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Saba Lepsveridze

Publications and source records attributed to Saba Lepsveridze.

7 recordsLinked to original sources
arXiv2026

Graphs with connectivity $3/4 - \varepsilon$ are globally synchronizing

We study synchronization in the Kuramoto model on finite graphs. We prove that there is an absolute constant $\eta>0$ such that every finite simple graph $G$ on $n$ vertices with minimum degree at least $(3/4-\eta)n$ has no local minima of the Kuramoto energy other than the fully synchronized states. This strictly improves the previous $3/4$ upper bound and refutes a conjecture of Bandeira, Kireeva, Maillard, and R\"odder.

math.CO↗
arXiv2026

Optimal Spectral Algorithms for Correlated Two-view Models in High Dimensions

We study high-dimensional inference in correlated two-view models, focusing on spectral methods for strong detection and weak recovery. We introduce a general framework, motivated by a TAP type heuristic from statistical physics, that provides a unified treatment of three canonical models: high-dimensional canonical correlation analysis, and the correlated spiked Wigner and Wishart models. Our main contribution is to construct explicit spectral algorithms in all three settings, that achieve strong detection and weak recovery down to the corresponding thresholds, where we prove matching information-theoretic lower bounds. Furthermore, our spectral procedures operate without knowledge of the model parameters, relying solely on the observed data. This demonstrates the optimality of spectral methods in these models and the broad statistical applicability of the framework.

math.ST↗
arXiv2026

Uniform Geodesic Drawings of Graphs

We study crossing numbers of dense graph drawings whose vertices are uniformly distributed either on the unit sphere or in a compact convex planar domain. We prove a sharp inequality for weighted geodesic drawings on $\mathbb S^2$ in a continuous setting: among all measurable edge arrangements of a fixed density, the amount of crossings is minimized by connecting pairs of points within a fixed distance threshold. We also prove a planar analogue for straight-line drawings in convex planar domains. We transfer these continuous results to finite graphs using a smoothing argument. In the small density limit, we recover the conjectured midrange crossing constant lower bound of $8/(9\pi^2)$ for this restricted model.

math.CO↗
arXiv2026

Sharp Threshold for the Convergence of Nonstationary Averaging

We study non-stationary averaging processes, where each term of a sequence is a weighted average of previous terms, namely $a_{n+1} = \sum_{j=1}^n p_n(j) a_j$. Our results extend classical theory in two distinct regimes. First, we prove a sharp threshold for convergence in the regime where the weights are bounded between two envelopes $(\log n)^{-\alpha} \le np_n(\cdot) \leq (\log n)^{\beta}$. We show that the sequence necessarily converges when $\alpha + \beta / 2 \leq 1$, while $\alpha + \beta / 2 > 1$ the convergence can fail. Second, we study complementary fixed shape regime, when $p_n$ is obtained by a fixed limiting density on $(0,1)$. We show that under mild regularity assumptions, the sequence converges.

math.PR↗
arXiv2025

Optimal Thresholds for Monotone Non-Boolean Functions

Let $[q] = \{0,1,\ldots,q-1\}$, let $\Delta[q]$ denote the simplex of probability measures on $[q]$, and let $\gamma$ denote the Lebesgue measure normalized on $\Delta[q]$. We prove that for any symmetric monotone function $f \colon[q]^n \to [q]$ and any $a \in [q]$ we have \begin{equation*} \gamma(\{\mu \in \Delta[q]\;\vert\;\mathbb{P}_{x\sim\mu^{\otimes n}}[f(x)=a] \in (\varepsilon,1-\varepsilon)\}) = O(1/\log n)\text{.} \end{equation*} We also show that this bound is tight. This improves Kalai and Mossel's previous bound of $O(\log \log n/\log n)$ and answers their question completely.

math.PR↗
arXiv2023

Uniacute Spherical Codes

A spherical $L$-code, where $L \subseteq [-1,\infty)$, consists of unit vectors in $\mathbb{R}^d$ whose pairwise inner products are contained in $L$. Determining the maximum cardinality $N_L(d)$ of an $L$-code in $\mathbb{R}^d$ is a fundamental question in discrete geometry and has been extensively investigated for various choices of $L$. Our understanding in high dimensions is generally quite poor. Equiangular lines, corresponding to $L = \{-α, α\}$, is a rare and notable solved case. Bukh studied an extension of equiangular lines and showed that $N_L(d) = O_L(d)$ for $L = [-1, -β] \cup \{α\}$ with $α,β> 0$ (we call such $L$-codes "uniacute"), leaving open the question of determining the leading constant factor. Balla, Dräxler, Keevash, and Sudakov proved a "uniform bound" showing $\limsup_{d\to\infty} N_L(d)/d \le 2p$ for $L = [-1, -β] \cup \{α\}$ and $p = \lfloor α/β\rfloor + 1$. For which $(α,β)$ is this uniform bound tight? We completely answer this question. We develop a framework for studying uniacute codes, including a global structure theorem showing that the Gram matrix has an approximate $p$-block structure. We also formulate a notion of "modular codes," which we conjecture to be optimal in high dimensions.

math.CO↗