On the Boundary Dieudonné–Pick Lemma

The class of holomorphic self-maps of a disk with a boundary fixed point is studied. For this class of functions, the famous Julia–Carathéodory theorem gives a sharp estimate of the angular derivative at the boundary fixed point in terms of the image of the interior point. In the case when additional information about the value of the derivative at the interior point is known, a sharp estimate of the angular derivative at the boundary fixed point is obtained. As a consequence, the sharpness of the boundary Dieudonné–Pick lemma is established and the class of the extremal functions is identified. An unimprovable strengthening of the Osserman general boundary lemma is also obtained.

Autonomous Toy Drone via Coresets for Pose Estimation

A coreset of a dataset is a small weighted set, such that querying the coreset provably yields a ( 1 + ε )-factor approximation to the original (full) dataset, for a given family of queries. This paper suggests accurate coresets ( ε = 0 ) that are subsets of the input for fundamental optimization problems. These coresets enabled us to implement a “Guardian Angel” system that computes pose-estimation in a rate > 20 frames per second. It tracks a toy quadcopter which guides guests in a supermarket, hospital, mall, airport, and so on. We prove that any set of n matrices in R d × d whose sum is a matrix S of rank r, has a coreset whose sum has the same left and right singular vectors as S, and consists of O ( d r ) = O ( d 2 ) matrices, independent of n. This implies the first (exact, weighted subset) coreset of O ( d 2 ) points to problems such as linear regression, PCA/SVD, and Wahba’s problem, with corresponding streaming, dynamic, and distributed versions. Our main tool is a novel usage of the Caratheodory Theorem for coresets, an algorithm that computes its set in time that is linear in its cardinality. Extensive experimental results on both synthetic and real data, companion video of our system, and open code are provided.

An Inductive Julia-Carathéodory Theorem for Pick Functions in Two Variables

Classically, Nevanlinna showed that functions from the complex upper half plane into itself which satisfy nice asymptotic conditions are parametrized by finite measures on the real line. Furthermore, the higher order asymptotic behaviour at infinity of a map from the complex upper half plane into itself is governed by the existence of moments of its representing measure, which was the key to his solution of the Hamburger moment problem. Agler and McCarthy showed that an analogue of the above correspondence holds between a Pick function f of two variables, an analytic function which maps the product of two upper half planes into the upper half plane, and moment-like quantities arising from an operator theoretic representation for f. We apply their ‘moment’ theory to show that there is a fine hierarchy of levels of regularity at infinity for Pick functions in two variables, given by the Löwner classes and intermediate Löwner classes of order N, which can be exhibited in terms of certain formulae akin to the Julia quotient.

Robust Tverberg and Colourful Carathéodory Results via Random Choice

We use the probabilistic method to obtain versions of the colourful Carathéodory theorem and Tverberg's theorem with tolerance.In particular, we give bounds for the smallest integer N = N(t,d,r) such that for any N points in ℝd, there is a partition of them into r parts for which the following condition holds: after removing any t points from the set, the convex hulls of what is left in each part intersect.We prove a bound N = rt + O($\sqrt{t}$) for fixed r,d which is polynomial in each parameters. Our bounds extend to colourful versions of Tverberg's theorem, as well as Reay-type variations of this theorem.