652 research outputs found
Nonnegative k-sums, fractional covers, and probability of small deviations
More than twenty years ago, Manickam, Mikl\'{o}s, and Singhi conjectured that
for any integers satisfying , every set of real numbers
with nonnegative sum has at least -element subsets whose
sum is also nonnegative. In this paper we discuss the connection of this
problem with matchings and fractional covers of hypergraphs, and with the
question of estimating the probability that the sum of nonnegative independent
random variables exceeds its expectation by a given amount. Using these
connections together with some probabilistic techniques, we verify the
conjecture for . This substantially improves the best previously
known exponential lower bound . In addition we prove
a tight stability result showing that for every and all sufficiently large
, every set of reals with a nonnegative sum that does not contain a
member whose sum with any other members is nonnegative, contains at least
subsets of cardinality with
nonnegative sum.Comment: 15 pages, a section of Hilton-Milner type result adde
Large matchings in uniform hypergraphs and the conjectures of Erdos and Samuels
In this paper we study conditions which guarantee the existence of perfect
matchings and perfect fractional matchings in uniform hypergraphs. We reduce
this problem to an old conjecture by Erd\H{o}s on estimating the maximum number
of edges in a hypergraph when the (fractional) matching number is given, which
we are able to solve in some special cases using probabilistic techniques.
Based on these results, we obtain some general theorems on the minimum
-degree ensuring the existence of perfect (fractional) matchings. In
particular, we asymptotically determine the minimum vertex degree which
guarantees a perfect matching in 4-uniform and 5-uniform hypergraphs. We also
discuss an application to a problem of finding an optimal data allocation in a
distributed storage system
Rate of convergence and Edgeworth-type expansion in the entropic central limit theorem
An Edgeworth-type expansion is established for the entropy distance to the
class of normal distributions of sums of i.i.d. random variables or vectors,
satisfying minimal moment conditions.Comment: Published in at http://dx.doi.org/10.1214/12-AOP780 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Chromatic PAC-Bayes Bounds for Non-IID Data: Applications to Ranking and Stationary -Mixing Processes
Pac-Bayes bounds are among the most accurate generalization bounds for
classifiers learned from independently and identically distributed (IID) data,
and it is particularly so for margin classifiers: there have been recent
contributions showing how practical these bounds can be either to perform model
selection (Ambroladze et al., 2007) or even to directly guide the learning of
linear classifiers (Germain et al., 2009). However, there are many practical
situations where the training data show some dependencies and where the
traditional IID assumption does not hold. Stating generalization bounds for
such frameworks is therefore of the utmost interest, both from theoretical and
practical standpoints. In this work, we propose the first - to the best of our
knowledge - Pac-Bayes generalization bounds for classifiers trained on data
exhibiting interdependencies. The approach undertaken to establish our results
is based on the decomposition of a so-called dependency graph that encodes the
dependencies within the data, in sets of independent data, thanks to graph
fractional covers. Our bounds are very general, since being able to find an
upper bound on the fractional chromatic number of the dependency graph is
sufficient to get new Pac-Bayes bounds for specific settings. We show how our
results can be used to derive bounds for ranking statistics (such as Auc) and
classifiers trained on data distributed according to a stationary {\ss}-mixing
process. In the way, we show how our approach seemlessly allows us to deal with
U-processes. As a side note, we also provide a Pac-Bayes generalization bound
for classifiers learned on data from stationary -mixing distributions.Comment: Long version of the AISTATS 09 paper:
http://jmlr.csail.mit.edu/proceedings/papers/v5/ralaivola09a/ralaivola09a.pd
Maximizing the number of nonnegative subsets
Given a set of real numbers, if the sum of elements of every subset of
size larger than is negative, what is the maximum number of subsets of
nonnegative sum? In this note we show that the answer is , settling a problem of Tsukerman.
We provide two proofs, the first establishes and applies a weighted version of
Hall's Theorem and the second is based on an extension of the nonuniform
Erd\H{o}s-Ko-Rado Theorem
A Note on the Manickam-Mikl\'os-Singhi Conjecture for Vector Spaces
Let be an -dimensional vector space over a finite field
. Define a real-valued weight function on the -dimensional
vector spaces of such that the sum of all weights is zero. Let the weight
of a subspace be the sum of the weights of the -dimensional subspaces
contained in . In 1988 Manickam and Singhi conjectured that if ,
then the number of -dimensional subspaces with nonnegative weight is at
least the number of -dimensional subspaces on a fixed -dimensional
subspace.
Recently, Chowdhury, Huang, Sarkis, Shahriari, and Sudakov proved the
conjecture of Manickam and Singhi for . We modify the technique used
by Chowdhury et al. to prove the conjecture for if is large.
Furthermore, if equality holds and , then the set of
-dimensional subspaces with nonnegative weight is the set of all
-dimensional subspaces on a fixed -dimensional subspace.Comment: 15 pages; this version fixes typos and some minor mistakes, also some
proofs got a bit more explicit for an easier understandin
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