arXiv subjects

Johanna Hennig

Publications and source records attributed to Johanna Hennig.

7 recordsLinked to original sources

arXiv2025

Topology of Out-of-Distribution Examples in Deep Neural Networks

As deep neural networks (DNNs) become increasingly common, concerns about their robustness do as well. A longstanding problem for deployed DNNs is their behavior in the face of unfamiliar inputs; specifically, these models tend to be overconfident and incorrect when encountering out-of-distribution (OOD) examples. In this work, we present a topological approach to characterizing OOD examples using latent layer embeddings from DNNs. Our goal is to identify topological features, referred to as landmarks, that indicate OOD examples. We conduct extensive experiments on benchmark datasets and a realistic DNN model, revealing a key insight for OOD detection. Well-trained DNNs have been shown to induce a topological simplification on training data for simple models and datasets; we show that this property holds for realistic, large-scale test and training data, but does not hold for OOD examples. More specifically, we find that the average lifetime (or persistence) of OOD examples is statistically longer than that of training or test examples. This indicates that DNNs struggle to induce topological simplification on unfamiliar inputs. Our empirical results provide novel evidence of topological simplification in realistic DNNs and lay the groundwork for topologically-informed OOD detection strategies.

cs.LG↗

arXiv2019

Translation functors and decomposition numbers for the periplectic Lie superalgebra $\mathfrak{p}(n)$

We study the category $\mathcal{F}_n$ of finite-dimensional integrable representations of the periplectic Lie superalgebra $\mathfrak{p}(n)$. We define an action of the Temperley--Lieb algebra with infinitely many generators and defining parameter $0$ on the category $\mathcal{F}_n$ by translation functors. We also introduce combinatorial tools, called weight diagrams and arrow diagrams for $\mathfrak{p}(n)$ resembling those for $\mathfrak{gl}(m|n)$. Using the Temperley--Lieb algebra action and the combinatorics of weight and arrow diagrams, we then calculate the multiplicities of standard and costandard modules in indecomposable projective modules and classify the blocks of $\mathcal{F}_n$. We also prove that indecomposable projective modules in this category are multiplicity-free.

math.RT↗

arXiv2018

The affine VW supercategory

We define the affine VW supercategory $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$, which arises from studying the action of the periplectic Lie superalgebra $\mathfrak{p}(n)$ on the tensor product $M\otimes V^{\otimes a}$ of an arbitrary representation $M$ with several copies of the vector representation $V$ of $\mathfrak{p}(n)$. It plays a role analogous to that of the degenerate affine Hecke algebras in the context of representations of the general linear group; the main obstacle was the lack of a quadratic Casimir element in $\mathfrak{p}(n)\otimes \mathfrak{p}(n)$. When $M$ is the trivial representation, the action factors through the Brauer supercategory $\mathit{s}\mathcal{B}\mathit{r}$. Our main result is an explicit basis theorem for the morphism spaces of $\mathit{s}\hspace{-0.7mm}\bigvee\mkern-15mu\bigvee$ and, as a consequence, of $\mathit{s}\mathcal{B}\mathit{r}$. The proof utilises the close connection with the representation theory of $\mathfrak{p}(n)$. As an application we explicitly describe the centre of all endomorphism algebras, and show that it behaves well under the passage to the associated graded and under deformation.

math.RT↗

arXiv2013

Simple, locally finite dimensional Lie algebras in positive characteristic

We prove two structure theorems for simple, locally finite dimensional Lie algebras over an algebraically closed field of characteristic $p$ which give sufficient conditions for the algebras to be of the form $[R^{(-)}, R^{(-)}] / (Z(R) \cap [R^{(-)}, R^{(-)}])$ or $[K(R, *), K(R, *)]$ for a simple, locally finite dimensional associative algebra $R$ with involution $*$. The first proves that a condition we introduce, known as locally nondegenerate, along with the existence of an ad-nilpotent element suffice. The second proves that a uniformly ad-integrable Lie algebra is of this type if the characteristic of the ground field is sufficiently large. Lastly we construct a simple, locally finite dimensional associative algebra $R$ with involution $*$ such that $K(R, *) \ne [K(R, *), K(R, *)]$ to demonstrate the necessity of considering the commutator in the first two theorems.

math.RA↗

arXiv2013

A Generalization of Lie's Theorem

We prove that in a locally finite dimensional Lie algebra L, any maximal, locally solvable subalgebra (Borel subalgebra) is the stabilizer of a maximal, generalized flag in an integrable, faithful module over L.

math.RT↗

arXiv2011

Presentations for the higher dimensional Thompson's groups nV

In his papers [2], [3] Brin introduced the higher dimensional Thompson groups nV which are generalizations to the Thompson's group V of self-homeomorphisms of the Cantor set and found a finite set of generators and relations in the case n = 2. We show how to generalize his construction to obtain a finite presentation for every positive integer n. As a corollary, we obtain another proof that the groups nV are simple (first proved by Brin in [4]).

math.GR↗

arXiv2010

The column group and its link invariants

The column group is a subgroup of the symmetric group on the elements of a finite blackboard birack generated by the column permutations in the birack matrix. We use subgroups of the column group associated to birack homomorphisms to define an enhancement of the integral birack counting invariant and give examples which show that the enhanced invariant is stronger than the unenhanced invariant.

math.GT↗