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Stefan Haller

Publications and source records attributed to Stefan Haller.

At least 19 recordsLinked to original sources
arXiv2025

On the eta invariant of (2,3,5) distributions

We consider the Rumin complex associated with a generic rank two distribution on a closed 5-manifold. The Rumin differential in middle degrees gives rise to a self-adjoint differential operator of Heisenberg order two. We study the eta function and the eta invariant of said operator, twisted by unitary flat vector bundles. For (2,3,5) nilmanifolds this eta invariant vanishes but the eta function is nontrivial, in general. We establish a formula expressing the eta function of (2,3,5) nilmanifolds in terms of more elementary functions.

math.DG↗
arXiv2025

Nonlinear Grassmannians: plain, decorated, augmented

Decorated and augmented nonlinear Grassmannians can be used to parametrize coadjoint orbits of classical diffeomorphism groups. We provide a general framework for decoration and augmentation functors that facilitates the construction of a smooth structure on decorated or augmented nonlinear Grassmannians. This permits to equip the corresponding coadjoint orbits with the structure of a smooth symplectic Frechet manifold. The coadjoint orbits obtained in this way are not new. Here, we provide a uniform description of their smooth structures.

math.DG↗
arXiv2024

A Bregman-Sinkhorn Algorithm for the Maximum Weight Independent Set Problem

We propose a scalable approximate algorithm for the NP-hard maximum-weight independent set problem, based on dual coordinate descent applied to a smoothed clique-cover LP relaxation. Our method, a variant of the Bregman/Sinkhorn algorithm, employs entropy smoothing with a novel duality-gap-based smoothing scheduling strategy that empirically outperforms standard feasibility scheduling for the relaxed problem. Our new projection to the primal feasible set enables accurate duality gap estimation. Combined with a basic primal heuristic leveraging reduced costs, our approach yields high-quality integer solutions. On real-world datasets, it efficiently finds high-quality approximate solutions for graphs with up to 882,000 nodes and 344 million edges within seconds.

math.OC↗
arXiv2024

A dual pair for the group of volume preserving diffeomorphisms

We use cotangent bundles of spaces of smooth embeddings to construct symplectic dual pairs involving the group of volume preserving diffeomorphisms. Via symplectic reduction we obtain descriptions of coadjoint orbits of this group in terms of nonlinear Grassmannians of augmented submanifolds. For codimension one embeddings these submanifolds are further constrained to the leaves of isodrastic foliations with finite codimensions.

math.SG↗
arXiv2023

Regularized determinants of the Rumin complex in irreducible unitary representations of the (2,3,5) nilpotent Lie group

We study the Rumin differentials of the 5-dimensional graded nilpotent Lie group that appears as osculating group of generic rank two distributions in dimension five. In irreducible unitary representations of this group, the Rumin differentials provide intriguing generalizations of the classical quantum harmonic oscillator. For the Schrodinger representations, we compute the spectrum and the zeta regularized determinant of each Rumin differential. In the generic representations, we evaluate their alternating product, i.e., the analytic torsion of the Rumin complex.

math.SP↗
arXiv2023

Weighted nonlinear flag manifolds as coadjoint orbits

A weighted nonlinear flag is a nested set of closed submanifolds, each submanifold endowed with a volume density. We study the geometry of Frechet manifolds of weighted nonlinear flags, in this way generalizing the weighted nonlinear Grassmannians. When the ambient manifold is symplectic, we use these nonlinear flags to describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms, orbits that consist of weighted isotropic nonlinear flags.

math.DG↗
arXiv2022

Graded hypoellipticity of BGG sequences

This article studies hypoellipticity on general filtered manifolds. We extend the Rockland criterion to a pseudodifferential calculus on filtered manifolds, construct a parametrix and describe its precise analytic structure. We use this result to study Rockland sequences, a notion generalizing elliptic sequences to filtered manifolds. The main application that we present is to the analysis of the Bernstein--Gelfand--Gelfand (BGG) sequences over regular parabolic geometries. We do this by generalizing the BGG machinery to more general filtered manifolds (in a non-canonical way) and show that the generalized BGG sequences are Rockland in a graded sense.

math.DG↗
arXiv2022

A Comparative Study of Graph Matching Algorithms in Computer Vision

The graph matching optimization problem is an essential component for many tasks in computer vision, such as bringing two deformable objects in correspondence. Naturally, a wide range of applicable algorithms have been proposed in the last decades. Since a common standard benchmark has not been developed, their performance claims are often hard to verify as evaluation on differing problem instances and criteria make the results incomparable. To address these shortcomings, we present a comparative study of graph matching algorithms. We create a uniform benchmark where we collect and categorize a large set of existing and publicly available computer vision graph matching problems in a common format. At the same time we collect and categorize the most popular open-source implementations of graph matching algorithms. Their performance is evaluated in a way that is in line with the best practices for comparing optimization algorithms. The study is designed to be reproducible and extensible to serve as a valuable resource in the future. Our study provides three notable insights: 1.) popular problem instances are exactly solvable in substantially less than 1 second and, therefore, are insufficient for future empirical evaluations; 2.) the most popular baseline methods are highly inferior to the best available methods; 3.) despite the NP-hardness of the problem, instances coming from vision applications are often solvable in a few seconds even for graphs with more than 500 vertices.

cs.CV↗
arXiv2022

Survey on Automated Short Answer Grading with Deep Learning: from Word Embeddings to Transformers

Automated short answer grading (ASAG) has gained attention in education as a means to scale educational tasks to the growing number of students. Recent progress in Natural Language Processing and Machine Learning has largely influenced the field of ASAG, of which we survey the recent research advancements. We complement previous surveys by providing a comprehensive analysis of recently published methods that deploy deep learning approaches. In particular, we focus our analysis on the transition from hand engineered features to representation learning approaches, which learn representative features for the task at hand automatically from large corpora of data. We structure our analysis of deep learning methods along three categories: word embeddings, sequential models, and attention-based methods. Deep learning impacted ASAG differently than other fields of NLP, as we noticed that the learned representations alone do not contribute to achieve the best results, but they rather show to work in a complementary way with hand-engineered features. The best performance are indeed achieved by methods that combine the carefully hand-engineered features with the power of the semantic descriptions provided by the latest models, like transformers architectures. We identify challenges and provide an outlook on research direction that can be addressed in the future

cs.CL↗
arXiv2021

Fusion Moves for Graph Matching

We contribute to approximate algorithms for the quadratic assignment problem also known as graph matching. Inspired by the success of the fusion moves technique developed for multilabel discrete Markov random fields, we investigate its applicability to graph matching. In particular, we show how fusion moves can be efficiently combined with the dedicated state-of-the-art dual methods that have recently shown superior results in computer vision and bio-imaging applications. As our empirical evaluation on a wide variety of graph matching datasets suggests, fusion moves significantly improve performance of these methods in terms of speed and quality of the obtained solutions. Our method sets a new state-of-the-art with a notable margin with respect to its competitors.

cs.CV↗
arXiv2021

Analytic torsion of generic rank two distributions in dimension five

We propose an analytic torsion for the Rumin complex associated with generic rank two distributions on closed 5-manifolds. This torsion behaves as expected with respect to Poincare duality and finite coverings. We establish anomaly formulas, expressing the dependence on the sub-Riemannian metric and the 2-plane bundle in terms of integrals over local quantities. For certain nilmanifolds we are able to show that this torsion coincides with the Ray-Singer analytic torsion, up to a constant.

math.DG↗
arXiv2020

Exact MAP-Inference by Confining Combinatorial Search with LP Relaxation

We consider the MAP-inference problem for graphical models, which is a valued constraint satisfaction problem defined on real numbers with a natural summation operation. We propose a family of relaxations (different from the famous Sherali-Adams hierarchy), which naturally define lower bounds for its optimum. This family always contains a tight relaxation and we give an algorithm able to find it and therefore, solve the initial non-relaxed NP-hard problem. The relaxations we consider decompose the original problem into two non-overlapping parts: an easy LP-tight part and a difficult one. For the latter part a combinatorial solver must be used. As we show in our experiments, in a number of applications the second, difficult part constitutes only a small fraction of the whole problem. This property allows to significantly reduce the computational time of the combinatorial solver and therefore solve problems which were out of reach before.

cs.CV↗
arXiv2020

A Primal-Dual Solver for Large-Scale Tracking-by-Assignment

We propose a fast approximate solver for the combinatorial problem known as tracking-by-assignment, which we apply to cell tracking. The latter plays a key role in discovery in many life sciences, especially in cell and developmental biology. So far, in the most general setting this problem was addressed by off-the-shelf solvers like Gurobi, whose run time and memory requirements rapidly grow with the size of the input. In contrast, for our method this growth is nearly linear. Our contribution consists of a new (1) decomposable compact representation of the problem; (2) dual block-coordinate ascent method for optimizing the decomposition-based dual; and (3) primal heuristics that reconstructs a feasible integer solution based on the dual information. Compared to solving the problem with Gurobi, we observe an up to~60~times speed-up, while reducing the memory footprint significantly. We demonstrate the efficacy of our method on real-world tracking problems.

cs.CV↗
arXiv2020

Nonlinear flag manifolds as coadjoint orbits

A nonlinear flag is a finite sequence of nested closed submanifolds. We study the geometry of Frechet manifolds of nonlinear flags, in this way generalizing the nonlinear Grassmannians. As an application we describe a class of coadjoint orbits of the group of Hamiltonian diffeomorphisms that consist of nested symplectic submanifolds, i.e., symplectic nonlinear flags.

math.DG↗
arXiv2020

The heat asymptotics on filtered manifolds

The short-time heat kernel expansion of elliptic operators provides a link between local and global features of classical geometries. For many geometric structures related to (non-)involutive distributions, the natural differential operators tend to be Rockland, hence hypoelliptic. In this paper we establish a universal heat kernel expansion for formally selfadjoint non-negative Rockland differential operators on general closed filtered manifolds. The main ingredient is the analysis of parametrices in a recently constructed calculus adapted to these geometric structures. The heat expansion implies that the new calculus, a more general version of the Heisenberg calculus, also has a non-commutative residue. Many of the well known implications of the heat expansion such as, the structure of the complex powers, the heat trace asymptotics, the continuation of the zeta function, as well as Weyl's law for the eigenvalue asymptotics, can be adapted to this calculus. Other consequences include a McKean-Singer type formula for the index of Rockland differential operators. We illustrate some of these results by providing a more explicit description of Weyl's law for Rumin-Seshadri operators associated with curved BGG sequences over 5-manifolds equipped with a rank two distribution of Cartan type.

math.DG↗
arXiv2020

Analytic eigenbranches in the semi-classical limit

We consider a one parameter family of Laplacians on a closed manifold and study the semi-classical limit of its analytically parametrized eigenvalues. Our results are analogous to a theorem for scalar Schrödinger operators on Euclidean space by Luc Hillairet and apply to geometric operators like Witten's Laplacian associated with a Morse function.

math.SP↗
arXiv2019

A dual pair for the contact group

Generalizing the canonical symplectization of contact manifolds, we construct an infinite dimensional non-linear Stiefel manifold of weighted embeddings into a contact manifold. This space carries a symplectic structure such that the contact group and the group of reparametrizations act in a Hamiltonian fashion with equivariant moment maps, respectively, giving rise to a dual pair, called the EPContact dual pair. Via symplectic reduction, this dual pair provides a conceptual identification of non-linear Grassmannians of weighted submanifolds with certain coadjoint orbits of the contact group. Moreover, the EPContact dual pair gives rise to singular solutions for the geodesic equation on the group of contact diffeomorphisms. For the projectivized cotangent bundle, the EPContact dual pair is closely related to the EPDiff dual pair due to Holm and Marsden, and leads to a geometric description of some coadjoint orbits of the full diffeomorphism group.

math.SG↗