arXiv subjects

Tatiana Brailovskaya

Publications and source records attributed to Tatiana Brailovskaya.

5 recordsLinked to original sources

arXiv2026

Spectral Recovery of Point Clouds from Noisy Geometric Graphs

We study the problem of recovering low-dimensional latent geometry from a random geometric graph generated by noisy, high-dimensional data. Specifically, we analyze the performance of a spectral embedding algorithm on the Signal+Noise Graph Model, in which vertices are associated to points perturbed by Gaussian noise, and edges are included for pairs whose inner product exceeds a specified alignment threshold. In the high-dimensional regime where the number $n$ of points and the ambient dimension $d$ both tend to infinity, we show that under a spectral gap condition, the top eigenvectors and eigenvalues of the graph's adjacency matrix can be used to approximately recover the point cloud up to an orthogonal transformation. We illustrate our results on point clouds sampled from nested spheres and high-dimensional sinusoid curves.

math.ST↗

arXiv2026

Eigenvalues of Brownian Motions on $\mathrm{GL}(N,\mathbb{C})$

We prove that the empirical law of eigenvalues of Brownian motion on the Lie Group $\mathrm{GL}(N,\mathbb{C})$ converges almost surely to a deterministic probability measure, characterized by a free stochastic differential equation. This fully resolves a conjecture made by Philippe Biane in 1997. Our analysis includes a family $\{B=B_{ρ,ζ}\colon |ζ|<ρ\}$ of nondegenerate diffusion processes on $\mathrm{GL}(N,\mathbb{C})$ whose laws are invariant under unitary conjugation, with initial distributions assumed to be uniformly bounded and invertible. The crux of our analysis is a strong quantitative approximation of Brownian motion $B(t)$ on $\mathrm{GL}(N,\mathbb{C})$ for small $t$ by a single increment $I+W(t)$, where $W=W_{ρ,ζ}$ is an elliptic Brownian motion in the Lie algebra $\mathfrak{gl}(N,\mathbb{C}) = \mathbb{M}_N(\mathbb{C})$. Specifically, for any $t\in[0,1]$ and $δ>0$, \[ \mathbb{P}\left(\|B(t)-I-W(t)\|\geq δ\right)\leq \left(C t/δ\right)^{N^{2/3}} \] for a constant $C=C_ρ$. Leveraging independence of multiplicative increments of the Brownian motion then allows us to use powerful (anti-)concentration tools for Gaussian matrices to complete the Hermitization procedure for convergence of eigenvalues.

math.PR↗

arXiv2025

Extremal random matrices with independent entries and matrix superconcentration inequalities

We prove nonasymptotic matrix concentration inequalities for the spectral norm of (sub)gaussian random matrices with centered independent entries that capture fluctuations at the Tracy-Widom scale. This considerably improves previous bounds in this setting due to Bandeira and Van Handel, and establishes the best possible tail behavior for random matrices with an arbitrary variance pattern. These bounds arise from an extremum problem for nonhomogeneous random matrices: among all variance patterns with a given sparsity parameter, the moments of the random matrix are maximized by block-diagonal matrices with i.i.d. entries in each block. As part of the proof, we obtain sharp bounds on large moments of Gaussian Wishart matrices.

math.PR↗

arXiv2024

Universality and sharp matrix concentration inequalities

We show that, under mild assumptions, the spectrum of a sum of independent random matrices is close to that of the Gaussian random matrix whose entries have the same mean and covariance. This nonasymptotic universality principle yields sharp matrix concentration inequalities for general sums of independent random matrices when combined with the Gaussian theory of Bandeira, Boedihardjo, and Van Handel. A key feature of the resulting theory is that it is applicable to a broad class of random matrix models that may have highly nonhomogeneous and dependent entries, which can be far outside the mean-field situation considered in classical random matrix theory. We illustrate the theory in applications to random graphs, matrix concentration inequalities for smallest singular values, sample covariance matrices, strong asymptotic freeness, and phase transitions in spiked models.

math.PR↗

arXiv2021

Tree trace reconstruction using subtraces

Tree trace reconstruction aims to learn the binary node labels of a tree, given independent samples of the tree passed through an appropriately defined deletion channel. In recent work, Davies, Rácz, and Rashtchian used combinatorial methods to show that $\exp(\mathcal{O}(k \log_{k} n))$ samples suffice to reconstruct a complete $k$-ary tree with $n$ nodes with high probability. We provide an alternative proof of this result, which allows us to generalize it to a broader class of tree topologies and deletion models. In our proofs, we introduce the notion of a subtrace, which enables us to connect with and generalize recent mean-based complex analytic algorithms for string trace reconstruction.

cs.DS↗