arXiv subjects

Afonso S. Bandeira

Publications and source records attributed to Afonso S. Bandeira.

At least 19 recordsLinked to original sources

arXiv2026

Matrix Concentration and Equivalent Operators on Fock Spaces

We develop a method for proving matrix concentration inequalities by identifying random matrices with associated deterministic operators acting on suitable Fock spaces and controlling norms of these operators and their restrictions to low-order subspaces. Applying this method, we obtain new strengthenings of the non-commutative Khintchine inequality that improve on the state of the art, in particular sharpening recent inequalities due to Bandeira, Boedihardjo, and van Handel (2023) quantifying intrinsic freeness of random matrices. Our proofs of these results are based on relatively simple operator algebra arguments and involve neither Gaussian interpolation nor explicit combinatorics of tracial moments. Further, our techniques apply equally well to several models of non-commutative random variables of interest in the literature, such as operator series constructed from $q$-Gaussian and $Γ$-independent systems of operators, treating all of these objects with the same method. We obtain new norm bounds for such operators both in the style of the non-commutative Khintchine inequality and in the style of Lehner's operator norm formula.

math.PR↗

arXiv2026

Convergence of Kikuchi matrices to $Γ$-independent and $q$-Gaussian limits

Kikuchi matrices are a family of structured matrices that were introduced to study problems involving tensors and hypergraphs. We show that, as the ambient dimension grows, dense random Kikuchi matrices have a limit described by a system of $Γ$-independent semicircular elements. This characterizes their limiting spectral distribution and yields improved bounds on their spectral norm, a key quantity in the analysis of algorithms for Tensor PCA. Finally, we show that, in an appropriate double limit, independent Kikuchi matrices converge to the $q$-Gaussian system, another central object in noncommutative probability.

math.PR↗

arXiv2026

The Hypergraph Moore Bound

The hypergraph Moore bound conjectured by Feige (2008) controls the size of the smallest even cover in a $k$-uniform hypergraph in terms of the average density of hyperedges. An even cover is a set of hyperedges covering each vertex an even number of times, generalizing the notion of a cycle in a graph, so the size of the smallest non-trivial even cover provides a notion of hypergraph girth. Recent work, starting from the breakthrough result of Guruswami, Kothari, and Manohar (2022) proved the conjecture up to polylogarithmic factors, whose exponents were later gradually improved. We give a simple proof of Feige's original hypergraph Moore bound conjecture for all $k \geq 3$, with no superfluous polylogarithmic factors. For the case of $k$ even, our proof roughly follows the proof of the graph Moore bound, but works with colored walks in a Kikuchi graph built from a hypergraph and controls their growth using the polynomial method. The argument is then extended to the case of $k$ odd by adapting a procedure in [GKM22].

math.CO↗

arXiv2026

Mathematics of Data Science

This book is about the mathematical foundations of data science. 1. Introduction 2. Curses, Blessings, and Surprises in High Dimensions 3. Singular Value Decomposition and Principal Component Analysis 4. Linear Regression and Regularization 5. Graphs, Networks, and Clustering 6. Nonlinear Dimension Reduction and Diffusion Maps 7. Linear Dimension Reduction via Random Projections 8. Optimization for Data Science 9. Classification 10. A Mathematical Introduction to Deep Learning 11. Large Sample Limit of Graph Laplacians 12. Community 13. Concentration of Measure and Gaussian Analysis 14. Matrix Concentration Inequalities 15. Compressive Sensing and Sparsity 16. Low-Rank Matrix Recovery

cs.LG↗

arXiv2026

Mixing of Glauber Dynamics on High Overlap Gibbs Measures

We show fast mixing of Glauber dynamics for certain quadratic Gibbs measures with large external fields. The main ingredient is an overlap condition that allows us to control correlation matrices uniformly over all pinnings, by controlling norms of small submatrices of the interaction matrix. Using stochastic localization, we then obtain a lower bound on the spectral gap and, consequently, polynomial-time mixing of Glauber dynamics. As a direct application, we consider the Sherrington-Kirkpatrick model, whose interaction matrix is a scaled GOE matrix. For this model, we show that for any fixed finite inverse temperature $β$, there exists a strength of external field $θ$, not depending on the size of the system, for which Glauber dynamics mixes in polynomial time (with high probability on the draw of the interaction matrix).

math.PR↗

arXiv2026

Matrix Discrepancy for Representations of Finite Groups

We prove the group version of the Matrix Spencer conjecture. For every finite group $G$, there exist signs $\varepsilon\in\{\pm1\}^G$ such that $$\left\| \sum_{g\in G} \varepsilon_gρ(g) \right\|\leq C\, \sqrt{|G|},$$ where $ρ$ is the left regular representation of $G$ and $C$ is a universal constant. This conjecture was posed in [BKMZ24], which settled it for simple groups; we establish it for all finite groups, combining the Peter--Weyl decomposition with the intrinsic-freeness inequalities of [BBvH23] in an iterated partial-coloring argument.

math.PR↗

arXiv2026

Randomstrasse101: Open Problems of 2025

Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2025, with the goal of easing academic referencing. The blog can currently be accessed at randomstrasse101.math.ethz.ch

math.PR↗

arXiv2026

Expander graphs are globally synchronizing

The Kuramoto model is fundamental to the study of synchronization. It consists of a collection of oscillators with interactions given by a network, which we identify respectively with vertices and edges of a graph. In this paper, we show that a graph with sufficient expansion must be globally synchronizing, meaning that a homogeneous Kuramoto model of identical oscillators on such a graph will converge to the fully synchronized state with all the oscillators having the same phase, for every initial state up to a set of measure zero. In particular, we show that for any $\varepsilon > 0$ and $p \geq (1 + \varepsilon) (\log n) / n$, the homogeneous Kuramoto model on the Erdős-Rényi random graph $G(n, p)$ is globally synchronizing with probability tending to one as $n$ goes to infinity. This improves on a previous result of Kassabov, Strogatz, and Townsend and solves a conjecture of Ling, Xu, and Bandeira. We also show that the model is globally synchronizing on any $d$-regular Ramanujan graph, and on typical $d$-regular graphs, for large enough degree $d$.

math.CO↗

arXiv2025

Matrix Concentration Inequalities and Free Probability II. Two-sided Bounds and Applications

The first paper in this series introduced a new family of nonasymptotic matrix concentration inequalities that sharply capture the spectral properties of very general random matrices in terms of an associated noncommutative model. These methods achieved matching upper and lower bounds for smooth spectral statistics, but only provided upper bounds for the spectral edges. Here we obtain matching lower bounds for the spectral edges, completing the theory initiated in the first paper. The resulting two-sided bounds enable the study of problems that require an exact determination of the spectral edges to leading order, which is fundamentally beyond the reach of classical matrix concentration inequalities. To illustrate their utility, we develop two general results that explain phase transitions of spectral outliers of a large class of nonhomogeneous random matrices. This enables us to elucidate phase transition phenomena that arise in diverse applications, including decoding node labels on graphs, tensor PCA, contextual stochastic block models, and centered sample covariance matrices.

math.PR↗

arXiv2025

Random Matrices, Intrinsic Freeness, and Sharp Non-Asymptotic Inequalities

Random matrix theory has played a major role in several areas of pure and applied mathematics, as well as statistics, physics, and computer science. This lecture aims to describe the intrinsic freeness phenomenon and how it provides new easy-to-use sharp non-asymptotic bounds on the spectrum of general random matrices. We will also present a couple of illustrative applications in high dimensional statistical inference. This article accompanies a lecture that will be given by the author at the International Congress of Mathematicians in Philadelphia in the Summer of 2026.

math.PR↗

arXiv2025

Exact threshold for approximate ellipsoid fitting of random points

We consider the problem $(\rm P)$ of exactly fitting an ellipsoid (centered at $0$) to $n$ standard Gaussian random vectors in $\mathbb{R}^d$, as $n, d \to \infty$ with $n / d^2 \to α> 0$. This problem is conjectured to undergo a sharp transition: with high probability, $(\rm P)$ has a solution if $α< 1/4$, while $(\rm P)$ has no solutions if $α> 1/4$. So far, only a trivial bound $α> 1/2$ is known to imply the absence of solutions, while the sharpest results on the positive side assume $α\leq η$ (for $η> 0$ a small constant) to prove that $(\rm P)$ is solvable. In this work we show a universality property for the minimal fitting error achievable by ellipsoids: we show that, to leading order, it coincides with the minimal error in a so-called "Gaussian equivalent" problem, for which the satisfiability transition can be rigorously analyzed. Our main results follow from this finding, and they are twofold. On the positive side, we prove that if $α< 1/4$, there exists an ellipsoid fitting all the points up to a small error, and that the lengths of its principal axes are bounded above and below. On the other hand, for $α> 1/4$, we show that achieving small fitting error is not possible if the length of the ellipsoid's shortest axis does not approach $0$ as $d \to \infty$ (and in particular there does not exist any ellipsoid fit whose shortest axis length is bounded away from $0$ as $d \to \infty$). To the best of our knowledge, our work is the first rigorous result characterizing the expected phase transition in ellipsoid fitting at $α= 1/4$. In a companion non-rigorous work, the second author and D. Kunisky give a general analysis of ellipsoid fitting using the replica method of statistical physics, which inspired the present work.

math.PR↗

arXiv2025

Average-case complexity in statistical inference: A puzzle-driven research seminar

These notes describe our experience with running a student seminar on average-case complexity in statistical inference using the jigsaw learning format at ETH Zurich in Fall of 2024. The jigsaw learning technique is an active learning technique where students work in groups on independent parts of the task and then reassemble the groups to combine all the parts together. We implemented this technique for the proofs of various recent research developments, combined with a presentation by one of the students in the beginning of the session. We describe our experience and thoughts on such a format applied in a student research seminar: including, but not limited to, higher engagement, more accessible talks by the students, and increased student participation in discussions. In the Appendix, we include all the exercises sheets for the topic, which may be of independent interest for courses on statistical-to-computational gaps and average-case complexity.

stat.OT↗

arXiv2025

Randomstrasse101: Open Problems of 2024

$\texttt{Randomstrasse101}$ is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2024, with the goal of easing academic referencing. The blog can currently be accessed at $\texttt{randomstrasse101.math.ethz.ch}$.

math.PR↗

arXiv2025

Matrix Chaos Inequalities and Chaos of Combinatorial Type

Matrix concentration inequalities and their recently discovered sharp counterparts provide powerful tools to bound the spectrum of random matrices whose entries are linear functions of independent random variables. However, in many applications in theoretical computer science and in other areas one encounters more general random matrix models, called matrix chaoses, whose entries are polynomials of independent random variables. Such models have often been studied on a case-by-case basis using ad-hoc methods that can yield suboptimal dimensional factors. In this paper we provide general matrix concentration inequalities for matrix chaoses, which enable the treatment of such models in a systematic manner. These inequalities are expressed in terms of flattenings of the coefficients of the matrix chaos. We further identify a special family of matrix chaoses of combinatorial type for which the flattening parameters can be computed mechanically by a simple rule. This allows us to provide a unified treatment of and improved bounds for matrix chaoses that arise in a variety of applications, including graph matrices, Khatri-Rao matrices, and matrices that arise in average case analysis of the sum-of-squares hierarchy.

math.PR↗

arXiv2025

The Lovász number of random circulant graphs

This paper addresses the behavior of the Lovász number for dense random circulant graphs. The Lovász number is a well-known semidefinite programming upper bound on the independence number. Circulant graphs, an example of a Cayley graph, are highly structured vertex-transitive graphs on integers modulo $n$, where the connectivity of pairs of vertices depends only on the difference between their labels. While for random circulant graphs the asymptotics of fundamental quantities such as the clique and the chromatic number are well-understood, characterizing the exact behavior of the Lovász number remains open. In this work, we provide upper and lower bounds on the expected value of the Lovász number and show that it scales as the square root of the number of vertices, up to a log log factor. Our proof relies on a reduction of the semidefinite program formulation of the Lovász number to a linear program with random objective and constraints via diagonalization of the adjacency matrix of a circulant graph by the discrete Fourier transform (DFT). This leads to a problem about controlling the norms of vectors with sparse Fourier coefficients, which we study using results on the restricted isometry property of subsampled DFT matrices.

math.CO↗

arXiv2024

Injectivity of ReLU networks: perspectives from statistical physics

When can the input of a ReLU neural network be inferred from its output? In other words, when is the network injective? We consider a single layer, $x \mapsto \mathrm{ReLU}(Wx)$, with a random Gaussian $m \times n$ matrix $W$, in a high-dimensional setting where $n, m \to \infty$. Recent work connects this problem to spherical integral geometry giving rise to a conjectured sharp injectivity threshold for $α= \frac{m}{n}$ by studying the expected Euler characteristic of a certain random set. We adopt a different perspective and show that injectivity is equivalent to a property of the ground state of the spherical perceptron, an important spin glass model in statistical physics. By leveraging the (non-rigorous) replica symmetry-breaking theory, we derive analytical equations for the threshold whose solution is at odds with that from the Euler characteristic. Furthermore, we use Gordon's min--max theorem to prove that a replica-symmetric upper bound refutes the Euler characteristic prediction. Along the way we aim to give a tutorial-style introduction to key ideas from statistical physics in an effort to make the exposition accessible to a broad audience. Our analysis establishes a connection between spin glasses and integral geometry but leaves open the problem of explaining the discrepancies.

cond-mat.dis-nn↗

arXiv2024

A Geometric Perspective on the Injective Norm of Sums of Random Tensors

Matrix concentration inequalities, intimately connected to the Non-Commutative Khintchine inequality, have been an important tool in both applied and pure mathematics. We study tensor versions of these inequalities, and establish non-asymptotic inequalities for the $\ell_p$ injective norm of random tensors with correlated entries. In certain regimes of $p$ and the tensor order, our tensor concentration inequalities are nearly optimal in their dimension dependencies. We illustrate our result with applications to problems including structured models of random tensors and matrices, tensor PCA, and connections to lower bounds in coding theory. Our techniques are based on covering number estimates as opposed to operator theoretic tools, which also provide a geometric proof of a weaker version of the Non-Commutative Khintchine inequality, motivated by a question of Talagrand.

math.PR↗

arXiv2024

Fitting an ellipsoid to a quadratic number of random points

We consider the problem $(\mathrm{P})$ of fitting $n$ standard Gaussian random vectors in $\mathbb{R}^d$ to the boundary of a centered ellipsoid, as $n, d \to \infty$. This problem is conjectured to have a sharp feasibility transition: for any $\varepsilon > 0$, if $n \leq (1 - \varepsilon) d^2 / 4$ then $(\mathrm{P})$ has a solution with high probability, while $(\mathrm{P})$ has no solutions with high probability if $n \geq (1 + \varepsilon) d^2 /4$. So far, only a trivial bound $n \geq d^2 / 2$ is known on the negative side, while the best results on the positive side assume $n \leq d^2 / \mathrm{polylog}(d)$. In this work, we improve over previous approaches using a key result of Bartl & Mendelson (2022) on the concentration of Gram matrices of random vectors under mild assumptions on their tail behavior. This allows us to give a simple proof that $(\mathrm{P})$ is feasible with high probability when $n \leq d^2 / C$, for a (possibly large) constant $C > 0$.

math.PR↗