arXiv subjects

Esther Ezra

Publications and source records attributed to Esther Ezra.

21 records · Page 2Linked to original sources

arXiv2012

Convex Hull of Points Lying on Lines in o(n log n) Time after Preprocessing

Motivated by the desire to cope with data imprecision, we study methods for taking advantage of preliminary information about point sets in order to speed up the computation of certain structures associated with them. In particular, we study the following problem: given a set L of n lines in the plane, we wish to preprocess L such that later, upon receiving a set P of n points, each of which lies on a distinct line of L, we can construct the convex hull of P efficiently. We show that in quadratic time and space it is possible to construct a data structure on L that enables us to compute the convex hull of any such point set P in O(n alpha(n) log* n) expected time. If we further assume that the points are "oblivious" with respect to the data structure, the running time improves to O(n alpha(n)). The analysis applies almost verbatim when L is a set of line-segments, and yields similar asymptotic bounds. We present several extensions, including a trade-off between space and query time and an output-sensitive algorithm. We also study the "dual problem" where we show how to efficiently compute the (<= k)-level of n lines in the plane, each of which lies on a distinct point (given in advance). We complement our results by Omega(n log n) lower bounds under the algebraic computation tree model for several related problems, including sorting a set of points (according to, say, their x-order), each of which lies on a given line known in advance. Therefore, the convex hull problem under our setting is easier than sorting, contrary to the "standard" convex hull and sorting problems, in which the two problems require Theta(n log n) steps in the worst case (under the algebraic computation tree model).

cs.CG↗

arXiv2012

Small-Size Relative (p,Epsilon)-Approximations for Well-Behaved Range Spaces

We present improved upper bounds for the size of relative (p,Epsilon)-approximation for range spaces with the following property: For any (finite) range space projected onto (that is, restricted to) a ground set of size n and for any parameter 1 <= k <= n, the number of ranges of size at most k is only nearly-linear in n and polynomial in k. Such range spaces are called "well behaved". Our bound is an improvement over the bound O(\log{(1/p)/\eps^2 p) introduced by Li etal. for the general case (where this bound has been shown to be tight in the worst case), when p << Epsilon. We also show that such small size relative (p,Epsilon)-approximations can be constructed in expected polynomial time. Our bound also has an interesting interpretation in the context of "p-nets": As observed by Har-Peled and Sharir, p-nets are special cases of relative (p,Epsilon)-approximations. Specifically, when Epsilon is a constant smaller than 1, their analysis implies that there are p-nets of size O(\log{(1/p)}/p) that are \emph{also} relative approximations. In this context our construction significantly improves this bound for well-behaved range spaces. Despite the progress in the theory of p-nets and the existence of improved bounds corresponding to the cases that we study, these bounds do not necessarily guarantee a bounded relative error. Lastly, we present several geometric scenarios of well-behaved range spaces, and show the resulting bound for each of these cases obtained as a consequence of our analysis. In particular, when Epsilon is a constant smaller than 1, our bound for points and axis-parallel boxes in two and three dimensions, as well as points and "fat" triangles in the plane, matches the optimal bound for p-nets.

cs.CG↗

arXiv2012

Active Learning Using Smooth Relative Regret Approximations with Applications

The disagreement coefficient of Hanneke has become a central data independent invariant in proving active learning rates. It has been shown in various ways that a concept class with low complexity together with a bound on the disagreement coefficient at an optimal solution allows active learning rates that are superior to passive learning ones. We present a different tool for pool based active learning which follows from the existence of a certain uniform version of low disagreement coefficient, but is not equivalent to it. In fact, we present two fundamental active learning problems of significant interest for which our approach allows nontrivial active learning bounds. However, any general purpose method relying on the disagreement coefficient bounds only fails to guarantee any useful bounds for these problems. The tool we use is based on the learner's ability to compute an estimator of the difference between the loss of any hypotheses and some fixed "pivotal" hypothesis to within an absolute error of at most $\eps$ times the

cs.LG↗