5,551 research outputs found
Shattering Thresholds for Random Systems of Sets, Words, and Permutations
This paper considers a problem that relates to the theories of covering
arrays, permutation patterns, Vapnik-Chervonenkis (VC) classes, and probability
thresholds. Specifically, we want to find the number of subsets of
[n]:={1,2,....,n} we need to randomly select, in a certain probability space,
so as to respectively "shatter" all t-subsets of [n]. Moving from subsets to
words, we ask for the number of n-letter words on a q-letter alphabet that are
needed to shatter all t-subwords of the q^n words of length n. Finally, we
explore the number of random permutations of [n] needed to shatter
(specializing to t=3), all length 3 permutation patterns in specified
positions. We uncover a very sharp zero-one probability threshold for the
emergence of such shattering; Talagrand's isoperimetric inequality in product
spaces is used as a key tool.Comment: 25 page
Set Systems and Families of Permutations with Small Traces
We study the maximum size of a set system on elements whose trace on any
elements has size at most . We show that if for some the
shatter function of a set system satisfies then ; this generalizes Sauer's Lemma on the size of
set systems with bounded VC-dimension. We use this bound to delineate the main
growth rates for the same problem on families of permutations, where the trace
corresponds to the inclusion for permutations. This is related to a question of
Raz on families of permutations with bounded VC-dimension that generalizes the
Stanley-Wilf conjecture on permutations with excluded patterns
Tight bounds on the maximum size of a set of permutations with bounded VC-dimension
The VC-dimension of a family P of n-permutations is the largest integer k
such that the set of restrictions of the permutations in P on some k-tuple of
positions is the set of all k! permutation patterns. Let r_k(n) be the maximum
size of a set of n-permutations with VC-dimension k. Raz showed that r_2(n)
grows exponentially in n. We show that r_3(n)=2^Theta(n log(alpha(n))) and for
every s >= 4, we have almost tight upper and lower bounds of the form 2^{n
poly(alpha(n))}. We also study the maximum number p_k(n) of 1-entries in an n x
n (0,1)-matrix with no (k+1)-tuple of columns containing all (k+1)-permutation
matrices. We determine that p_3(n) = Theta(n alpha(n)) and that p_s(n) can be
bounded by functions of the form n 2^poly(alpha(n)) for every fixed s >= 4. We
also show that for every positive s there is a slowly growing function
zeta_s(m) (of the form 2^poly(alpha(m)) for every fixed s >= 5) satisfying the
following. For all positive integers n and B and every n x n (0,1)-matrix M
with zeta_s(n)Bn 1-entries, the rows of M can be partitioned into s intervals
so that at least B columns contain at least B 1-entries in each of the
intervals.Comment: 22 pages, 4 figures, correction of the bound on r_3 in the abstract
and other minor change
An Active Learning Algorithm for Ranking from Pairwise Preferences with an Almost Optimal Query Complexity
We study the problem of learning to rank from pairwise preferences, and solve
a long-standing open problem that has led to development of many heuristics but
no provable results for our particular problem. Given a set of
elements, we wish to linearly order them given pairwise preference labels. A
pairwise preference label is obtained as a response, typically from a human, to
the question "which if preferred, u or v?u,v\in V{n\choose 2}$ possibilities only. We present an active learning algorithm for
this problem, with query bounds significantly beating general (non active)
bounds for the same error guarantee, while almost achieving the information
theoretical lower bound. Our main construct is a decomposition of the input
s.t. (i) each block incurs high loss at optimum, and (ii) the optimal solution
respecting the decomposition is not much worse than the true opt. The
decomposition is done by adapting a recent result by Kenyon and Schudy for a
related combinatorial optimization problem to the query efficient setting. We
thus settle an open problem posed by learning-to-rank theoreticians and
practitioners: What is a provably correct way to sample preference labels? To
further show the power and practicality of our solution, we show how to use it
in concert with an SVM relaxation.Comment: Fixed a tiny error in theorem 3.1 statemen
Class Proportion Estimation with Application to Multiclass Anomaly Rejection
This work addresses two classification problems that fall under the heading
of domain adaptation, wherein the distributions of training and testing
examples differ. The first problem studied is that of class proportion
estimation, which is the problem of estimating the class proportions in an
unlabeled testing data set given labeled examples of each class. Compared to
previous work on this problem, our approach has the novel feature that it does
not require labeled training data from one of the classes. This property allows
us to address the second domain adaptation problem, namely, multiclass anomaly
rejection. Here, the goal is to design a classifier that has the option of
assigning a "reject" label, indicating that the instance did not arise from a
class present in the training data. We establish consistent learning strategies
for both of these domain adaptation problems, which to our knowledge are the
first of their kind. We also implement the class proportion estimation
technique and demonstrate its performance on several benchmark data sets.Comment: Accepted to AISTATS 2014. 15 pages. 2 figure
Unification and Logarithmic Space
We present an algebraic characterization of the complexity classes Logspace
and NLogspace, using an algebra with a composition law based on unification.
This new bridge between unification and complexity classes is inspired from
proof theory and more specifically linear logic and Geometry of Interaction.
We show how unification can be used to build a model of computation by means
of specific subalgebras associated to finite permutations groups. We then prove
that whether an observation (the algebraic counterpart of a program) accepts a
word can be decided within logarithmic space. We also show that the
construction can naturally represent pointer machines, an intuitive way of
understanding logarithmic space computing
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