arXiv subjects

Jakob Lindner

Publications and source records attributed to Jakob Lindner.

4 recordsLinked to original sources

arXiv2026

Universal $x$-$y$ duality in equivariant topological recursion and quartic Kontsevich model

Topological recursion is a universal procedure that generates an infinite family of multi-differentials with different interpretations in various areas of mathematics. Recently, a new understanding of topological recursion has emerged from $x$-$y$ duality, which relates two such families through explicit formulae. We study $x$-$y$ duality, its parametrised version, and symplectic duality in the setting of equivariant topological recursion. We explicitly analyse the pole structure of the resulting dual families and provide recursive formulae for the different types of polar parts. In contrast to classical topological recursion, the dual family has additional poles at fixed points of the group action, and at anti-diagonals determined by the group action. We show that the partition function $Z_P(E)=\int_{H_N} e^{-N\,\mathrm{Tr}(EΦ^2+P(Φ))}dΦ$ of the Kontsevich matrix model with arbitrary potential $P$, where $dΦ$ is the Lebesgue measure on Hermitian $N\times N$ matrices and $E$ a positive matrix, admits a Fock space realisation related to the character expansion of the IZHC integral. The Fock space framework provides two mutually $X$-$Y$ dual families of multi-differentials relative to an implicitly defined spectral curve $(X,Y)$. The solution $Y(X)$ succeeds for the quartic Kontsevich model with potential $P(Φ)=\fracλ{4}Φ^4$. We identify a network of duality transformations and symplectic shifts which relate the quartic Kontsevich model with the $\mathbb{Z}/2\mathbb{Z}$-equivariant spectral curve. The proposed construction yields a conjectural all-genus solution to the quartic Kontsevich model, which reproduces recursive structures previously established for genus $g\leq 1$.

math-ph↗

arXiv2024

Question: How do Large Language Models perform on the Question Answering tasks? Answer:

Large Language Models (LLMs) have been showing promising results for various NLP-tasks without the explicit need to be trained for these tasks by using few-shot or zero-shot prompting techniques. A common NLP-task is question-answering (QA). In this study, we propose a comprehensive performance comparison between smaller fine-tuned models and out-of-the-box instruction-following LLMs on the Stanford Question Answering Dataset 2.0 (SQuAD2), specifically when using a single-inference prompting technique. Since the dataset contains unanswerable questions, previous work used a double inference method. We propose a prompting style which aims to elicit the same ability without the need for double inference, saving compute time and resources. Furthermore, we investigate their generalization capabilities by comparing their performance on similar but different QA datasets, without fine-tuning neither model, emulating real-world uses where the context and questions asked may differ from the original training distribution, for example swapping Wikipedia for news articles. Our results show that smaller, fine-tuned models outperform current State-Of-The-Art (SOTA) LLMs on the fine-tuned task, but recent SOTA models are able to close this gap on the out-of-distribution test and even outperform the fine-tuned models on 3 of the 5 tested QA datasets.

cs.CL↗

arXiv2022

One Model, Any CSP: Graph Neural Networks as Fast Global Search Heuristics for Constraint Satisfaction

We propose a universal Graph Neural Network architecture which can be trained as an end-2-end search heuristic for any Constraint Satisfaction Problem (CSP). Our architecture can be trained unsupervised with policy gradient descent to generate problem specific heuristics for any CSP in a purely data driven manner. The approach is based on a novel graph representation for CSPs that is both generic and compact and enables us to process every possible CSP instance with one GNN, regardless of constraint arity, relations or domain size. Unlike previous RL-based methods, we operate on a global search action space and allow our GNN to modify any number of variables in every step of the stochastic search. This enables our method to properly leverage the inherent parallelism of GNNs. We perform a thorough empirical evaluation where we learn heuristics for well known and important CSPs from random data, including graph coloring, MaxCut, 3-SAT and MAX-k-SAT. Our approach outperforms prior approaches for neural combinatorial optimization by a substantial margin. It can compete with, and even improve upon, conventional search heuristics on test instances that are several orders of magnitude larger and structurally more complex than those seen during training.

cs.AI↗

arXiv2022

Two Approaches For a Perturbative Expansion in Blobbed Topological Recursion

In this paper we continue the perturbative analysis of the quartic Kontsevich model. We investigate meromorphic functions $Ω^{(0)}_m$ with $m=1,2$, that obey blobbed topological recursion. We calculate their expansions and check their equivalence to sums of ribbon graph weights, which are obtained with common methods of perturbation theory in QFT, up to fifth order in the coupling using Mathematica. Furthermore, we provide a catalog of permutation pairs $(α,σ)$, which encode all 5660 vacuum ribbon graphs that contribute to the free energy $\mathcal{F}^{(g)}$ with genus $g\geq 0$ up to fifth order and begin to expand upon the used methods to also consider ribbon graphs of general correlation functions $G_{\dots}$. This is a first step towards automation of the calculation of ribbon graph expansions in the quartic Kontsevich model.

hep-th↗