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Max Horn

Publications and source records attributed to Max Horn.

31 records · Page 2Linked to original sources
arXiv2020

Path Imputation Strategies for Signature Models of Irregular Time Series

The signature transform is a 'universal nonlinearity' on the space of continuous vector-valued paths, and has received attention for use in machine learning on time series. However, real-world temporal data is typically observed at discrete points in time, and must first be transformed into a continuous path before signature techniques can be applied. We make this step explicit by characterising it as an imputation problem, and empirically assess the impact of various imputation strategies when applying signature-based neural nets to irregular time series data. For one of these strategies, Gaussian process (GP) adapters, we propose an extension~(GP-PoM) that makes uncertainty information directly available to the subsequent classifier while at the same time preventing costly Monte-Carlo (MC) sampling. In our experiments, we find that the choice of imputation drastically affects shallow signature models, whereas deeper architectures are more robust. Next, we observe that uncertainty-aware predictions (based on GP-PoM or indicator imputations) are beneficial for predictive performance, even compared to the uncertainty-aware training of conventional GP adapters. In conclusion, we have demonstrated that the path construction is indeed crucial for signature models and that our proposed strategy leads to competitive performance in general, while improving robustness of signature models in particular.

cs.LG↗
arXiv2019

Neural Persistence: A Complexity Measure for Deep Neural Networks Using Algebraic Topology

While many approaches to make neural networks more fathomable have been proposed, they are restricted to interrogating the network with input data. Measures for characterizing and monitoring structural properties, however, have not been developed. In this work, we propose neural persistence, a complexity measure for neural network architectures based on topological data analysis on weighted stratified graphs. To demonstrate the usefulness of our approach, we show that neural persistence reflects best practices developed in the deep learning community such as dropout and batch normalization. Moreover, we derive a neural persistence-based stopping criterion that shortens the training process while achieving comparable accuracies as early stopping based on validation loss.

cs.LG↗
arXiv2019

Kac-Moody symmetric spaces

In the present article we introduce and study a class of topological reflection spaces that we call Kac-Moody symmetric spaces. These generalize Riemannian symmetric spaces of non-compact type. We observe that in a non-spherical Kac-Moody symmetric space there exist pairs of points that do not lie on a common geodesic; however, any two points can be connected by a chain of geodesic segments. We moreover classify maximal flats in Kac-Moody symmetric spaces and study their intersection patterns, leading to a classification of global and local automorphisms. Unlike Riemannian symmetric spaces, non-spherical non-affine irreducible Kac-Moody symmetric spaces also admit an invariant causal structure. For causal and anti-causal geodesic rays with respect to this structure we find a notion of asymptoticity, which allows us to define a future and past boundary of such Kac-Moody symmetric space. We show that these boundaries carry a natural polyhedral structure and are cellularly isomorphic to the halves of the geometric realization of the twin buildings of the underlying split real Kac-Moody group. We also show that every automorphism of the symmetric space is uniquely determined by the induced cellular automorphism of the future and past boundary. The invariant causal structure on a non-spherical non-affine irreducible Kac-Moody symmetric space gives rise to an invariant pre-order on the underlying space, and thus to a subsemigroup of the Kac-Moody group. We conclude that while in some aspects Kac-Moody symmetric spaces closely resemble Riemannian symmetric spaces, in other aspects they behave similarly to ordered affine hovels, their non-Archimedean cousins.

math.GR↗
arXiv2019

Machine learning for early prediction of circulatory failure in the intensive care unit

Intensive care clinicians are presented with large quantities of patient information and measurements from a multitude of monitoring systems. The limited ability of humans to process such complex information hinders physicians to readily recognize and act on early signs of patient deterioration. We used machine learning to develop an early warning system for circulatory failure based on a high-resolution ICU database with 240 patient years of data. This automatic system predicts 90.0% of circulatory failure events (prevalence 3.1%), with 81.8% identified more than two hours in advance, resulting in an area under the receiver operating characteristic curve of 94.0% and area under the precision-recall curve of 63.0%. The model was externally validated in a large independent patient cohort.

cs.LG↗
arXiv2017

Decompositions of Kac-Moody groups

Let $G$ be a split (minimal) Kac-Moody group over $\mathbb{R}$ or $\mathbb{C}$ with maximal torus $T$, and let $θ$ be a Cartan-Chevalley involution of $G$, twisted by complex conjugation, and satisfying that $θ(T)=T$. Furthermore, let $K$ be the subgroup fixed by $θ$, and $τ:G\to G, g\mapsto gθ(g)^{-1}$. Let $A:=τ(T)$. In this note, we show resp. revisit that $G$ admits a (refined) Iwasawa decompositions $G=UAK$. We also show that if $G$ is of non-spherical type, then it never admits a polar decomposition $G=τ(G)K$ nor a Cartan decompositions $G=KAK$. This has implications for the geometrical structure of the Kac-Moody symmetric space $G/K \cong τ(G)$.

math.GR↗
arXiv2015

Spin covers of maximal compact subgroups of Kac-Moody groups and spin-extended Weyl groups

Let G be a split real Kac-Moody group of arbitrary type and let K be its maximal compact subgroup, i.e. the subgroup of elements fixed by a Cartan-Chevalley involution of G. We construct non-trivial spin covers of K, thus confirming a conjecture by Damour and Hillmann (arXiv:0906.3116). For irreducible simply laced diagrams and for all spherical diagrams these spin covers are two-fold central extensions of K. For more complicated irreducible diagrams these spin covers are central extensions by a finite 2-group of possibly larger cardinality. Our construction is amalgam-theoretic and makes use of the generalized spin representations of maximal compact subalgebras of split real Kac-Moody algebras studied in arXiv:1110.5576. Our spin covers contain what we call spin-extended Weyl groups which admit a presentation by generators and relations obtained from the one for extended Weyl groups by relaxing the condition on the generators so that only their eighth powers are required to be trivial.

math.GR↗
arXiv2014

Generalized spin representations. Part 2: Cartan-Bott periodicity for the split real En series

In this article we analyze the quotients of the maximal compact subalgebras of the split real Kac-Moody algebras of the En series resulting from the generalized spin representations introduced in part 1. It turns out that these quotients satisfy a Cartan-Bott periodicity. Our findings are also meaningful in the finite-dimensional cases of A2 + A1, A4, D5, E6, E7, E8, where it turns out that the generalized spin representation is injective. Consequently the observed Cartan-Bott periodicity provides a structural explanation for the seemingly sporadic isomorphism types of the maximal compact Lie subalgebras of the split real Lie algebras of types E6, E7, E8.

math.RA↗
arXiv2013

Second cohomology of Lie rings and the Schur multiplier

We exhibit an explicit construction for the second cohomology group $H^2(L, A)$ for a Lie ring $L$ and a trivial $L$-module $A$. We show how the elements of $H^2(L, A)$ correspond one-to-one to the equivalence classes of central extensions of $L$ by $A$, where $A$ now is considered as an abelian Lie ring. For a finite Lie ring $L$ we also show that $H^2(L, \C^*) \cong M(L)$, where $M(L)$ denotes the Schur multiplier of $L$. These results match precisely the analogue situation in group theory.

math.GR↗
arXiv2013

The construction of finite solvable groups revisited

We describe a new approach towards the systematic construction of finite groups up to isomorphism. This approach yields a practical algorithm for the construction of finite solvable groups up to isomorphism. We report on a GAP implementation of this method for finite solvable groups and exhibit some sample applications.

math.GR↗
arXiv2010

Abstract involutions of algebraic groups and of Kac-Moody groups

Based on the second author's thesis in this article we provide a uniform treatment of abstract involutions of algebraic groups and of Kac-Moody groups using twin buildings, RGD systems, and twisted involutions of Coxeter groups. Notably we simultaneously generalize the double coset decompositions established by Springer and by Helminck-Wang for algebraic groups and by Kac-Wang for certain Kac-Moody groups, we analyze the filtration studied by Devillers-Muhlherr in the context of arbitrary involutions, and we answer a structural question on the combinatorics of involutions of twin buildings raised by Bennett-Gramlich-Hoffman-Shpectorov.

math.GR↗
arXiv2009

Iwasawa decompositions of split Kac-Moody groups

We characterize all fields F for which a group with an F-locally split root group datum admits an Iwasawa decomposition. This class of groups in particular includes the split semisimple algebraic groups and the split Kac-Moody groups.

math.GR↗
arXiv2007

Intransitive geometries and fused amalgams

We study geometries that arise from the natural $G_2(K)$ action on the geometry of one-dimensional subspaces, of nonsingular two-dimensional subspaces, and of nonsingular three-dimensional subspaces of the building geometry of type $C_3(K)$ where $K$ is a perfect field of characteristic 2. One of these geometries is intransitive in such a way that the non-standard geometric covering theory by the first and the last author is not applicable. In this paper we introduce the concept of fused amalgams in order to extend the geometric covering theory so that it applies to that geometry. This yields an interesting new amalgamation result for the group $G_2(K)$.

math.GR↗