Georgia Institute of Technology College of Sciences

The main result in this talk concerns a new fast algorithm to solve a minimal problem with many spurious solutions that arises as a relaxation of a geometric optimization problem. The algorithm recovers relative camera pose from points and lines in multiple views. Solvers like this are the backbone of structure-from-motion techniques that estimate 3D structures from 2D image sequences.  

Our methodology is general and applicable in areas other than computer vision. The ingredients come from algebra, geometry, numerical methods, and applied statistics. Our fast implementation relies on a homotopy continuation optimized for our setting and a machine-learned neural network.

(This covers joint works with Tim Duff, Ricardo Fabbri, Petr Hruby, Kathlen Kohn, Tomas Pajdla, and others.

The talk is suitable for both professors and students.)

The Erdős–Faber–Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$.  In joint work with Dong Yeap Kang, Daniela Kühn, Abhishek Methuku, and Deryk Osthus, we proved this conjecture for every sufficiently large $n$.  In this talk, I will present the history of this conjecture and sketch our proof in a special case.

In the first part of the talk, we discuss the use of a penalized Huber M-estimator for high-dimensional linear regression. We explain how a fairly straightforward analysis yields high-probability error bounds that hold even when the additive errors are heavy-tailed. However, the parameter governing the shape of the Huber loss must be chosen in relation to the scale of the error distribution. We discuss how to use an adaptive technique, based on Lepski's method, to overcome the difficulties traditionally faced by applying Huber M-estimation in a context where both location and scale are unknown.

In the second part of the talk, we turn to a more complicated setting where both the covariates and responses may be heavy-tailed and/or adversarially contaminated. We show how to modify the Huber regression estimator by first applying an appropriate "filtering" procedure to the data based on the covariates. We prove that in low-dimensional settings, this filtered Huber regression estimator achieves near-optimal error rates. We further show that the commonly used least trimmed squares and least absolute deviation estimators may similarly be made robust to contaminated covariates via the same covariate filtering step. This is based on joint work with Ankit Pensia and Varun Jog.