11,496 research outputs found

### Riesz bases of exponentials on multiband spectra

Let $S$ be the union of finitely many disjoint intervals on the real line.
Suppose that there are two real numbers $\alpha, \beta$ such that the length of
each interval belongs to $Z \alpha + Z \beta$. We use quasicrystals to
construct a discrete set of real frequencies such that the corresponding system
of exponentials is a Riesz basis in the space $L^2(S)$.Comment: 5 pages, to appear in Proceedings of the American Mathematical
Societ

### 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 $V$ of $n$
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?$for two elements$u,v\in V$. We
assume possible non-transitivity paradoxes which may arise naturally due to
human mistakes or irrationality. The goal is to linearly order the elements
from the most preferred to the least preferred, while disagreeing with as few
pairwise preference labels as possible. Our performance is measured by two
parameters: The loss and the query complexity (number of pairwise preference
labels we obtain). This is a typical learning problem, with the exception that
the space from which the pairwise preferences is drawn is finite, consisting of${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

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