arXiv subjects
Publications and source records attributed to Manuel Fernandez V.
We prove an exact isoperimetric inequality for cubes and tori under the $\ell_{\infty} metric. As a corollary, we recover the celebrated grid edge-isoperimetric inequality of Bollobás-Leader.
Let $A=(ξ_{ij})$ be an $n\times n$ random matrix with independent, not necessarily identically distributed, real entries satisfying \[ \mathbb Eξ_{ij}=0,\qquad \mathbb Eξ_{ij}^{2}=1,\qquad \sup_{z\in\mathbb R}\mathbb P(|ξ_{ij}-z| 0$ and $b\in(0,1)$. We prove that, for every $δ\in(0,1)$, there are constants $c,C>0$, depending only on $a,b,δ$, such that \[ \mathbb P\left( s_{n+1-l}(A)>Ct\frac l{\sqrt n} \right) \le \exp\!\left(-c\min\{tl,n\}\right) \] for every $t\ge1$ and every $1\le l\le(1-δ)n$. Thus, with no moment assumption beyond variance, all but a fixed proportion of the largest singular values satisfy the optimal upper bound of order $l/\sqrt n$ with an exponential upper-tail estimate. Combined with the lower bound of the rectangular least singular value bound, this gives $s_{n+1-l}(A)\asymp l/\sqrt n$ with failure probability exponentially small in $l$. The same argument gives the rectangular scale $\sqrt{N+1}-\sqrt{n-l+1}$ for $N\times n$ matrices whenever $N-n+l\le(1-δ)N$.
We study $\ell_0$ isoperimetry for a convex body $K\subset \mathbb{R}^n$, $n\ge2$. For a Borel set $S\subset K$, let $\partial_0^K S$ be the set of points in $K \setminus S$ that can be reached from $S$ by changing at most one coordinate (i.e. the $\ell_0$ boundary of $S$). Suppose that, for some unconditional convex body $Q \subset \mathbb{R}^n$, numbers $r,R>0$, and possibly different centers $x_0,y_0$, \[ x_0+rQ \subset K\subset y_0+RQ. \] Writing $s=\text{vol}(S)/\text{vol}(K)$, we prove that whenever $0 0$ is an absolute constant. Consequently, the associated $\ell_0$-isoperimetric coefficient is at least $cr/(n^2R)$. Previous direct lower bounds were only known for $\ell_2$ and $\ell_\infty$ regularity whereas our lower bound holds directly for any $Q$-regularity, where $Q$ is an unconditional convex body. Compared to $\ell_2$ and $\ell_\infty$ regularity, our lower bound result improves upon the previously best known lower bounds, for any $s$, by a factor of $n$. As an application of our result, we give improved mixing time bounds for the Coordinate Hit and Run walk (CHAR). Our proof of the lower bound is based on a modification of the method of canonical paths applied to a continuous Hamming graph over our convex body. Our construction of canonical paths can be viewed as a suitable coordinate discretization of certain mass transport maps from $S$ to $S^c$. We also give complementary upper-bounds for any $Q$-regularity, with an overall factor of $n$ gap between the two.
We study sparse threshold random geometric graphs generated by high-dimensional spherical or Gaussian latent vectors. Although each edge has marginal probability $p$, shared latent variables make the adjacency entries dependent. At the connectivity scale $np=Ω(\log n)$, the spherical adjacency matrix satisfies, with high probability,$\|A-\mathbb E A\|_{\mathrm{op}}=O\left(\sqrt{np\log n}+npτ\right)$, where $τ$ is the cap threshold; an analogous estimate holds for Gaussian vectors after controlling radial fluctuations. This sharpens the spectral bound in Liu, Mohanty, Schramm, and Yang (2023) under weaker assumptions and strengthens the global-synchronization guarantee of Abdalla, Bandeira, and Invernizzi (2024) for the homogeneous Kuramoto model. The leading eigenspace also estimates the latent geometry. When $np\gg\log n$, vector and relative Gram-matrix errors vanish for$\log(1/p)\ll d\ll np\log(1/p)/\log n$ in the spherical model and $\log^2(1/p)\log n\ll d\ll np\log(1/p)/\log n$ in the Gaussian model, improving the recovery conditions of Li and Schramm (2023). For the Gaussian mixture block model introduced there, a polynomial-time semidefinite program gives, to our knowledge, the first exact-recovery guarantee at the connectivity scale in a moderate-separation regime. At much larger separation, fixed edge density creates isolated vertices and makes exact recovery impossible. Our reusable decoupling and matrix concentration framework avoids trace-moment methods and applies broadly to random graph models with latent vectors.
We study the community detection problem in the non-uniform hypergraph stochastic block model (HSBM), where hyperedges of varying sizes coexist. This setting captures higher-order and multi-view interactions and raises a fundamental question: can multiple uniform hypergraph layers below the detection threshold be combined to enable weak recovery? We answer this question by establishing a Kesten--Stigum-type bound for weak recovery in a general class of non-uniform HSBMs with $r$ blocks, generated according to multiple symmetric probability tensors. In the case $r=2$, we show that weak recovery is possible whenever the sum of the signal-to-noise ratios across all uniform hypergraph layers exceeds one, thereby confirming the positive part of a conjecture in (Chodrow et al., 2023). Moreover, we provide a polynomial-time spectral algorithm that achieves this threshold via an optimally weighted non-backtracking operator. For the unweighted non-backtracking matrix, our spectral method attains a different algorithmic threshold, also conjectured in (Chodrow et al., 2023). Our approach develops a spectral theory for weighted non-backtracking operators on non-uniform hypergraphs, including a precise characterization of outlier eigenvalues and eigenvector overlaps. We introduce a novel Ihara--Bass formula tailored to weighted non-uniform hypergraphs, which yields an efficient low-dimensional representation and leads to a provable spectral reconstruction algorithm. Taken together, these results provide a principled and computationally efficient approach to clustering in non-uniform hypergraphs, and highlight the role of optimal weighting in aggregating heterogeneous higher-order interactions.
Let $A$ be an $N\times n$ random matrix whose entries are coordinates of an isotropic log-concave random vector in $\mathbb{R}^{Nn}$. We prove sharp lower tail estimates for the smallest singular value of $A$ in the following cases: (1) when $N=n$ and $A$ is drawn from an unconditional distribution, with no independence assumption; (2) when the columns of $A$ are independent and $N\geq n$; (3) when $A$ is sufficiently tall, that is $N\geq (1+λ)n$ for any positive constant $λ$.
Let $A \in \mathbb{R}^{n \times (n - d)}$ be a random matrix with independent uniformly anti-concentrated entries satisfying $\mathbb{E}\lvert A\rvert_{HS}^2 \leq Kn(n-d)$ and let $H$ be the subspace spanned by the columns of $A$. Let $X \in \mathbb{R}^n$ be a random vector with uniformly anti-concentrated entries. We show that when $1 \leq d \leq λn/\log n$ the distance between between $X$ and $H$ satisfies the following following small ball estimate: \[ \Pr\left( \text{dis}(X,H) \leq t\sqrt{d} \right) \leq (Ct)^{d} + e^{-cn}, \] for some constants $λ,c,C > 0$. This extends the distance theorems of Rudelson and Vershynin, Livshyts, and Livshyts,Tikhomirov, and Vershynin by dropping any identical distribution assumptions about the entries of $X$ and $A$. Furthermore it can be applied to prove numerous results about random matrices in the inhomogenous setting. These include lower tail estimates on the smallest singular value of rectangular matrices and upper tail estimates on the smallest singular value of square matrices. To obtain a distance theorem for inhomogenous rectangular matrices we introduce a new tool for this new general ensemble of random matrices, Randomized Logarithmic LCD, a natural combination of the Randomized LCD, used in study of smallest singular values of inhomogenous square matrices, and of the Logarithmic LCD, used in the study of no-gaps delocalization of eigenvectors and the smallest singular values of Hermitian random matrices.
In this paper we prove that the $\ell_0$ isoperimetric coefficient for any axis-aligned cubes, $ψ_{\mathcal{C}}$, is $Θ(n^{-1/2})$ and that the isoperimetric coefficient for any measurable body $K$, $ψ_K$, is of order $O(n^{-1/2})$. As a corollary we deduce that axis-aligned cubes essentially "maximize" the $\ell_0$ isoperimetric coefficient: There exists a positive constant $q > 0$ such that $ψ_K \leq q \cdot ψ_{\mathcal{C}}$, whenever $\mathcal{C}$ is an axis-aligned cube and $K$ is any measurable set. Lastly, we give immediate applications of our results to the mixing time of Coordinate-Hit-and-Run for sampling points uniformly from convex bodies.
The clique chromatic number of a graph is the smallest number of colors in a vertex coloring so that no maximal clique is monochromatic. In this paper, we determine the order of magnitude of the clique chromatic number of the random graph G_{n,p} for most edge-probabilities p in the range n^{-2/5} \ll p \ll 1. This resolves open problems and questions of Lichev, Mitsche and Warnke as well as Alon and Krievelevich. One major proof difficulty stems from high-degree vertices, which prevent maximal cliques in their neighborhoods: we deal with these vertices by an intricate union bound argument, that combines the probabilistic method with new degree counting arguments in order to enable Janson's inequality. This way we determine the asymptotics of the clique chromatic number of G_{n,p} in some ranges, and discover a surprising new phenomenon that contradicts earlier predictions for edge-probabilities p close to n^{-2/5}.