14 research outputs found
A Spectral Bound on Hypergraph Discrepancy
Let be a -regular hypergraph on vertices and edges.
Let be the incidence matrix of and let us denote
. We show that the
discrepancy of is . As a corollary, this
gives us that for every , the discrepancy of a random -regular hypergraph
with vertices and edges is almost surely as
grows. The proof also gives a polynomial time algorithm that takes a hypergraph
as input and outputs a coloring with the above guarantee.Comment: 18 pages. arXiv admin note: substantial text overlap with
arXiv:1811.01491, several changes to the presentatio
Extending the Centerpoint Theorem to Multiple Points
The centerpoint theorem is a well-known and widely used result in discrete geometry. It states that for any point set P of n points in R^d, there is a point c, not necessarily from P, such that each halfspace containing c contains at least n/(d+1) points of P. Such a point c is called a centerpoint, and it can be viewed as a generalization of a median to higher dimensions. In other words, a centerpoint can be interpreted as a good representative for the point set P. But what if we allow more than one representative? For example in one-dimensional data sets, often certain quantiles are chosen as representatives instead of the median.
We present a possible extension of the concept of quantiles to higher dimensions. The idea is to find a set Q of (few) points such that every halfspace that contains one point of Q contains a large fraction of the points of P and every halfspace that contains more of Q contains an even larger fraction of P. This setting is comparable to the well-studied concepts of weak epsilon-nets and weak epsilon-approximations, where it is stronger than the former but weaker than the latter. We show that for any point set of size n in R^d and for any positive alpha_1,...,alpha_k where alpha_1 <= alpha_2 <= ... <= alpha_k and for every i,j with i+j <= k+1 we have that (d-1)alpha_k+alpha_i+alpha_j <= 1, we can find Q of size k such that each halfspace containing j points of Q contains least alpha_j n points of P. For two-dimensional point sets we further show that for every alpha and beta with alpha <= beta and alpha+beta <= 2/3 we can find Q with |Q|=3 such that each halfplane containing one point of Q contains at least alpha n of the points of P and each halfplane containing all of Q contains at least beta n points of P. All these results generalize to the setting where P is any mass distribution. For the case where P is a point set in R^2 and |Q|=2, we provide algorithms to find such points in time O(n log^3 n)
Optimal Approximation of Zonoids and Uniform Approximation by Shallow Neural Networks
We study the following two related problems. The first is to determine to
what error an arbitrary zonoid in can be approximated in the
Hausdorff distance by a sum of line segments. The second is to determine
optimal approximation rates in the uniform norm for shallow ReLU neural
networks on their variation spaces. The first of these problems has been solved
for , but when a logarithmic gap between the best upper and
lower bounds remains. We close this gap, which completes the solution in all
dimensions. For the second problem, our techniques significantly improve upon
existing approximation rates when , and enable uniform approximation
of both the target function and its derivatives
Discrepancy, chaining and subgaussian processes
We show that for a typical coordinate projection of a subgaussian class of
functions, the infimum over signs is asymptotically smaller than the
expectation over signs as a function of the dimension , if the canonical
Gaussian process indexed by is continuous. To that end, we establish a
bound on the discrepancy of an arbitrary subset of using
properties of the canonical Gaussian process the set indexes, and then obtain
quantitative structural information on a typical coordinate projection of a
subgaussian class.Comment: Published in at http://dx.doi.org/10.1214/10-AOP575 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Boosting Simple Learners
Boosting is a celebrated machine learning approach which is based on the idea
of combining weak and moderately inaccurate hypotheses to a strong and accurate
one. We study boosting under the assumption that the weak hypotheses belong to
a class of bounded capacity. This assumption is inspired by the common
convention that weak hypotheses are "rules-of-thumbs" from an "easy-to-learn
class". (Schapire and Freund '12, Shalev-Shwartz and Ben-David '14.) Formally,
we assume the class of weak hypotheses has a bounded VC dimension. We focus on
two main questions: (i) Oracle Complexity: How many weak hypotheses are needed
in order to produce an accurate hypothesis? We design a novel boosting
algorithm and demonstrate that it circumvents a classical lower bound by Freund
and Schapire ('95, '12). Whereas the lower bound shows that
weak hypotheses with -margin are sometimes
necessary, our new method requires only weak
hypothesis, provided that they belong to a class of bounded VC dimension.
Unlike previous boosting algorithms which aggregate the weak hypotheses by
majority votes, the new boosting algorithm uses more complex ("deeper")
aggregation rules. We complement this result by showing that complex
aggregation rules are in fact necessary to circumvent the aforementioned lower
bound. (ii) Expressivity: Which tasks can be learned by boosting weak
hypotheses from a bounded VC class? Can complex concepts that are "far away"
from the class be learned? Towards answering the first question we identify a
combinatorial-geometric parameter which captures the expressivity of
base-classes in boosting. As a corollary we provide an affirmative answer to
the second question for many well-studied classes, including half-spaces and
decision stumps. Along the way, we establish and exploit connections with
Discrepancy Theory.Comment: A minor revision according to STOC review
Recommended from our members
Discrete Geometry
A number of important recent developments in various branches of discrete geometry were presented at the workshop. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics or algebraic geometry. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry