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Thomas A. Hogan

Publications and source records attributed to Thomas A. Hogan.

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Stochastic Tverberg theorems and their applications in multi-class logistic regression, data separability, and centerpoints of data

We present new stochastic geometry theorems that give bounds on the probability that $m$ random data classes all contain a point in common in their convex hulls. We apply these stochastic separation theorems to obtain bounds on the probability of existence of maximum likelihood estimators in multinomial logistic regression. We also discuss connections to condition numbers for analysis of steepest descent algorithms in logistic regression and to the computation of centerpoints of data clouds.

math.PR

Tverberg theorems over discrete sets of points

This paper discusses Tverberg-type theorems with coordinate constraints (i.e., versions of these theorems where all points lie within a subset $S \subset \mathbb{R}^d$ and the intersection of convex hulls is required to have a non-empty intersection with $S$). We determine the $m$-Tverberg number, when $m \geq 3$, of any discrete subset $S$ of $\mathbb{R}^2$ (a generalization of an unpublished result of J.-P. Doignon). We also present improvements on the upper bounds for the Tverberg numbers of $\mathbb{Z}^3$ and $\mathbb{Z}^j \times \mathbb{R}^k$ and an integer version of the well-known positive-fraction selection lemma of J. Pach.

math.MG

Universal Decision-Based Black-Box Perturbations: Breaking Security-Through-Obscurity Defenses

We study the problem of finding a universal (image-agnostic) perturbation to fool machine learning (ML) classifiers (e.g., neural nets, decision tress) in the hard-label black-box setting. Recent work in adversarial ML in the white-box setting (model parameters are known) has shown that many state-of-the-art image classifiers are vulnerable to universal adversarial perturbations: a fixed human-imperceptible perturbation that, when added to any image, causes it to be misclassified with high probability Kurakin et al. [2016], Szegedy et al. [2013], Chen et al. [2017a], Carlini and Wagner [2017]. This paper considers a more practical and challenging problem of finding such universal perturbations in an obscure (or black-box) setting. More specifically, we use zeroth order optimization algorithms to find such a universal adversarial perturbation when no model information is revealed-except that the attacker can make queries to probe the classifier. We further relax the assumption that the output of a query is continuous valued confidence scores for all the classes and consider the case where the output is a hard-label decision. Surprisingly, we found that even in these extremely obscure regimes, state-of-the-art ML classifiers can be fooled with a very high probability just by adding a single human-imperceptible image perturbation to any natural image. The surprising existence of universal perturbations in a hard-label black-box setting raises serious security concerns with the existence of a universal noise vector that adversaries can possibly exploit to break a classifier on most natural images.

cs.LG

Tverberg-Type Theorems with Trees and Cycles as (Nerve) Intersection Patterns

Tverberg's theorem says that a set with sufficiently many points in $\mathbb{R}^d$ can always be partitioned into $m$ parts so that the $(m-1)$-simplex is the (nerve) intersection pattern of the convex hulls of the parts. The main results of our paper demonstrate that Tverberg's theorem is but a special case of a more general situation. Given sufficiently many points, all trees and cycles can also be induced by at least one partition of a point set.

math.MG