arXiv subjects

Balázs Szegedy

Publications and source records attributed to Balázs Szegedy.

At least 19 recordsLinked to original sources

arXiv2026

An inverse theorem for all finite abelian groups via nilmanifolds

We prove a first inverse theorem for Gowers norms on all finite abelian groups that uses only nilmanifolds (rather than possibly more general nilspaces). This makes progress toward confirming the Jamneshan--Tao conjecture. The correlating function in our theorem is a projected nilsequence, obtained as the fiber-wise average of a nilsequence defined on a boundedly-larger abelian group extending the original abelian group. This result is tight in the following sense: we prove also that $k$-step projected nilsequences of bounded complexity are genuine obstructions to having small Gowers $U^{k+1}$-norm. This inverse theorem relies on a new result concerning compact finite-rank (CFR) nilspaces, which is the main contribution in this paper: every $k$-step CFR nilspace is a factor of a $k$-step nilmanifold. This new connection between the classical theory of nilmanifolds and the more recent theory of nilspaces has applications beyond arithmetic combinatorics. We illustrate this with an application in topological dynamics, by proving the following result making progress on a question of Jamneshan, Shalom and Tao: every minimal $\mathbb{Z}^ω$-system of order $k$ is a factor of an inverse limit of $\mathbb{Z}^ω$-polynomial orbit systems of order $k$, these being natural generalizations of nilsystems alternative to translational systems.

math.DS↗

arXiv2026

The Jamneshan-Tao conjecture for finite abelian groups of bounded rank

We confirm the Jamneshan-Tao conjecture for finite abelian groups of rank at most a fixed integer $R$ (i.e. finite abelian groups generated by at most $R$ elements), by proving an inverse theorem for 1-bounded functions of non-trivial Gowers norm on such groups, concluding that such a function must correlate non-trivially with a nilsequence of bounded complexity.

math.GR↗

arXiv2025

Free nilspaces, double-coset nilspaces, and Gowers norms

Compact finite-rank nilspaces have become central in the nilspace approach to higher-order Fourier analysis, notably through their role in a general form of the inverse theorem for the Gowers norms. This paper studies these nilspaces per se, and in connection with further refinements of this inverse theorem that have been conjectured recently. Our first main result states that every compact finite-rank nilspace is obtained by taking a free nilspace (a nilspace based on an abelian group of the form $\mathbb{Z}^{r}\times \mathbb{R}^s$) and quotienting this by a discrete group action of a specific type, describable in terms of polynomials. We call these group actions "higher-order lattice actions", as they generalize actions of lattices in $\mathbb{Z}^r\times \mathbb{R}^s$. The second main result (which relies on the first one) represents every compact finite-rank nilspace as a double-coset space $K\backslash G / Γ$ where $G$ is a nilpotent Lie group of a specific kind. Our third main result extends the aforementioned results to $k$-step compact nilspaces (not necessarily of finite rank), by representing any such nilspace as a quotient of infinite products of free nilspaces and also as double coset spaces $K\backslash G/Γ$ where $G$ is a degree-$k$ nilpotent pro-Lie group. These results open the study of compact nilspaces to areas more classical than nilspace theory, such as the theory of topological group actions. The results also require developing the theory of topological non-compact nilspaces, for which we provide groundwork in this paper. Applications include new inverse theorems for Gowers norms on any finite abelian group. These theorems are purely group theoretic in that the correlating harmonics are based on double-coset spaces. This yields progress towards the Jamneshan-Tao conjecture.

math.CO↗

arXiv2025

Mesterséges Intelligencia Kutatások Magyarországon

Artificial intelligence (AI) has undergone remarkable development since the mid-2000s, particularly in the fields of machine learning and deep learning, driven by the explosive growth of large databases and computational capacity. Hungarian researchers recognized the significance of AI early on, actively participating in international research and achieving significant results in both theoretical and practical domains. This article presents some key achievements in Hungarian AI research. It highlights the results from the period before the rise of deep learning (the early 2010s), then discusses major theoretical advancements in Hungary after 2010. Finally, it provides a brief overview of AI-related applied scientific achievements from 2010 onward.

cs.GL↗

arXiv2025

Spectral algorithms in higher-order Fourier analysis

Our goal is to provide simple and practical algorithms in higher-order Fourier analysis which are based on spectral decompositions of operators. We propose a general framework for such algorithms and provide a detailed analysis of the quadratic case. Our results reveal new spectral aspects of the theory underlying higher-order Fourier analysis. Along these lines, we prove new inverse and regularity theorems for the Gowers norms based on higher-order character decompositions. Using these results, we prove a spectral inverse theorem and a spectral regularity theorem in quadratic Fourier analysis.

math.CO↗

arXiv2024

Impact of physicality on network structure

The emergence of detailed maps of physical networks, like the brain connectome, vascular networks, or composite networks in metamaterials, whose nodes and links are physical entities, have demonstrated the limits of the current network science toolset. Link physicality imposes a non-crossing condition that affects both the evolution and the structure of a network, in a way that the adjacency matrix alone -- the starting point of all graph-based approaches -- cannot capture. Here, we introduce a meta-graph that helps us discover an exact mapping between linear physical networks and independent sets, a central concept in graph theory. The mapping allows us to analytically derive both the onset of physical effects and the emergence of a jamming transition, and show that physicality impacts the network structure even when the total volume of the links is negligible. Finally, we construct the meta-graphs of several real physical networks, which allows us to predict functional features, such as synapse formation in the brain connectome, that agree with empirical data. Overall, our results show that, in order to understand the evolution and behavior of real complex networks, the role of physicality must be fully quantified.

cond-mat.stat-mech↗

arXiv2024

Dynamic Memory Based Adaptive Optimization

Define an optimizer as having memory $k$ if it stores $k$ dynamically changing vectors in the parameter space. Classical SGD has memory $0$, momentum SGD optimizer has $1$ and Adam optimizer has $2$. We address the following questions: How can optimizers make use of more memory units? What information should be stored in them? How to use them for the learning steps? As an approach to the last question, we introduce a general method called "Retrospective Learning Law Correction" or shortly RLLC. This method is designed to calculate a dynamically varying linear combination (called learning law) of memory units, which themselves may evolve arbitrarily. We demonstrate RLLC on optimizers whose memory units have linear update rules and small memory ($\leq 4$ memory units). Our experiments show that in a variety of standard problems, these optimizers outperform the above mentioned three classical optimizers. We conclude that RLLC is a promising framework for boosting the performance of known optimizers by adding more memory units and by making them more adaptive.

cs.LG↗

arXiv2023

On the inverse theorem for Gowers norms in abelian groups of bounded torsion

In recent work, Jamneshan, Shalom and Tao proved an inverse theorem for the Gowers $U^{k+1}$-norm on finite abelian groups of fixed torsion $m$, where the final correlating harmonic is a polynomial phase function of degree at most $C(k,m)$. They also posed a related central question, namely, whether the bound $C$ can be reduced to the optimal value $k$ for every $m$. We make progress on this question using nilspace theory. First we connect the question to the study of finite nilspaces whose structure groups have torsion $m$. Then we prove one of the main results of this paper: a primary decomposition theorem for finite nilspaces, extending the Sylow decomposition in group theory. Thus we give an analogue for nilspaces of an ergodic-theoretic Sylow decomposition in the aforementioned work of Jamneshan-Shalom-Tao. We deduce various consequences which illustrate the following general idea: the primary decomposition enables a reduction of higher-order Fourier analysis in the $m$-torsion setting to the case of abelian $p$-groups. These consequences include a positive answer to the question of Jamneshan-Shalom-Tao when $m$ is squarefree, and also a new relation between uniformity norms and certain generalized cut norms on products of abelian groups of coprime orders. Another main result in this paper is a positive answer to the above central question for the $U^3$-norm, proving that $C(2,m)=2$ for all $m$. Finally, we give a partial answer to the question for all $k$ and $m$, proving an inverse theorem involving extensions of polynomial phase functions which were introduced by the third-named author, known as projected phase polynomials of degree $k$. A notable aspect is that this inverse theorem implies that of Jamneshan-Shalom-Tao, while involving projected phase polynomials of degree $k$, which are genuine obstructions to having small $U^{k+1}$-norm.

math.CO↗

arXiv2023

Subgraph densities in Markov spaces

We generalize subgraph densities, arising in dense graph limit theory, to Markov spaces (symmetric measures on the square of a standard Borel space). More generally, we define an analogue of the set of homomorphisms in the form of a measure on maps of a finite graph into a Markov space. The existence of such homomorphism measures is not always guaranteed, but can be established under rather natural smoothness conditions on the Markov space and sparseness conditions on the graph. This continues a direction in graph limit theory in which such measures are viewed as limits of graph sequences.

math.PR↗

arXiv2023

On measure-preserving $\mathbb{F}_p^ω$-systems of order $k$

Building on previous work in the nilspace-theoretic approach to the study of Host-Kra factors of measure-preserving systems, we prove that every ergodic $\mathbb{F}_p^ω$-system of order $k$ is a factor of an Abramov $\mathbb{F}_p^ω$-system of order $k$. This answers a question of Jamneshan, Shalom and Tao.

math.DS↗

arXiv2022

On higher-order Fourier analysis in characteristic $p$

In this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb{F}_p$, with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call $p$-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite $p$-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian $p$-groups. The applications include a description of the Host-Kra factors of ergodic $\mathbb{F}_p^ω$-systems as $p$-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the $k$-th Host-Kra factor is an Abramov system of order $\leq k$, extending a result of Bergelson-Tao-Ziegler that holds for $k< p$. We illustrate the utility of $p$-homogeneous nilspaces also by showing that the above-mentioned structure theorem yields a new proof of the Tao-Ziegler inverse theorem for Gowers norms on $\mathbb{F}_p^n$.

math.DS↗

arXiv2022

A refinement of Cauchy-Schwarz complexity

We introduce a notion of complexity for systems of linear forms, called sequential Cauchy-Schwarz complexity, which is parametrized by two positive integers $k,\ell$ and refines the notion of Cauchy-Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy-Schwarz complexity at most $(k,\ell)$ then any average of 1-bounded functions over this system is controlled by the $2^{1-\ell}$-th power of the Gowers $U^{k+1}$-norms of the functions. For $\ell=1$ this agrees with Cauchy-Schwarz complexity, but for $\ell>1$ there are systems that have sequential Cauchy-Schwarz complexity at most $(k,\ell)$ whereas their Cauchy-Schwarz complexity is greater than $k$. Our main application illustrates this with systems over a prime field $\mathbb{F}_p$ that are denoted by $Φ_{k,M}$ and can be viewed as $M$-dimensional arithmetic progressions of length $k$. For each $M\geq 2$ we prove that $Φ_{k,M}$ has sequential Cauchy-Schwarz complexity at most $(k-2,|Φ_{k,M}|)$ (where $|Φ_{k,M}|$ is the number of forms in the system), whereas the Cauchy-Schwarz complexity of $Φ_{k,M}$ can be greater than $k-2$. Thus we obtain polynomial true-complexity bounds for $Φ_{k,M}$ with exponent $2^{-|Φ_{k,M}|}$. A recent general theorem of Manners, proved independently with different methods, implies a similar application but with different polynomial true-complexity bounds, as explained in the paper. In separate work, we use our application to give a new proof of the inverse theorem for Gowers norms on $\mathbb{F}_p^n$, and related results concerning ergodic actions of $\mathbb{F}_p^ω$.

math.CO↗

arXiv2022

On $\mathbb{F}_2^ω$-affine-exchangeable probability measures

For any standard Borel space $B$, let $\mathcal{P}(B)$ denote the space of Borel probability measures on $B$. In relation to a difficult problem of Aldous in exchangeability theory, and in connection with arithmetic combinatorics, Austin raised the question of describing the structure of affine-exchangeable probability measures on product spaces indexed by the vector space $\mathbb{F}_2^ω$, i.e., the measures in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ that are invariant under the coordinate permutations on $B^{\mathbb{F}_2^ω}$ induced by all affine automorphisms of $\mathbb{F}_2^ω$. We answer this question by describing the extreme points of the space of such affine-exchangeable measures. We prove that there is a single structure underlying every such measure, namely, a random infinite-dimensional cube (sampled using Haar measure adapted to a specific filtration) on a group that is a countable power of the 2-adic integers. Indeed, every extreme affine-exchangeable measure in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ is obtained from a $\mathcal{P}(B)$-valued function on this group, by a vertex-wise composition with this random cube. The consequences of this result include a description of the convex set of affine-exchangeable measures in $\mathcal{P}(B^{\mathbb{F}_2^ω})$ equipped with the vague topology (when $B$ is a compact metric space), showing that this convex set is a Bauer simplex. We also obtain a correspondence between affine-exchangeability and limits of convergent sequences of (compact-metric-space valued) functions on vector spaces $\mathbb{F}_2^n$ as $n\to\infty$. Via this correspondence, we establish the above-mentioned group as a general limit domain valid for any such sequence.

math.PR↗

arXiv2022

Regularity and inverse theorems for uniformity norms on compact abelian groups and nilmanifolds

We prove a general form of the regularity theorem for uniformity norms, and deduce an inverse theorem for these norms which holds for a class of compact nilspaces including all compact abelian groups, and also nilmanifolds; in particular we thus obtain the first non-abelian versions of such theorems. We derive these results from a general structure theorem for cubic couplings, thereby unifying these results with the Host-Kra Ergodic Structure Theorem. A unification of this kind had been propounded as a conceptual prospect by Host and Kra. Our work also provides new results on nilspaces. In particular, we obtain a new stability result for nilspace morphisms. We also strengthen a result of Gutman, Manners and Varjú, by proving that a $k$-step compact nilspace of finite rank is a toral nilspace (in particular, a connected nilmanifold) if and only if its $k$-dimensional cube set is connected. We also prove that if a morphism from a cyclic group of prime order into a compact finite-rank nilspace is sufficiently balanced (i.e. equidistributed in a certain quantitative and multidimensional sense), then the nilspace is toral. As an application of this, we obtain a new proof of a refinement of the Green-Tao-Ziegler inverse theorem.

math.CO↗

arXiv2021

Typicality and entropy of processes on infinite trees

Consider a uniformly sampled random $d$-regular graph on $n$ vertices. If $d$ is fixed and $n$ goes to $\infty$ then we can relate typical (large probability) properties of such random graph to a family of invariant random processes (called "typical" processes) on the infinite $d$-regular tree $T_d$. This correspondence between ergodic theory on $T_d$ and random regular graphs is already proven to be fruitful in both directions. This paper continues the investigation of typical processes with a special emphasis on entropy. We study a natural notion of micro-state entropy for invariant processes on $T_d$. It serves as a quantitative refinement of the notion of typicality and is tightly connected to the asymptotic free energy in statistical physics. Using entropy inequalities, we provide new sufficient conditions for typicality for edge Markov processes. We also extend these notions and results to processes on unimodular Galton-Watson random trees.

math.PR↗

arXiv2021

Multigraph limits, unbounded kernels, and Banach space decorated graphs

We present a construction that allows us to define a limit object of Banach space decorated graph sequences in a generalized homomorphism density sense. This general functional analytic framework provides a universal language for various combinatorial limit notions. In particular it makes it possible to assign limit objects to multigraph sequences that are convergent in the sense of node-and-edge homomorphism numbers, and it generalizes the limit theory for graph sequences with compact decorations.

math.CO↗

arXiv2021

Random homomorphisms into the orthogonality graph

Subgraph densities have been defined, and served as basic tools, both in the case of graphons (limits of dense graph sequences) and graphings (limits of bounded-degree graph sequences). While limit objects have been described for the "middle ranges", the notion of subgraph densities in these limit objects remains elusive. We define subgraph densities in the orthogonality graphs on the unit spheres in dimension $d$, under appropriate sparsity condition on the subgraphs. These orthogonality graphs exhibit the main difficulties of defining subgraphs the "middle" range, and so we expect their study to serve as a key example to defining subgraph densities in more general Markov spaces. The problem can also be formulated as defining and computing random orthogonal representations of graphs. Orthogonal representations have played a role in information theory, optimization, rigidity theory and quantum physics, so to study random ones may be of interest from the point of view of these applications as well.

math.CO↗

arXiv2021

On Sidorenko's conjecture for determinants and Gaussian Markov random fields

We study a class of determinant inequalities that are closely related to Sidorenko's famous conjecture (Also conjectured by Erd\H os and Simonovits in a different form). Our main result can also be interpreted as an entropy inequality for Gaussian Markov random fields (GMRF). We call a GMRF on a finite graph $G$ homogeneous if the marginal distributions on the edges are all identical. We show that if $G$ is bipartite then the differential entropy of any homogeneous GMRF on $G$ is at least $|E(G)|$ times the edge entropy plus $|V(G)|-2|E(G)|$ times the point entropy. We also show that in the case of non-negative correlation on edges, the result holds for an arbitrary graph $G$. The connection between Sidorenko's conjecture and GMRF's is established via a large deviation principle on high dimensional spheres combined with graph limit theory. Connection with Ihara zeta function and the number of spanning trees is also discussed.

math.CO↗