arXiv subjects

Dániel Varga

Publications and source records attributed to Dániel Varga.

17 recordsLinked to original sources

arXiv2026

A unit-distance graph in the plane with independence ratio below 1/4

We prove that there exists a finite unit-distance graph in the plane with independence ratio strictly smaller than 1/4, answering a question of Erdős. Our proof closely follows the framework of Matolcsi, Ruzsa, Varga, and Zsámboki, based on the geometric fractional chromatic number, but adds a carefully chosen two-vertex augmentation that pushes their 27-vertex construction from geometric fractional chromatic number $4$ to a value strictly larger than 4. This disproves their Conjecture 1, and implies that the fractional chromatic number of the plane is strictly larger than 4. The proof can be made fully constructive, but the resulting finite graph has an enormous number of vertices.

math.CO↗

arXiv2026

Improved bounds for the double cap conjecture

In 1974, Witsenhausen asked for the maximum possible density $α_n$ of a measurable subset $A$ of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ such that $A$ contains no pair of orthogonal vectors. For $n=3$, the best known lower bound is $1 - 1/\sqrt{2} = 0.29289\dots$, obtained from the natural "double cap" construction of two opposite spherical caps, which is conjectured to be optimal for all $n$ by Gil Kalai. In this paper, we use a novel approach to establish an upper bound of $α_3\le 0.2953$, improving the previous best known bound $0.2977$ due to Bekker et al. (2025). Our approach combines harmonic-analytic arguments with the geometric fractional chromatic number of finite graphs, recently introduced by Ambrus et al. (2024). In this framework, any finite subset of the sphere yields an upper bound for $α_n$, and we obtain our bound by identifying an appropriate 33-element point set through a large-scale computer search. The same method can also be used in higher dimensions to yield potential improvements of the best known bounds.

math.CO↗

arXiv2026

Cold-Atom Buoy: A Differential Magnetic Sensing Technique in Cold Quadrupole Traps

We present a differential technique for vector magnetic sensing based on a cold-atom cloud in a magnetic quadrupole trap. An external homogeneous magnetic field displaces the trap center in a direction and magnitude proportional to the field. By reversing the quadrupole polarity between experimental shots and comparing the resulting cloud positions, we extract a differential displacement signal that is free from common-mode effects such as gravity and weak magnetic-field inhomogeneities. The signal is directionally proportional to the external field and requires only absorption imaging, without spectroscopic interrogation. Assuming micron-scale position resolution, the technique enables field resolution at the milli-Gauss level. It offers a practical tool for field compensation in magnetically sensitive experimental stages, bridging operational regimes from Earth-level fields to atomic magnetometry. A straightforward extension to full three-dimensional sensing is possible with only a minimal addition to standard cold-atom infrastructure.

quant-ph↗

arXiv2025

The Structure of Relation Decoding Linear Operators in Large Language Models

This paper investigates the structure of linear operators introduced in Hernandez et al. [2023] that decode specific relational facts in transformer language models. We extend their single-relation findings to a collection of relations and systematically chart their organization. We show that such collections of relation decoders can be highly compressed by simple order-3 tensor networks without significant loss in decoding accuracy. To explain this surprising redundancy, we develop a cross-evaluation protocol, in which we apply each linear decoder operator to the subjects of every other relation. Our results reveal that these linear maps do not encode distinct relations, but extract recurring, coarse-grained semantic properties (e.g., country of capital city and country of food are both in the country-of-X property). This property-centric structure clarifies both the operators' compressibility and highlights why they generalize only to new relations that are semantically close. Our findings thus interpret linear relational decoding in transformer language models as primarily property-based, rather than relation-specific.

cs.CL↗

arXiv2025

On the density of Kravitz sets

We show that for a subset $A$ of the cyclic group of prime order $p>3$, if the sumset $A+A-2A$ is not the whole group, then $|A|\le \frac27\,p$. Besides combinatorial arguments, we utilize a general technique involving linear programming.

math.NT↗

arXiv2025

Triplets of Mutually Unbiased Bases

We initiate a systematic study of triplets of mutually unbiased bases (MUBs). We show that in $\mathbb{C}^d$ each MUB-triplet is characterized by a $d\times d\times d$ object that we call a Hadamard cube. We describe the basic properties of Hadamard cubes, and show how an MUB-triplet can be reconstructed from such a cube, up to unitary equivalence. We also present an algebraic identity which is conjectured to hold for all MUB-triplets in dimension 6. If true, it would imply the long-standing conjecture of Zauner that the maximum number of MUBs in dimension 6 is three.

math.CO↗

arXiv2025

Demonstration of strong coupling of a subradiant atom array to a cavity vacuum

By considering linear scattering of laser-driven cold atoms inside an undriven high-finesse optical resonator, we experimentally demonstrate effects unique to a strongly coupled vacuum field. Arranging the atoms in an incommensurate lattice with respect to the radiation wavelength, the Bragg scattering into the cavity can be suppressed by destructive interference: the atomic array is subradiant to the cavity mode under transverse illumination. We show however, that strong collective coupling leads to a drastic modification of the excitation spectrum, as evidenced by well-resolved vacuum Rabi splitting in the intensity of the fluctuations. Furthermore, we demonstrate a significant polarization rotation in the linear scattering off the subradiant array via Raman scattering induced by the strongly coupled vacuum field.

quant-ph↗

arXiv2025

Diverse beam search to find densest-known planar unit distance graphs

This paper addresses the problem of determining the maximum number of edges in a unit distance graph (UDG) of $n$ vertices using computer search. An unsolved problem of Paul Erdős asks the maximum number of edges $u(n)$ a UDG of $n$ vertices can have. Those UDGs that attain $u(n)$ are called "maximally dense." In this paper, we seek to demonstrate a computer algorithm to generate dense UDGs for vertex counts up to at least 100. Via beam search with an added visitation metric, our algorithm finds all known maximally dense UDGs up to isomorphism at the push of a button. In addition, for $15 < n$, where $u(n)$ is unknown, i) the algorithm finds all previously published densest UDGs up to isomorphism for $15 < n \le 30$, and ii) the rate of growth of $u(n)/n$ remains similar for $30 < n$. The code and database of over 60 million UDGs found by our algorithm can be found at https://codeberg.org/zsamboki/dbs-udg.

math.CO↗

arXiv2025

The fractional chromatic number of the plane is at least 4

We prove that the fractional chromatic number $χ_f(\mathbb R^2)$ of the unit distance graph of the Euclidean plane is greater than or equal to $4$. Interestingly, however, we cannot present a finite subgraph $G$ of the plane such that $χ_f(G)\ge 4$. Instead, we utilize the concept of the geometric fractional chromatic number $χ_{gf}(G)$, which was introduced recently in connection with density bounds for 1-avoiding sets. First, as $G$ ranges over finite subgraphs of the plane, we establish that the supremum of $χ_f(G)$ is the same as that of $χ_{gf}(G)$. The proof exploits the amenability of the group of Euclidean transformations in dimension 2 and, as such, we do not know whether the analogous statement holds in higher dimensions. We then present a specific planar unit distance graph $G$ on 27 vertices such that $χ_{gf}(G)=4$, and conclude $χ_f(\mathbb R^2)\ge 4$ as a corollary. As another main result we show that the finitary fractional chromatic number and the Hall ratio of the plane are equal. As a consequence, we conclude that there exist finite unit distance graphs with independence ratio $\frac{1}{4}+\varepsilon$, while we conjecture that the value $\frac{1}{4}$ cannot be reached.

math.CO↗

arXiv2025

Piercing intersecting convex sets

Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $γ>0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $γ|\mathcal A|$ sets in $\mathcal A$ or $γ|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.

math.CO↗

arXiv2023

Mode Combinability: Exploring Convex Combinations of Permutation Aligned Models

We explore element-wise convex combinations of two permutation-aligned neural network parameter vectors $Θ_A$ and $Θ_B$ of size $d$. We conduct extensive experiments by examining various distributions of such model combinations parametrized by elements of the hypercube $[0,1]^{d}$ and its vicinity. Our findings reveal that broad regions of the hypercube form surfaces of low loss values, indicating that the notion of linear mode connectivity extends to a more general phenomenon which we call mode combinability. We also make several novel observations regarding linear mode connectivity and model re-basin. We demonstrate a transitivity property: two models re-based to a common third model are also linear mode connected, and a robustness property: even with significant perturbations of the neuron matchings the resulting combinations continue to form a working model. Moreover, we analyze the functional and weight similarity of model combinations and show that such combinations are non-vacuous in the sense that there are significant functional differences between the resulting models.

cs.LG↗

arXiv2023

The density of planar sets avoiding unit distances

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erdős that the density of any measurable planar set avoiding unit distances cannot exceed $1/4$. Our argument implies the upper bound of $0.2470$.

math.MG↗

arXiv2023

Piercing the chessboard

We consider the minimum number of lines $h_n$ and $p_n$ needed to intersect or pierce, respectively, all the cells of the $n \times n$ chessboard. Determining these values can also be interpreted as a strengthening of the classical plank problem for integer points. Using the symmetric plank theorem of K. Ball, we prove that $h_n = \lceil \frac n 2 \rceil$ for each $n \geq 1$. Studying the piercing problem, we show that $0.7n \leq p_n \leq n-1$ for $n\geq 3$, where the upper bound is conjectured to be sharp. The lower bound is proven by using the linear programming method, whose limitations are also demonstrated.

math.CO↗

arXiv2022

Negative Sampling in Variational Autoencoders

Modern deep artificial neural networks have achieved great success in the domain of computer vision and beyond. However, their application to many real-world tasks is undermined by certain limitations, such as overconfident uncertainty estimates on out-of-distribution data or performance deterioration under data distribution shifts. Several types of deep learning models used for density estimation through probabilistic generative modeling have been shown to fail to detect out-of-distribution samples by assigning higher likelihoods to anomalous data. We investigate this failure mode in Variational Autoencoder models, which are also prone to this, and improve upon the out-of-distribution generalization performance of the model by employing an alternative training scheme utilizing negative samples. We present a fully unsupervised version: when the model is trained in an adversarial manner, the generator's own outputs can be used as negative samples. We demonstrate empirically the effectiveness of the approach in reducing the overconfident likelihood estimates of out-of-distribution inputs on image data.

cs.LG↗

arXiv2021

Similarity and Matching of Neural Network Representations

We employ a toolset -- dubbed Dr. Frankenstein -- to analyse the similarity of representations in deep neural networks. With this toolset, we aim to match the activations on given layers of two trained neural networks by joining them with a stitching layer. We demonstrate that the inner representations emerging in deep convolutional neural networks with the same architecture but different initializations can be matched with a surprisingly high degree of accuracy even with a single, affine stitching layer. We choose the stitching layer from several possible classes of linear transformations and investigate their performance and properties. The task of matching representations is closely related to notions of similarity. Using this toolset, we also provide a novel viewpoint on the current line of research regarding similarity indices of neural network representations: the perspective of the performance on a task.

cs.LG↗

arXiv2020

Visualizing Transfer Learning

We provide visualizations of individual neurons of a deep image recognition network during the temporal process of transfer learning. These visualizations qualitatively demonstrate various novel properties of the transfer learning process regarding the speed and characteristics of adaptation, neuron reuse, spatial scale of the represented image features, and behavior of transfer learning to small data. We publish the large-scale dataset that we have created for the purposes of this analysis.

cs.CV↗

arXiv2018

Gradient Regularization Improves Accuracy of Discriminative Models

Regularizing the gradient norm of the output of a neural network with respect to its inputs is a powerful technique, rediscovered several times. This paper presents evidence that gradient regularization can consistently improve classification accuracy on vision tasks, using modern deep neural networks, especially when the amount of training data is small. We introduce our regularizers as members of a broader class of Jacobian-based regularizers. We demonstrate empirically on real and synthetic data that the learning process leads to gradients controlled beyond the training points, and results in solutions that generalize well.

cs.LG↗