied by
the classical Johnson bound.
The remaining results concern the standard notion of list-decoding, and help
clarify the combinatorial landscape of list-decoding:
1. We give a short simple proof, over all fixed alphabets, of the
above-mentioned lower bound. Earlier, this bound
followed from a complicated, more general result of Blinovsky.
2. We show that one {\em cannot} improve the
lower bound via techniques based on identifying the zero-rate regime for list
decoding of constant-weight codes.
3. We show a "reverse connection" showing that constant-weight codes for list
decoding imply general codes for list decoding with higher rate.
4. We give simple second moment based proofs of tight (up to constant
factors) lower bounds on the list-size needed for list decoding random codes
and random linear codes from errors as well as erasures.Comment: 28 pages. Extended abstract in RANDOM 201
Center-based Clustering under Perturbation Stability
Clustering under most popular objective functions is NP-hard, even to
approximate well, and so unlikely to be efficiently solvable in the worst case.
Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at
bypassing this computational barrier by using properties of instances one might
hope to hold in practice. In particular, they argue that instances in practice
should be stable to small perturbations in the metric space and give an
efficient algorithm for clustering instances of the Max-Cut problem that are
stable to perturbations of size . In addition, they conjecture that
instances stable to as little as O(1) perturbations should be solvable in
polynomial time. In this paper we prove that this conjecture is true for any
center-based clustering objective (such as -median, -means, and
-center). Specifically, we show we can efficiently find the optimal
clustering assuming only stability to factor-3 perturbations of the underlying
metric in spaces without Steiner points, and stability to factor
perturbations for general metrics. In particular, we show for such instances
that the popular Single-Linkage algorithm combined with dynamic programming
will find the optimal clustering. We also present NP-hardness results under a
weaker but related condition
Thresholds for Extreme Orientability
Multiple-choice load balancing has been a topic of intense study since the
seminal paper of Azar, Broder, Karlin, and Upfal. Questions in this area can be
phrased in terms of orientations of a graph, or more generally a k-uniform
random hypergraph. A (d,b)-orientation is an assignment of each edge to d of
its vertices, such that no vertex has more than b edges assigned to it.
Conditions for the existence of such orientations have been completely
documented except for the "extreme" case of (k-1,1)-orientations. We consider
this remaining case, and establish:
- The density threshold below which an orientation exists with high
probability, and above which it does not exist with high probability.
- An algorithm for finding an orientation that runs in linear time with high
probability, with explicit polynomial bounds on the failure probability.
Previously, the only known algorithms for constructing (k-1,1)-orientations
worked for k<=3, and were only shown to have expected linear running time.Comment: Corrected description of relationship to the work of LeLarg
Interactive Channel Capacity Revisited
We provide the first capacity approaching coding schemes that robustly
simulate any interactive protocol over an adversarial channel that corrupts any
fraction of the transmitted symbols. Our coding schemes achieve a
communication rate of over any
adversarial channel. This can be improved to for
random, oblivious, and computationally bounded channels, or if parties have
shared randomness unknown to the channel.
Surprisingly, these rates exceed the interactive channel capacity bound
which [Kol and Raz; STOC'13] recently proved for random errors. We conjecture
and to be the optimal rates for their respective settings
and therefore to capture the interactive channel capacity for random and
adversarial errors.
In addition to being very communication efficient, our randomized coding
schemes have multiple other advantages. They are computationally efficient,
extremely natural, and significantly simpler than prior (non-capacity
approaching) schemes. In particular, our protocols do not employ any coding but
allow the original protocol to be performed as-is, interspersed only by short
exchanges of hash values. When hash values do not match, the parties backtrack.
Our approach is, as we feel, by far the simplest and most natural explanation
for why and how robust interactive communication in a noisy environment is
possible
Probing Quarkonium Production Mechanisms with Jet Substructure
We use fragmenting jet functions (FJFs) in the context of quarkonia to study
the production channels predicted by NRQCD (3S_1^(1), 3S_1^(8), 1S_0^(8),
3P_J^(8)). We choose a set of FJFs that give the probability to find a
quarkonium with a given momentum fraction inside a cone-algorithm jet with
fixed cone size and energy. This observable gives several lever arms that allow
one to distinguish different production channels. In particular, we show that
at fixed momentum fraction the individual production mechanisms have distinct
behaviors as a function of the the jet energy. As a consequence of this fact,
we arrive at the robust prediction that if the depolarizing 1S_0^(8) matrix
element dominates, then the gluon FJF will diminish with increasing energy for
fixed momentum fraction, z, and z > 0.5.Comment: 13 pages, 6 figures; v2: Typos fixed, figures updated and new figure
added to reflect nontrivial error correlation in long-distance matrix element
determination which leads to stronger prediction for our observables; v3:
Operational discussion of fragmenting jet function expanded and figure typo
fixe
Multiple Gamma Function and Its Application to Computation of Series
The multiple gamma function , defined by a recurrence-functional
equation as a generalization of the Euler gamma function, was originally
introduced by Kinkelin, Glaisher, and Barnes around 1900. Today, due to the
pioneer work of Conrey, Katz and Sarnak, interest in the multiple gamma
function has been revived. This paper discusses some theoretical aspects of the
function and their applications to summation of series and infinite
products.Comment: 20 pages, Latex, uses kluwer.cls, will appear in The Ramanujan
Journa
Differentially Private Data Analysis of Social Networks via Restricted Sensitivity
We introduce the notion of restricted sensitivity as an alternative to global
and smooth sensitivity to improve accuracy in differentially private data
analysis. The definition of restricted sensitivity is similar to that of global
sensitivity except that instead of quantifying over all possible datasets, we
take advantage of any beliefs about the dataset that a querier may have, to
quantify over a restricted class of datasets. Specifically, given a query f and
a hypothesis H about the structure of a dataset D, we show generically how to
transform f into a new query f_H whose global sensitivity (over all datasets
including those that do not satisfy H) matches the restricted sensitivity of
the query f. Moreover, if the belief of the querier is correct (i.e., D is in
H) then f_H(D) = f(D). If the belief is incorrect, then f_H(D) may be
inaccurate.
We demonstrate the usefulness of this notion by considering the task of
answering queries regarding social-networks, which we model as a combination of
a graph and a labeling of its vertices. In particular, while our generic
procedure is computationally inefficient, for the specific definition of H as
graphs of bounded degree, we exhibit efficient ways of constructing f_H using
different projection-based techniques. We then analyze two important query
classes: subgraph counting queries (e.g., number of triangles) and local
profile queries (e.g., number of people who know a spy and a computer-scientist
who know each other). We demonstrate that the restricted sensitivity of such
queries can be significantly lower than their smooth sensitivity. Thus, using
restricted sensitivity we can maintain privacy whether or not D is in H, while
providing more accurate results in the event that H holds true
Bicriteria Network Design Problems
We study a general class of bicriteria network design problems. A generic
problem in this class is as follows: Given an undirected graph and two
minimization objectives (under different cost functions), with a budget
specified on the first, find a <subgraph \from a given subgraph-class that
minimizes the second objective subject to the budget on the first. We consider
three different criteria - the total edge cost, the diameter and the maximum
degree of the network. Here, we present the first polynomial-time approximation
algorithms for a large class of bicriteria network design problems for the
above mentioned criteria. The following general types of results are presented.
First, we develop a framework for bicriteria problems and their
approximations. Second, when the two criteria are the same %(note that the cost
functions continue to be different) we present a ``black box'' parametric
search technique. This black box takes in as input an (approximation) algorithm
for the unicriterion situation and generates an approximation algorithm for the
bicriteria case with only a constant factor loss in the performance guarantee.
Third, when the two criteria are the diameter and the total edge costs we use a
cluster-based approach to devise a approximation algorithms --- the solutions
output violate both the criteria by a logarithmic factor. Finally, for the
class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms
for a number of bicriteria problems using dynamic programming. We show how
these pseudopolynomial-time algorithms can be converted to fully
polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur
Weak lensing calibration of mass bias in the REFLEX+BCS X-ray galaxy cluster catalogue
The use of large, X-ray selected galaxy cluster catalogues for cosmological
analyses requires a thorough understanding of the X-ray mass estimates. Weak
gravitational lensing is an ideal method to shed light on such issues, due to
its insensitivity to the cluster dynamical state. We perform a weak lensing
calibration of 166 galaxy clusters from the REFLEX and BCS cluster catalogue
and compare our results to the X-ray masses based on scaled luminosities from
that catalogue. To interpret the weak lensing signal in terms of cluster
masses, we compare the lensing signal to simple theoretical Navarro-Frenk-White
models and to simulated cluster lensing profiles, including complications such
as cluster substructure, projected large-scale structure, and Eddington bias.
We find evidence of underestimation in the X-ray masses, as expected, with
stat. sys. for our best-fit model. The biases in cosmological parameters in a
typical cluster abundance measurement that ignores this mass bias will
typically exceed the statistical errors.Comment: 13 pages, 5 figures. Revised to address referee comment
Ramsey games with giants
The classical result in the theory of random graphs, proved by Erdos and
Renyi in 1960, concerns the threshold for the appearance of the giant component
in the random graph process. We consider a variant of this problem, with a
Ramsey flavor. Now, each random edge that arrives in the sequence of rounds
must be colored with one of R colors. The goal can be either to create a giant
component in every color class, or alternatively, to avoid it in every color.
One can analyze the offline or online setting for this problem. In this paper,
we consider all these variants and provide nontrivial upper and lower bounds;
in certain cases (like online avoidance) the obtained bounds are asymptotically
tight.Comment: 29 pages; minor revision
- …